Multistage Hypothesis Tests for the Mean of a Normal Distribution

Multistage Hyp othesi s T ests for the Mean o f a Normal Dis tribution ∗ Xinjia Chen Octob er 2008 Abstract In this pap er, we ha ve developed new multistage tests which g uaran tee pr escribed level of power and are more efficient than prev io us tests in ter ms of a verage sampling n umber and the num b er of sampling op erations. Without trunca tion, the maximum sampling nu mbers of our testing pla ns ar e absolutely b ounded. Based on geometrical arguments, w e have derived extremely tigh t bo unds for the operating characteristic function. T o reduce the computatio na l complexity for the relev an t integrals, we prop ose adaptive scanning algorithms which are no t only useful for present hypothesis testing problem but also for o ther pro blem area s. 1 In tro duction Consider a Gaussian random v ariable X with mean µ and v ariance σ 2 . In many applications, it is an imp ortan t problem to d e termine wh e ther the mean µ is less or greater than a pr e scrib ed v alue γ based on i.i.d. rand om samples X 1 , X 2 , · · · of X . Su c h problem can b e put in to the setting of testing hyp othesis H 0 : µ ≤ µ 0 v ersus H 1 : µ > µ 1 with µ 0 = γ − εσ and µ 1 = γ + εσ , where ε is a p ositiv e num b er sp ecifying th e wid th of the indifference zone ( µ 0 , µ 1 ). It is usually required that the size of the Type I error is no greater than α ∈ (0 , 1) and the size of the Typ e I I err o r is no greater than β ∈ (0 , 1). That is, Pr { Reject H 0 | µ } ≤ α, ∀ µ ∈ ( −∞ , µ 0 ] (1) Pr { Accept H 0 | µ } ≤ β , ∀ µ ∈ [ µ 1 , ∞ ) . (2) The h yp othesis testing pr o blem describ ed ab o v e has b een extensiv ely s tu died in th e framewo rk of sequen tial probabilit y r a tio test (SPR T), which was established by W al d [4] during the p erio d of second world w ar of last cen tury . The SPR T suffers fr o m several drawbac ks. First, th e sampling ∗ The author had b een previously working with Louisiana State U niv ersit y at Baton Rouge, LA 70803 , USA, and is n o w with Department of Electrical Engineering, Southern Un i versit y and A &M College, Baton Rouge, LA 70813, U SA; Email: c henxinjia@gmail.com 1 n umb e r of S PR T is a random num b er which is n o t b ounded. Ho wev er, to b e useful, the maxim um sampling n umb e r of an y testing plan should b e b ounded by a deterministic n um b er. Although this can be fixed b y forced termination (see, e.g., [3] and the references therein), the prescrib ed lev el of p o w er ma y not b e ensured as a result of tr uncati on. Second, the num b er of sampling op eratio ns of SPR T is as large as the n umber of samples. In practice, it is us uall y muc h more economical to tak e a batc h of samples at a time instead o f one b y one. Thir d , the efficie ncy of SPR T is optimal only for the endp oin ts of the indifference zone. F or other parametric v alues, the SPR T can b e extremely inefficien t. Needless to sa y , a truncated version of SPR T ma y suffer fr o m the same problem due to the partial use of the b oundary of SPR T. Third, wh e n the v ariance σ 2 is not av ailable, a w eigh ting function needs to b e constructed so that the testing pr o blem can b e fit in to th e framew ork of SPR T. The construction of s u c h w eigh ting fun c tion is a d i fficult task and sev erely limit the efficiency of the resultan t test plan. In this pap er, to ov ercome the limitations of existing tests for the mean of a normal distribu - tion, w e ha ve established a new class testing plans ha ving the follo wing feat ur e s: i) Th e testing has a fin it e n umb e r of s tages an d thus the cost of sampling op erat ions is reduced as compared to SPR T. ii) The sampling n umber is absolutely b ounded without truncation. iii) T he pr escrib ed lev el of p o w er is rigorously guaran teed. iv) The testing is not only efficient for th e end points of indifference zone, but also efficien t for other parametric v alues. v) Ev en the v ariance σ 2 is unknown, our test plans do not require any w eigh ting function. In general, our testing plans consist of s stages. F or ℓ = 1 , · · · , s , the sample size of the ℓ -th stage is n ℓ . F or the ℓ -th stage , a decision v ariable D ℓ is defined b y using samples X 1 , · · · , X n ℓ suc h that D ℓ assumes only three p ossible v alues 0 , 1 and 2 with the f o llo wing notion: (i) Samplin g is contin ued until D ℓ 6 = 0 for s o me ℓ ∈ { 1 , · · · , s } . Since th e sampling m ust b e terminated at or b efore the s - th stage, it is required th a t D s 6 = 0. F or simplicit y of notations, w e also define D 0 = 0. (ii) Th e null hyp othesis H 0 is accepted at the ℓ -th stage if D ℓ = 1 and D i = 0 f or 1 ≤ i < ℓ . (iii) Th e null h yp othesis H 0 is rejected at the ℓ -th stage if D ℓ = 2 and D i = 0 f or 1 ≤ i < ℓ . As will b e seen in the our sp ecific te sting plans, the sample sizes n 1 < n 2 < · · · , n s and decision v aria bles D 1 , · · · , D s dep end on the paramet ers α, β , µ 0 , µ 1 and other parameters s uc h as th e risk tuning p ar ameter ζ and the samp le size incr emental factor ρ . The requiremen ts of p o wer can b e satisfied by determinin g an app ropriate v alue of ζ via b isec tion searc h. F or this p u rp o se, w e ha v e derived, by a geometrica l approac h, readily computable b ounds for the ev aluation of th e op erating c haracteristic (OC ) function. The remaind e r of th e pap er is organized as follo ws. I n Section 2, we presen t our app roac h for testing the mean of a normal d istribution in the con text of kno wing the v ariance σ 2 . In Section 3, w e describ e our metho d for for testing the mean of a normal distribution for situations that the v aria nce σ 2 is n o t a v ailable. Section 4 discusses the ev aluati on of OC functions. In Sectio n, we prop ose adaptive scanning algorithms for inte gration, summation, zero findin g and optimization. These new metho ds are u seful for our curr en t p roblem and other problem areas. S ec tion 6 is the 2 conclusion. All p r oofs of theorems are giv en in App endices. Throughout this pap er, we shall u se the follo wing notations. The ceiling f unction is denoted b y ⌈ . ⌉ (i.e ., ⌈ x ⌉ represents th e smallest intege r no less than x ). The gamma fu ncti on is denoted b y Γ( . ). Th e inv erse cosine function taking v alues o n [0 , π ] is den o ted by arccos( . ). The inv erse tangen t function taking v alues on  − π 2 , π 2  is denoted b y arctan( . ). W e use the notation Pr { . | θ } to indicate that the asso ciate d rand om samples X 1 , X 2 , · · · are parameterized by θ . The parameter θ in Pr { . | θ } ma y b e dropp ed w h enev er this ca n b e d o ne without introdu c ing confusion. Th e other notations w ill b e made clear as we pro ceed. 2 T esting the Mean of a Normal Distribution with Kno wn V ari- ance F or δ ∈ (0 , 1), let Z δ > 0 b e the critical v alue of a n o rmal distribu ti on with zero mea n and u nit v aria nce, i.e., Φ( Z δ ) = 1 √ 2 π R ∞ Z δ e − x 2 2 dx = δ . In situations that the v ariance σ 2 is kno wn, our testing plan, dev elop ed in [2], is d e scrib ed as follo ws. Theorem 1 L et ζ > 0 and ρ > 0 . L et n 1 < n 2 < · · · < n s b e the asc ending ar r angement of al l distinct elements of nl ( Z ζ α + Z ζ β ) 2 4 ε 2 (1 + ρ ) i − τ m : i = 1 , · · · , τ o , wher e τ is a p ositive inte ge r. D efine a ℓ = ε √ n ℓ − Z ζ β , b ℓ = Z ζ α − ε √ n ℓ for ℓ = 1 , · · · , s − 1 , and a s = b s = Z ζ α −Z ζ β 2 . Define X n ℓ = P n ℓ i =1 X i n ℓ , T ℓ = √ n ℓ ( X n ℓ − γ ) σ , D ℓ =        1 for T ℓ ≤ a ℓ , 2 for T ℓ > b ℓ , 0 else for ℓ = 1 , · · · , s . Then, b oth (1) and (2) ar e guar ante e d pr ovide d that 0 < ζ ≤ 1 τ . Mor e over, the OC function Pr { A c c ept H 0 | µ } is monotonic al ly de cr e asing with r esp e ct to µ ∈ ( −∞ , µ 0 ) ∪ ( µ 1 , ∞ ) . T o compute tight b ound s for the OC function, we h a ve the follo wing result. Theorem 2 L et U and V b e i nd ep endent Gaussian r ando m variables with zer o me an and varianc e unity. Define ϕ ( θ, ζ , α, β ) = Φ ( √ n 1 θ − b 1 ) + s X ℓ =2 Pr  b ℓ − √ n ℓ θ ≤ U ≤ k ℓ V − √ n ℓ θ + r n ℓ n ℓ − 1 b ℓ − 1  − s X ℓ =2 Pr  b ℓ − √ n ℓ θ ≤ U ≤ k ℓ V − √ n ℓ θ + r n ℓ n ℓ − 1 a ℓ − 1  with k ℓ = q n ℓ n ℓ − 1 − 1 , ℓ = 2 , · · · , s . Then, Pr { A c c ept H 0 | µ = θ σ + γ } > 1 − ϕ ( θ , ζ , α, β ) for any θ ∈ ( −∞ , − ε ] and Pr { A c c ept H 0 | µ = θ σ + γ } < ϕ ( − θ , ζ , β , α ) for any θ ∈ [ ε, ∞ ) . 3 See App endix A for a pro of. As can b e seen f r o m the pro of of Theorem 2, w e ha ve P s ℓ =1 Pr { D ℓ − 1 = 0 , D ℓ = 2 | µ 0 } = ϕ ( µ 0 − γ σ , ζ , α, β ) and P s ℓ =1 Pr { D ℓ − 1 = 0 , D ℓ = 1 | µ 1 } = ϕ ( γ − µ 1 σ , ζ , β , α ) . By making use of such results and a b isec tion searc h method, w e can determine an app ropriate v alue of ζ so that b oth (1) and (2) are guaran teed. With regard to the distribu ti on of s amp le num b er n , w e hav e, for ℓ = 1 , · · · , s − 1, Pr { n > n ℓ } ≤ Pr { a ℓ < T ℓ ≤ b ℓ } = Pr { T ℓ ≤ b ℓ } − Pr { T ℓ ≤ a ℓ } = Pr { U + θ √ n ℓ ≤ b ℓ } − Pr { U + θ √ n ℓ ≤ a ℓ } = Φ ( b ℓ − √ n ℓ θ ) − Φ ( a ℓ − √ n ℓ θ ) , where U is a Gaussian rand o m v ariable with zero mean and un it v ariance. 3 T esting the Mean of a Normal Distribution with Unkn o wn V ariance F or δ ∈ (0 , 1), let t n,δ b e the critical v alue of Stud en t’s t -distribu tio n with n degrees of freedom. Namely , t n,δ is a n umber satisfying Z ∞ t n,δ Γ( n +1 2 ) √ nπ Γ( n 2 )  1 + x 2 n  − n +1 2 = δ. In situations that the v a riance σ 2 is unknown, our testing plan, develo p ed in [2], is describ ed as follo ws. Theorem 3 L et ζ > 0 a nd ρ > 0 . L et n ∗ b e the minimum inte ger n such that t n − 1 ,ζ α + t n − 1 ,ζ β ≤ 2 ε √ n − 1 . L et n 1 < n 2 < · · · < n s b e the asc ending arr angement of al l distinct e lements of {⌈ n ∗ (1 + ρ ) i − τ ⌉ : i = 1 , · · · , τ } , wher e τ is a p ositive inte ger. Define a ℓ = ε √ n ℓ − 1 − t n ℓ − 1 ,ζ β , b ℓ = t n ℓ − 1 ,ζ α − ε √ n ℓ − 1 for ℓ = 1 , · · · , s − 1 , a nd a s = b s = t n s − 1 ,ζ α − t n s − 1 ,ζ β 2 . Define X n ℓ = P n ℓ i =1 X i n ℓ , b σ n ℓ = s P n ℓ i =1 ( X i − X n ℓ ) 2 n ℓ − 1 , b T ℓ = √ n ℓ ( X n ℓ − γ ) b σ n ℓ , D ℓ =        1 for b T ℓ ≤ a ℓ , 2 for b T ℓ > b ℓ , 0 else for ℓ = 1 , · · · , s . Then, b oth (1) a nd (2) ar e guar a nte e d if ζ > 0 is sufficiently smal l. Mor e over, the OC function Pr { A c c ept H 0 | µ } is monotonic al ly de cr e asing with r esp e ct to µ ∈ ( −∞ , µ 0 ) ∪ ( µ 1 , ∞ ) . T o obtain tight b ounds f or the OC fun ct ion, the follo wing r e sult is useful. Theorem 4 L et U, V and Y ℓ , Z ℓ , ℓ = 2 , · · · , s b e indep endent r ando m variables such that U, V p ossess i dentic al normal distributions with zer o me an and unit varianc e and that Y ℓ , Z ℓ p ossess chi-squar e distributions of n ℓ − 1 − 1 and n ℓ − n ℓ − 1 − 1 de gr e es of fr e e dom r esp e ctively. Define 4 k ℓ = q n ℓ n ℓ − 1 − 1 , c ℓ = a ℓ √ n ℓ − 1 and d ℓ = b ℓ √ n ℓ − 1 for ℓ = 1 , · · · , s . De fine P ( θ , ζ , α, β ) = P s ℓ =1 P ℓ wher e P 1 = Pr n b T 1 > b 1 o and P ℓ =                Pr n d ℓ √ V 2 + Y ℓ + Z ℓ < U + √ n ℓ θ ≤ k ℓ V + d ℓ − 1 q n ℓ Y ℓ n ℓ − 1 o − Pr n d ℓ √ V 2 + Y ℓ + Z ℓ < U + √ n ℓ θ ≤ k ℓ V + c ℓ − 1 q n ℓ Y ℓ n ℓ − 1 o for d ℓ ≥ 0 , Pr n a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 o + Pr n | d ℓ | √ V 2 + Y ℓ + Z ℓ ≤ U − √ n ℓ θ < k ℓ V − d ℓ − 1 q n ℓ Y ℓ n ℓ − 1 o − Pr n | d ℓ | √ V 2 + Y ℓ + Z ℓ ≤ U − √ n ℓ θ < k ℓ V − c ℓ − 1 q n ℓ Y ℓ n ℓ − 1 o for d ℓ < 0 for ℓ = 2 , · · · , s . Then, Pr { A c c e pt H 0 | µ = θ σ + γ } ≥ 1 − P ( θ , ζ , α, β ) for any θ ∈ ( −∞ , − ε ] and Pr { A c c ept H 0 | µ = θ σ + γ } ≤ P ( − θ , ζ , β , α ) for any θ ∈ [ ε, ∞ ) . See App endix B for a p roof. As can b e seen fr o m the p roof of Th e orem 4, we ha v e P s ℓ =1 Pr { D ℓ − 1 = 0 , D ℓ = 2 | µ 0 } = P ( µ 0 − γ σ , ζ , α, β ) and P s ℓ =1 Pr { D ℓ − 1 = 0 , D ℓ = 1 | µ 1 } = P ( γ − µ 1 σ , ζ , β , α ). By making use of such r esu lt s and a b isection searc h metho d, we can determine an appropriate v alue of ζ so that b oth (1) and (2) are guaran teed. With regard to the d ist rib u tio n of the sample num b er n , we ha v e Pr { n > n ℓ } < Pr { a ℓ < b T ℓ ≤ b ℓ } for ℓ = 1 , · · · , s − 1, wher e the p robabilit y can b e expressed in terms of th e well-kno wn non -central t -distribution. 4 Ev aluation of OC F unctions In this secti on, we shall d e monstrate that the ev aluatio n of OC functions of tests describ ed in preceding discussion can b e reduced to the compu t ation of the probabilit y of a certain domain including t wo indep endent standard Gaussian v ariables. In this regard, our first general resu lt is as follo ws. Theorem 5 L et U and V b e two indep endent Gaussian r andom variables with zer o me an and unit varianc e. L et D b e a two-dimensional c onvex domain which c ontains the origin (0 , 0) . Supp ose the set of b ounda ry p oints of D c an b e expr esse d as B = { ( r, φ ) : r = B ( φ ) , φ ∈ A } i n p ola r c o or dinates, wher e B ( φ ) is a Riemann inte gr able function on set A . Then, Pr { ( U, V ) ∈ D } = 1 − 1 2 π Z A exp  − B 2 ( φ ) 2  dφ. See App endix C for a pro of. F or situations that the domain do es not con tain the origin (0 , 0), w e need to in tro duce the concept of visibility for bou n dary p oin ts of a tw o-dimensional domain D . The in tuitive notion of such concept is that a b oundary p oin t of D is visible if it can b e seen b y an observ er at the origin. Th e precise definition is as f o llo ws. Definition 1 A b ounda ry p oint, ( u, v ) , of do main D is said to b e visible if { ( q u, q v ) : 0 < q < 1 } ∩ D is empty . Otherwise, such a b oundary p oint is said to b e invisible. 5 Based on the concept of visib i lit y , w e h a ve derived the follo wing general result. Theorem 6 L et U and V b e two indep e nd ent Gaussian r andom variables with zer o me an and unit varianc e. L et D b e a two-dimensional c onvex domain which do es not c ontain the origin (0 , 0) . Supp ose the set of v i sible b oundary p oints of D c an b e expr esse d a s B v = { ( r , φ ) : r = B v ( φ ) , φ ∈ A v } in p olar c o or dinates, wher e B v ( φ ) is a Riema nn inte g r able function o n set A v . Supp ose the set of invisible b oundary p o ints of D c an b e expr esse d as B i = { ( r , φ ) : r = B i ( φ ) , φ ∈ A i } in p olar c o or dinates, wher e B i ( φ ) is a Riemann inte gr able function on set A i . Then, Pr { ( U, V ) ∈ D } = 1 2 π  Z A v exp  − B 2 v ( φ ) 2  dφ − Z A i exp  − B 2 i ( φ ) 2  dφ  . See App endix D f o r a pro of. As can b e seen fr o m Theorem 2, the ev al uation of OC functions of test plans d esig ned for the case that the v ariance σ 2 is known can b e red u ce d to the computation of pr o babilities of the form Pr { h ≤ U ≤ k V + g } . F or fast compu t ation of such probabilities, we ha v e deriv ed, b a sed on Theorems 5 and 6, the follo wing result. Theorem 7 L et k > 0 . L et U and V b e indep endent Gaussian r andom variables with zer o me a n and unit varianc e. Define Ψ h ( φ ) = 1 2 π exp  − h 2 2 cos 2 φ  , Ψ g,k ( φ ) = 1 2 π exp  − g 2 2(1+ k 2 ) cos 2 φ  , φ k = arctan ( k ) and φ R = ar ctan  h − g kh  . Then, Pr { h ≤ U ≤ k V + g } =          R π + φ k + φ R π / 2 Ψ g ,k ( φ ) dφ − R π + φ R π / 2 Ψ h ( φ ) dφ for max( g , h ) < 0 , 1 − R π + φ R π / 2 Ψ h ( φ ) dφ − R 3 π / 2 φ k + φ R Ψ g ,k ( φ ) dφ for h ≤ 0 ≤ g , R π / 2 φ R Ψ h ( φ ) dφ − R π / 2 φ k + φ R Ψ g ,k ( φ ) dφ else . See App endix E for a pro of. As can b e seen from T heo rem 4, the ev aluation of OC functions of test plans d esig ned for the case that the v ariance σ 2 is unknown can b e reduced to the compu ta tion of pr o babilities of th e typ e P r n λ √ V 2 + Y + Z ≤ U − ϑ < k V +  √ Y o with λ > 0, where Y and Z are c hi-square random v ariables indep endent with U and V . The ev aluat ion of su c h probab ilities is describ ed as follo ws. Define multiv ariate f unctions P ( y , z ) and P ( y , z ) so that P ( y , z ) =    Pr n λ q V 2 + y + z ≤ U − ϑ ≤ k V +  √ y o if  ≥ 0 , Pr n λ q V 2 + y + z ≤ U − ϑ ≤ k V +  √ y o if  < 0 P ( y , z ) =    Pr n λ p V 2 + y + z ≤ U − ϑ ≤ k V +  √ y o if  ≥ 0 , Pr n λ p V 2 + y + z ≤ U − ϑ ≤ k V +  √ y o if  < 0 for 0 < y ≤ y , 0 < z ≤ z . Then, Pr { λ √ V 2 + Y + Z ≤ U − ϑ ≤ k V +  √ Y , Y ∈ [ y , y ] , Z ∈ [ z , z ] } is smaller than Pr  Y ∈ [ y , y ]  × Pr { Z ∈ [ z , z ] } × P ( y , z ) and is greater than Pr  Y ∈ [ y , y ]  × Pr { Z ∈ [ z , z ] } × P ( y , z ). F or an y ǫ ∈ (0 , 1), w e can determine, via bisection searc h, p ositiv e 6 n umb e rs y min < y max and z min < z max suc h that Pr { Y < y min } < ǫ 4 , Pr { Y > y max } < ǫ 4 , Pr { Z < z min } < ǫ 4 and Pr { Z > z max } < ǫ 4 . By p artitioning the set { ( y , z ) : y ∈ [ y min , y max ] , z ∈ [ z min , z max ] } as sub-d omains { ( y , z ) : y ∈ [ y i , y i ] , z ∈ [ z i , z i ] } , i = 1 , · · · , m and ev aluating P i = Pr { Y ∈ [ y i , y i ] } × Pr { Z ∈ [ z i , z i ] } × P ( y i , z i ) and P i = Pr { y i ≤ Y ≤ y i } × P r { Z ∈ [ z i , z i ] } × P ( y i , z i ) for i = 1 , · · · , m , we h a v e X i P i < Pr n λ p V 2 + Y + Z ≤ U − ϑ ≤ k V +  √ Y o < ǫ + X i P i . The b ounds can b e refined by further partitio ning the sub-do m ains. F or efficiency , we can split the sub- domain with the large s t gap b et ween the upp er b ound P i and low er b ound P i in every additional partition. It can b e seen that the proba bilities like P ( y i , z i ) and P ( y i , z i ) ar e of the s ame type as Pr { ( U, V ) ∈ D } , where D = { ( u, v ) : √ λv 2 + h ≤ u − ϑ ≤ k v + g } with k > 0 , λ > 0 , h ≥ 0 and k 2 6 = λ . F or fas t computation of s uc h probabilities, we have derived, ba sed on Theorems 5 a nd 6, the following res ult s. Theorem 8 Define ∆ = h ( k 2 − λ ) + λg 2 , u A = λg − k √ ∆ λ − k 2 + ϑ, u B = λg + k √ ∆ λ − k 2 + ϑ, v A = gk − √ ∆ λ − k 2 , v B = gk + √ ∆ λ − k 2 , φ A = arccos  u A √ u 2 A + v 2 A  , φ B = arccos  u B √ u 2 B + v 2 B  , φ m = ar ctan  q h λ | ϑ 2 − h |  , φ λ = arctan  1 √ λ  , φ k = ar ctan ( k ) , Ψ ϑ,g, k ( φ ) = 1 2 π exp  − ( ϑ + g ) 2 2(1+ k 2 ) cos 2 φ  and Υ ϑ,λ,h ( φ ) = 1 2 π exp    − ( ϑ 2 − h ) 2 2 h ϑ co s φ + p ( h + λh − λϑ 2 ) co s 2 φ + λ ( ϑ 2 − h ) i 2    . Then, Pr { ( U, V ) ∈ D } =                    I np for k 2 < λ, g > √ h, ∆ ≥ 0 , I pp for k 2 < λ, 0 < g ≤ √ h, ∆ ≥ 0 , I n for k 2 > λ, g k > √ ∆, I p for k 2 > λ, g k ≤ √ ∆, 0 else wher e I np =                    I np , 1 for ϑ + h u B − ϑ ≥ 0 , I np , 2 for ϑ + h u B − ϑ < 0 ≤ ϑ + h u A − ϑ , I np , 3 for ϑ + h u A − ϑ < 0 ≤ ϑ + √ h, I np , 4 for ϑ + √ h < 0 ≤ ϑ + g , I np , 5 for ϑ + g < 0 I n =                    I n , 1 for ϑ ≥ 0 , I n , 2 for ϑ < 0 ≤ ϑ + h u A − ϑ , I n , 3 for ϑ + h u A − ϑ < 0 ≤ ϑ + √ h, I n , 4 for ϑ + √ h < 0 ≤ ϑ + g , I n , 5 for ϑ + g < 0 I pp =        I pp , 1 for ϑ + h u B − ϑ ≥ 0 , I pp , 2 for ϑ + h u B − ϑ < 0 ≤ ϑ + h u A − ϑ , I pp , 3 for ϑ + h u A − ϑ < 0 I p =        I p , 1 for ϑ ≥ 0 , I p , 2 for ϑ < 0 ≤ ϑ + h u A − ϑ , I p , 3 for ϑ + h u A − ϑ < 0 7 with I np , 1 = Z π + φ B π − φ A Υ( φ ) dφ − Z φ k + φ B φ k − φ A Ψ( φ ) dφ, I np , 2 = Z π + φ m π − φ A Υ( φ ) dφ − Z φ m φ B Υ( φ ) dφ − Z φ k + φ B φ k − φ A Ψ( φ ) dφ, I np , 3 = Z π + φ m π − φ m Υ( φ ) dφ − Z φ m φ B Υ( φ ) dφ − Z φ m φ A Υ( φ ) dφ − Z φ k + φ B φ k − φ A Ψ( φ ) dφ, I np , 4 = 1 − Z φ k + φ B φ k − φ A Ψ( φ ) dφ − Z 2 π − φ A φ B Υ( φ ) dφ, I np , 5 = Z φ k − φ A +2 π φ k + φ B Ψ( φ ) dφ − Z 2 π − φ A φ B Υ( φ ) dφ, I n , 1 = Z π + φ λ π − φ A Υ( φ ) dφ − Z π 2 φ k − φ A Ψ( φ ) dφ, I n , 2 = Z π + φ m π − φ A Υ( φ ) dφ − Z φ m φ λ Υ( φ ) dφ − Z π 2 φ k − φ A Ψ( φ ) dφ, I n , 3 = Z π + φ m π − φ m Υ( φ ) dφ − Z φ m φ λ Υ( φ ) dφ − Z φ m φ A Υ( φ ) dφ − Z π 2 φ k − φ A Ψ( φ ) dφ, I n , 4 = 1 − Z π 2 φ k − φ A Ψ( φ ) dφ − Z 2 π − φ A φ λ Υ( φ ) dφ, I n , 5 = Z φ k − φ A +2 π π 2 Ψ( φ ) dφ − Z 2 π − φ A φ λ Υ( φ ) dφ, I pp , 1 = Z π + φ B π + φ A Υ( φ ) dφ − Z φ k + φ B φ k + φ A Ψ( φ ) dφ, I pp , 2 = Z π + φ m π + φ A Υ( φ ) dφ − Z φ m φ B Υ( φ ) dφ − Z φ k + φ B φ k + φ A Ψ( φ ) dφ, I pp , 3 = Z φ k + φ A φ k + φ B Ψ( φ ) dφ − Z φ A φ B Υ( φ ) dφ, I p , 1 = Z φ k + φ A π 2 Ψ( φ ) dφ + Z π + φ λ π + φ A Υ( φ ) dφ, I p , 2 = Z φ k + φ A π 2 Ψ( φ ) dφ + Z π + φ m π + φ A Υ( φ ) dφ − Z φ m φ λ Υ( φ ) dφ, I p , 3 = Z φ k + φ A π 2 Ψ( φ ) dφ − Z φ A φ λ Υ( φ ) dφ. See App endix F for a pro of. In T he o rem 8, for simplicity of nota t ions, we have abbrevia ted Ψ ϑ,g, k ( φ ) and Υ ϑ,λ,h ( φ ) as Ψ ( φ ) a nd Υ( φ ) resp ectiv ely . 5 Adaptiv e Scanning Algorithms As ca n b e seen from last section, we need to frequently ev aluate integrals inv olving functions like Υ( . ) and Ψ( . ). Clea r ly , there are no c losed-form solutio ns for this type of in tegr als. Although existing num erica l 8 int egr ation metho d can be applied to o bt ain approximations for such in tegra ls, the a ccuracy of integration is not clear ly known. Since our c oncern is the risk o f making wro ng decisio ns in hypothesis testing, the quantification of in tegra t ion is crucial. Motiv ated b y this co nsideration, we have developed an adaptive scanning metho d for fast integration. Moreover, we have extended the metho d to summation, zero finding and optimization. 5.1 In tegration of Con tin uous F unctions The integrals in volv ed in h yp othesis testing can b e a ddressed in the general framework of computing I ( a, b ) = R b a f ( x ) dx by a numerical metho d. Exis t ing metho ds are quadr ature rules. A q uadrature r ule is an approximation of the definite int egr al o f a function, usually stated as a weight ed sum o f a function v alues a t sp ecified p oin ts within the domain o f in tegra tions. More for ma lly , a q uadrature rule pro ceeds as follows: (i) Partition the interv a l [ a, b ] by gr id p oin ts a = x 0 < x 1 < · · · < x n = b . (ii) E v aluate f ( x i ) , i = 0 , 1 , · · · , n . (iii) Co nstruct an estimate b I ( a, b ) for I ( a, b ) a s a weighted sum o f f ( x i ) , i = 0 , 1 , · · · , n . W ell known q ua drature r ules ar e r ectangle rule, trapez ium rule, Simpson’s rule, Romberg’s metho d, Gaussian q uadrature rule, Clenshaw-Curtis quadratur e rule, Newton-Cote formula, Richardson extr apola- tion, etc. One of the most frequently used method is the comp osite Simpson’s r ule. Suppo se that the interv al [ a, b ] is split up in n subin terv als, with n an even num ber . Then, the co mposite Simpson’s rule is given by Z b a f ( x ) dx ≈ h 3   f ( x 0 ) + 2 n 2 − 1 X j =1 f ( x 2 j ) + 4 n 2 X j =1 f ( x 2 j − 1 ) + f ( x n )   , where x j = a + j h, j = 0 , 1 , · · · , n − 1 , n and h = b − a n ; in pa rticular, x 0 = a and x n = b . It is widely r e c ognized that an ass essmen t of the ac c uracy is a n essential pa rt o f any numerical metho d. Spec ific a lly , given ε > 0, a crucia l question is how to ens ure | b I ( a, b ) − I ( a, b ) | ≤ ε ? The err or c o mmit ted b y the comp osite Simpso n’s rule is b ounded (in absolute v alue) by h 4 180 ( b − a ) max ζ ∈ [ a,b ]   f 4 ( ζ )   . In Simpson’s rule, it is not clear how to c ho ose the step length. If the step length is too small, the computation is to o slow. O n the other hand, a lar ge step length may cause intolerable error of the computation. Although the err or b ound ca n be e xpressed in ter ms of the fo urth deriv ative, to guara ntee the accura cy , we need to b ound the fourth deriv ative ov er the who le integration r ange [ a, b ]. The b ounding is not ea sy and ca n b e extremely conser v ative. It is not hard to se e that other qua drature rules suffer similar drawbacks as the Simpso n’s r ule . T o ov ercome such drawbacks, we prop ose a new approa c h so that the a c c uracy requir emen t can b e rig orously guaranteed under mild conditions. A salient feature of our approach is that, ins tea d of partition the int erv al [ a , b ], we sequentially and a daptiv ely p erform integration over subinterv als of the overall interv al. F or e a c h subinterv a l, making use of deriv ative informatio n, we for ce the integration to meet a certain accuracy r equiremen t. Sta rting fr om the left endp oin t of interv al [ a, b ], we determine an initial [ u 1 , v 1 ] with 9 u 1 = a such that the difference b et w een I ( u 1 , v 1 ) = R v 1 u 1 f ( x ) dx a nd its estimate b I ( u 1 , v 1 ) is no grea ter than ε b − a ( v 1 − u 2 ). Then, we determine next subin terv al [ u 2 , v 2 ] as the form u 2 = v 1 , v 2 = min { b, v 1 + ( v 1 − u 1 )2 ℓ } , with ℓ tak en as th e lar gest integer no gr eater than 1 to ensure tha t the difference betw een I ( u 2 , v 2 ) = R v 2 u 2 f ( x ) dx and its estimate b I ( u 2 , v 2 ) is no gre a ter than ε b − a ( v 2 − u 2 ). F or i > 1, given interv al [ u i , v i ], we determine next subinterv al [ u i +1 , v i +1 ] as the for m u i +1 = v i , v i +1 = min { b, v i + ( v i − u i )2 ℓ } , with ℓ taken as the larg est integer no gr eater than 1 to ensure that the difference b et ween I ( u i +1 , v i +1 ) = R v i +1 u i +1 f ( x ) dx and its estimate b I ( u i +1 , v i +1 ) is no greater than ε b − a ( v i +1 − u i +1 ). W e repea t this pr ocess un til v i = b for some i . Finally , the ov erall estimate b I ( a, b ) for I ( a, b ) is given by b I ( a, b ) = X i b I ( u i , v i ) , which ensur es that | b I ( a, b ) − I ( a, b ) | ≤ X i    b I ( u i , v i ) − I ( u i , v i )    ≤ X i ε b − a ( v i − u i ) = ε. Since the ab ov e pro cess of in tegra tion is like sca nning the interv al of integration, we call the metho d a s Adaptive Scanning Algorithm (ASA). The adaptive na tur e of the algorithm can b e see n from the dyna m ic choice of the length of subin terv al [ u i , v i ]. T o fo rmally des cribe our ASA, let I ( u, v ) = R v u f ( x ) dx and b I ( u, v ) b e an estimate of I ( u , v ) for a ≤ u ≤ v ≤ b . Assume that | b I ( u, v ) − I ( u, v ) | → 0 a s | u − v | → 0. Let η = ε b − a . Assume that we have a metho d for testing the truth of | b I ( u, v ) − I ( u, v ) | ≤ η ( v − u ) without knowing I ( u, v ). Our ASA pro ceeds as follows. ⋄ Cho ose initial step length ∆ as a pos itive num ber less than b − a 2 . ⋄ Let b I ( a, b ) ← 0 , η ← ε b − a and u ← a . ⋄ While u + ∆ < b , do the following: ⋆ Let st ← 0 and ℓ ← 2 ; ⋆ While st = 0, do the following: ∗ Let ℓ ← ℓ − 1 and ∆ ← 2 ℓ ∆ . ∗ If u + ∆ < b , let v ← u + ∆ . Otherwise, let v ← b . ∗ Ev aluate b I ( u, v ). ∗ T est the tr ut h of | b I ( u, v ) − I ( u , v ) | ≤ η ( v − u ) without knowledge of I ( u, v ). ∗ If | b I ( u, v ) − I ( u , v ) | ≤ η ( v − u ) is true, let b I ( a, b ) ← b I ( a, b ) + b I ( u, v ) a nd st ← 1 , u ← v . ⋄ Return b I ( a, b ) a s an estimate for I ( a, b ). Under the assumption that | b I ( u, v ) − I ( u, v ) | → 0 as | u − v | → 0, it can b e readily shown that | b I ( a, b ) − I ( a, b ) | ≤ ε is guar an teed after execution o f the algor ithm. This is b ecause | b I ( a, b ) − I ( a, b ) | is no greater than the summation of | b I ( u, v ) − I ( u , v ) | over all subinterv als ( u , v ) g enerated to cov er [ a, b ]. As can b e seen from the descr ipt ion of ASA, a critica l issue is to co nstruct b I ( u, v ) and test the truth of | b I ( u, v ) − I ( u , v ) | ≤ η ( v − u ) without any k no wledge of I ( u, v ). Our g eneral metho d for a ddressing this 10 issue is as follows. Let I ( u , v ) a nd I ( u, v ) b e lo wer and upp er bounds of I ( u, v ) respectively . Namely , I ( u, v ) ≤ I ( u, v ) = R v u f ( x ) dx ≤ I ( u, v ). Assume that I ( u, v ) − I ( u, v ) → 0 as | u − v | → 0. In many cases, the low er and upper b ounds ca n b e obtained from T aylor series ex pansion formu la. T o test the truth of | b I ( u, v ) − I ( u , v ) | ≤ η ( v − u ), we prop ose to make use of the following relatio nship I ( u, v ) − η ( v − u ) ≤ b I ( u, v ) ≤ I ( u, v ) + η ( v − u ) = ⇒ | b I ( u, v ) − I ( u, v ) | ≤ η ( v − u ) . (3) T o construc t es t imate b I ( u, v ) for I ( u, v ), we recommend to take b I ( u, v ) = 1 2 [ I ( u, v ) + I ( u, v )] or b I ( u, v ) = v − u 6  f ( u ) + 4 f ( u + v 2 ) + f ( v )  based on Simpson’s approximation rule. Assuming that the firs t deriv ative f ′ ( x ) of f ( x ) exists for all x ∈ [ a , b ], making use of (3), T aylor’s ser ies expansion fo rm ula, a nd Simpso n’s approximation r ule, we hav e der ived the following metho ds in Theorem 9 for co nstructing b I ( u, v ) and testing the truth o f | b I ( u, v ) − I ( u, v ) | ≤ η ( v − u ) without a n y k nowledge of I ( u, v ). Theorem 9 L et u, v and w b e t h r e e r e al num b ers su ch t h at w − u = v − w = h > 0 . L et I ( u, v ) = R v u f ( x ) dx and b I ( u, v ) = h 3 [ f ( u ) + 4 f ( w ) + f ( v )] . Then, the fol lowing statements hold tru e. (I) | b I ( u, v ) − I ( u, v ) | ≤ η ( v − u ) pr ovid e d that 3 κ − 6 η h ≤ f ( u ) + f ( v ) − 2 f ( w ) h ≤ 3 κ + 6 η h , wher e κ = 1 2 [min x ∈ [ u,w ] f ′ ( x ) + min x ∈ [ w ,v ] f ′ ( x )] and κ = 1 2 [max x ∈ [ u,w ] f ′ ( x ) + max x ∈ [ w ,v ] f ′ ( x )] . (II) | b I ( u, v ) − I ( u, v ) | ≤ η ( v − u ) pr ovid e d that f ( x ) is a c onc av e function of x ∈ [ u , v ] and that − 12 η h ≤ f ( u ) + f ( v ) − 2 f ( w ) h ≤ 3 4 [ f ′ ( v ) − f ′ ( u )] + 12 η h . (III) | b I ( u, v ) − I ( u , v ) | ≤ η ( v − u ) pr ovi de d t hat f ( x ) is a c onvex function of x ∈ [ u, v ] and that 3 4 [ f ′ ( v ) − f ′ ( u )] − 12 η h ≤ f ( u ) + f ( v ) − 2 f ( w ) h ≤ 12 η h . In the cas e that the co n v exity or co nca vity of f ( x ) are hard to deter mine, one may compute the b ounds of the firs t deriv ative of f ( x ) and apply s ta temen t (I) of Theorem 9 to ASA. F or e xample, the deriv atives of elliptical functions ca n b e easily b ounded, and th us one ca n use statement (I) for the purp ose o f integration. The applicatio ns o f statemen ts (I) a nd (I I) of The o rem 9 depend on the conv exity or concavit y of f ( x ). T o determine the c o n vexit y or concavit y of f ( x ), we can find the inflexion p oin ts from the equation f ′′ ( x ) = 0, which can frequent ly b e reduced to a quadr atic equatio n of x . Spe c ially , this is true for normal distribution, Gamma distribution, Beta distributio n, Studen t’s t -distr ibutio n, and F -distribution, etc. Once the inflexio n p oint s are obtained, the interv al o f integration can be decomp osed as subin terv als so that f ( x ) is completely convex or concave in each subinterv al. 5.2 Summation of Discrete F unctions In parallel to the problem of computing I ( a, b ) = R b a f ( x ) dx , a similar pr oblem is the computatio n of the discrete summation S ( a, b ) = P b k = a f ( k ), where a, b, k are in teger s. Let S ( u, v ) a nd S ( u, v ) b e the low er 11 and upp er b ounds of S ( u, v ) = P v k = u f ( k ) r espectively . W e can easily mo dif y the ASA of in tegra tion for computing S ( a, b ) a s follows. ⋄ Cho ose initial step length ∆ as a pos itive integer les s than b − a 2 . ⋄ Let b S ( a, b ) ← 0 , η ← ε b − a +1 and u ← a . ⋄ While u + ∆ < b , do the following: ⋆ Let st ← 0 and ℓ ← 2 ; ⋆ While st = 0, do the following: ∗ Let ℓ ← ℓ − 1 and ∆ ← ⌈ 2 ℓ ∆ ⌉ . ∗ If u + ∆ < b , let v ← u + ∆ . Otherwise, let v ← b . ∗ If u + 1 < v , ev aluate S ( u, v ) and S ( u, v ). ∗ If u + 1 < v a nd S ( u, v ) − S ( u, v ) ≤ 2 η ( v − u + 1), let b S ( a, b ) ← b S ( a, b ) + 1 2 [ S ( u, v ) − S ( u, v )] and st ← 1. ∗ If u + 1 = v , let b S ( a, b ) ← b S ( a, b ) + f ( u ) + f ( v ) and st ← 1. ∗ If st = 1, let u ← v + 1 . ⋄ Return b S ( a, b ) as an estima te for S ( a, b ). Clearly , | b S ( a, b ) − S ( a, b ) | ≤ ε is guaranteed after the execution o f the ab o ve algor ithm. A key ro utine is to ca lculate the low er and upper b ounds of S ( u, v ) = P v k = u f ( k ). F o r this purp ose, we hav e esta blished in [1] the following re s ult s. Theorem 1 0 L et u < v b e t wo inte gers. Define r u = f ( u +1) f ( u ) , r v = f ( v − 1) f ( v ) , r u,v = f ( u ) f ( v ) and j = u + v − u − (1 − r u,v )(1 − r v ) − 1 1+ r u,v (1 − r u )(1 − r v ) − 1 . Defin e α ( i ) = ( i + 1 − u ) h 1 + ( i − u )( r u − 1) 2 i and β ( i ) = ( v − i ) h 1 + ( v − i − 1)( r v − 1) 2 i . The fol lowing statements hold tru e: (I): If f ( k + 1) − f ( k ) ≤ f ( k ) − f ( k − 1) for u < k < v , then ( v − u + 1)[ f ( u ) + f ( v )] 2 ≤ v X k = u f ( k ) ≤ α ( i ) f ( u ) + β ( i ) f ( v ) (4) for u < i < v . The minimum gap b etwe en the lower and upp er b ounds is achieve d at i such that ⌊ j ⌋ ≤ i ≤ ⌈ j ⌉ . (II): If f ( k + 1) − f ( k ) ≥ f ( k ) − f ( k − 1) for u < k < v , t h en ( v − u + 1)[ f ( u ) + f ( v )] 2 ≥ v X k = u f ( k ) ≥ α ( i ) f ( u ) + β ( i ) f ( v ) for u < i < v . The minimum gap b etwe en the lower and upp er b ounds is achieve d at i such that ⌊ j ⌋ ≤ i ≤ ⌈ j ⌉ . T o inv estigate conditions like f ( k + 1 ) − f ( k ) ≤ f ( k ) − f ( k − 1) o r f ( k + 1) − f ( k ) ≥ f ( k ) − f ( k − 1), we ca n find the inflexion p oints fr om equatio n f ( k + 1) − f ( k ) = f ( k ) − f ( k − 1), which in many cases ca n be reduced to a quadra tic equation of k . Specially , this is tr ue for binomial distribution, negative binomial distribution, P ois s on distribution and hyper-ge o metrical distr ibutio n, etc. Onc e the inflexion p oint s are obtained, we can decomp ose the r ange of summa tio n as subsets so that f ( k ) is completely co n v ex or concave in each subset. 12 5.3 Zero Finding T o determine the co n v exity or co nca vity of f ( x ), we need to find zero s of the s econd deriv ative f ′′ ( x ). F or a function lik e Υ ( . ), there exits no analytic solution. Motiv ated by s uc h situation, w e pro pose a ge neral metho d for finding the zeros of f ( x ) for x ∈ [ a, b ]. Sine the zeros can b e obtained co nsecutiv ely , this problem can b e reduced to the following generic problem: Suppo se that f ( a ) < 0 and f ( x ) is contin uous for x ∈ [ a , b ]. Determine whether f ( x ) ha s at lea st o ne ro ot in [ a, b ]. In the case that f ( x ) has at least one ro ot in [ a, b ], find the smallest ro ot x ∈ [ a, b ] such that f ( x ) = 0 . Assume that, for a ny interv a l [ u , v ] ⊆ [ a, b ], it is p ossible to compute an upper b ound g ( u, v ) such that f ( x ) ≤ g ( u, v ) for any x ∈ [ u, v ] and tha t the upper bo und c o n verges to f ( x ) as the interv al width v − u tends to 0. Le t η > 0 be an extremely small n um b er, i.e. η = 10 − 15 . Our algorithm for zero finding pro ceeds a s follows: ⋄ Cho ose initial step length ∆ as a n umber b et ween η and b − a 2 . ⋄ Let F ← 0 , T ← 0 and a ← u . ⋄ While F = T = 0, do the following: ⋆ Let st ← 0 and ℓ ← 2 ; ⋆ While st = 0, do the following: ∗ Let ℓ ← ℓ − 1 and ∆ ← ∆ 2 ℓ . ∗ If u + ∆ < b , let v ← u + ∆ and T ← 0. Otherwis e, let v ← b and T ← 1. ∗ If g ( u, v ) < 0 , le t s t ← 1 and u ← v . ∗ If ∆ < η , let st ← 1 a nd F ← 1. ⋄ If F = 1, re tur n x = u + v 2 as the sma llest r oot in [ a, b ] such that f ( x ) = 0. Otherwise if F = 0 , declar e that f ( x ) ha s no ro ot on [ a , b ]. The ab o ve algorithm declares x = u + v 2 as an estimate of the smallest ro ot bas e d on the observ ation that f ( x ) < 0 for all x ∈ [ a, u ] a nd that g ( u, v ) ≥ 0. Since v − u < η ≈ 0 and g ( u, v ) → f ( u + v 2 ) as v − u → 0, it is reaso nable to b elieve that the smallest ro ot is close to u + v 2 . In the c a se that f ( x ) has mor e than one ro ots in [ a , b ], the above algor ith m can b e rep eatedly used to find all the zero s. It should be no ted that this algor ithm is actually adapted from our A da ptive Maximum Che cking Algo rithm (AMCA) established in [1 ]. 5.4 Finding Maxim um Clearly , finding the zeros of function f ( x ) is closely related to the problem of finding the minim um or maximum of f ( x ). Our AMCA ca n b e adapted for finding the maximum of f ( x ) for x ∈ [ a, b ]. F rom our previous pap er [1], it can be seen that our AMCA is a computationa l metho d to determine whether a function f ( x ) is smaller than a pre s cribed num b er for e very v alue o f x in in terv al [ a, b ]. Suppos e that we have a low er bo und L and an uppe r b ound U for max x ∈ [ a,b ] f ( x ). Then, we can apply our AMCA and a bis e ction sear c h method to deter m ine the exact v alue o f max x ∈ [ a,b ] f ( x ). O ne wa y to find a low er bo und L is to compute n v alues of f ( x ) and take the maximum a s L . Once a low er b ound L is obtained, one can find a n upp er b ound U as the form U = L 2 k , where the p ositiv e num b er k can b e determined as the minimu m integer by our AMCA s uc h that L 2 k > max x ∈ [ a,b ] f ( x ). Of course, there are so me other metho ds for finding L a nd U . 13 6 Conclusion In this pap er, we have develop ed new multistage s ampling schemes for testing the mean of a normal distribution. Our sa mpling schemes hav e absolutely bounded n umber of samples. Our test plans are significantly more efficient than previous tests, while rig orously guara n teeing prescr ibed level o f p o wer. In contrast to existing tests , o ur test plans inv olve no pr obabilit y r atio a nd weighting function. The ev aluation of op erating characteristic functions of our tests ca n b e readily a ccomplished by using tight b ounds derived from a geo metrical appro ac h. W e hav e es tablished ada pt ive sca nnin g metho ds for integration, summation, zero finding and optimization, which are useful for our current problem and other fields. A Pro of of Theorem 2 T o show Theorem 2, the following lemma is us ef ul. Lemma 1 L et m < n b e two p ositive int e gers. L et X 1 , X 2 , · · · , X n b e i.i.d. normal ra ndom variables with c ommon me an µ and varianc e σ 2 . L et X k = P k i =1 X i k for k = 1 , · · · , n . L et X m,n = P n i = m +1 X i n − m . Define U = √ n ( X n − µ ) σ , V = r m ( n − m ) n X m − X m,n σ , Y = 1 σ 2 m X i =1  X i − X m  2 , Z = 1 σ 2 n X i = m +1  X i − X m,n  2 . Then, U, V , Y , Z ar e indep endent r andom variables such that b oth U and V ar e normal ly distribut e d with zer o me an and varianc e 1 , Y p ossesses a chi -squar e distribution of de gr e e m − 1 , and Z p ossesses a chi- squar e distribution of de gr e e n − m − 1 . Mor e over, P n i =1 ( X i − X n ) 2 = σ 2 ( Y + Z + V 2 ) . Pro o f . Observ ing that R 1 = √ m ( X m − µ ) σ and R 2 = √ n − m ( X m,n − µ ) σ are indep enden t Gaussian random v ariables with z e r o mea n a nd unit v ariance and that U , V can be o bt ained fro m R 1 , R 2 by an orthogo nal transformatio n " U V # =   p m n q n − m n q n − m n − p m n   " R 1 R 2 # , we hav e that U and V are also indep enden t Gaussian ra ndo m v ariables with zer o mea n and unit v ariance. Since R 1 , R 2 , Y , Z ar e indep enden t, we hav e that U, V , Y , Z ar e indep enden t. F or simplicity of notations , let S n = P n i =1 ( X i − X n ) 2 and S m,n = P n i = m ( X i − X m,n ) 2 . Using identit y S n = P n i =1 X 2 i − nX 2 n , we have P m i =1 X 2 i = S m + mX 2 m , P n i = m +1 X 2 i = S m,n + ( n − m ) X 2 m,n and S n = n X i =1 X 2 i − n X 2 n = m X i =1 X 2 i + n X i = m +1 X 2 i − n  m X m + ( n − m ) X m,n n  2 = S m + m X 2 m + S m,n + ( n − m ) X 2 m,n − n  m X m + ( n − m ) X m,n n  2 = S m + S m,n + m ( n − m ) n ( X m − X m,n ) 2 = m X i =1 ( X i − X m ) 2 + n X i = m +1 ( X i − X m,n ) 2 + m ( n − m ) n ( X m − X m,n ) 2 = σ 2  Y + Z + V 2  . ✷ 14 Now we a re in a po sition to prov e the theorem. By Lemma 1 and some algebr aic o perations, we hav e U + r n − m m V = √ n ( X m − µ ) σ , ( X m − µ ) σ = 1 √ n U + r n − m m V ! , √ m ( X m − γ ) σ = r m n U + √ nθ + r n − m m V ! , √ n ( X n − γ ) σ = U + √ nθ . F or ℓ = 1, we have Pr { Re ject H 0 , n = n 1 | µ = θ σ + γ } = Pr { D ℓ = 2 | µ = θ σ + γ } ≤ Pr { T 1 > b 1 } = Pr  U + √ n 1 θ > b 1  = Φ( √ n 1 θ − b 1 ) for any θ ∈ ( −∞ , − ε ]. F or 1 < ℓ ≤ s , since a ℓ − 1 ≤ b ℓ − 1 , we have Pr { Reject H 0 , n = n ℓ | µ = θ σ + γ } < Pr { D ℓ − 1 = 0 , D ℓ = 2 | µ = θ σ + γ } = Pr { a ℓ − 1 < T ℓ − 1 ≤ b ℓ − 1 , T ℓ > b ℓ } = Pr { T ℓ − 1 ≤ b ℓ − 1 , T ℓ > b ℓ } − Pr { T ℓ − 1 ≤ a ℓ − 1 , T ℓ > b ℓ } = Pr  r n ℓ − 1 n ℓ ( U + √ n ℓ θ + k ℓ V ) ≤ b ℓ − 1 , U + √ n ℓ θ > b ℓ  − Pr  r n ℓ − 1 n ℓ ( U + √ n ℓ θ + k ℓ V ) ≤ a ℓ − 1 , U + √ n ℓ θ > b ℓ  = Pr  b ℓ − √ n ℓ θ ≤ U ≤ k ℓ V − √ n ℓ θ + r n ℓ n ℓ − 1 b ℓ − 1  − Pr  b ℓ − √ n ℓ θ ≤ U ≤ k ℓ V − √ n ℓ θ + r n ℓ n ℓ − 1 a ℓ − 1  for any θ ∈ ( −∞ , − ε ]. It follows that Pr { Accept H 0 , n = n ℓ | µ = θ σ + γ } = 1 − P s ℓ =1 Pr { Reject H 0 , n = n ℓ | µ = θ σ + γ } > 1 − ϕ ( θ , ζ , α, β ) for any θ ∈ ( −∞ , − ε ]. By sy mm etry , we have Pr { Accept H 0 , n = n ℓ | µ = θ σ + γ } < ϕ ( − θ , ζ , β , α ) fo r any θ ∈ [ ε , ∞ ). This completes the pro of of the theorem. B Pro of of Theorem 4 By Lemma 1 , we have b T ℓ − 1 √ n ℓ − 1 − 1 = r n ℓ − 1 n ℓ U + √ n ℓ θ + k ℓ V √ Y ℓ , b T ℓ √ n ℓ − 1 = U + √ n ℓ θ √ V 2 + Y ℓ + Z ℓ for 1 < ℓ ≤ s . W e shall fo cus on the ca s e of µ ≤ γ − εσ , s ince the case of µ ≤ γ + εσ ca n b e dealt with symmetrically . F or ℓ = 1 , we have Pr { Reject H 0 , n = n 1 } ≤ P 1 for a n y θ ∈ ( −∞ , − ε ]. F o r 1 < ℓ ≤ s , we hav e Pr { Reject H 0 , n = n ℓ | µ = θ σ + γ } < Pr { D ℓ − 1 = 0 , D ℓ = 2 | µ = θ σ + γ } = Pr ( a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 , b T ℓ √ n ℓ − 1 > d ℓ ) = Pr  a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 , U + √ n ℓ θ √ V 2 + Y ℓ + Z ℓ > d ℓ  for any θ ∈ ( −∞ , − ε ]. In the case of d ℓ ≥ 0, sinc e c ℓ − 1 ≤ d ℓ − 1 , it is eviden t that Pr  a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 , U + √ n ℓ θ √ V 2 + Y ℓ + Z ℓ > d ℓ  = Pr ( c ℓ − 1 s n ℓ Y ℓ n ℓ − 1 < U + √ n ℓ θ + k ℓ V ≤ d ℓ − 1 s n ℓ Y ℓ n ℓ − 1 , U + √ n ℓ θ √ V 2 + Y ℓ + Z ℓ > d ℓ ) = P ℓ 15 for any θ ∈ ( −∞ , − ε ]. In the case of d ℓ < 0, we hav e Pr  a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 , U + √ n ℓ θ √ V 2 + Y ℓ + Z ℓ > d ℓ  = Pr n a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 o − Pr  a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 , U + √ n ℓ θ √ V 2 + Y ℓ + Z ℓ ≤ d ℓ  = Pr n a ℓ − 1 < b T ℓ − 1 ≤ b ℓ − 1 o − Pr ( − d ℓ − 1 s n ℓ Y ℓ n ℓ − 1 < U − √ n ℓ θ + k ℓ V ≤ − c ℓ − 1 s n ℓ Y ℓ n ℓ − 1 , U − √ n ℓ θ √ V 2 + Y ℓ + Z ℓ ≥ − d ℓ ) = P ℓ for any θ ∈ ( −∞ , − ε ]. It follows that Pr { Accept H 0 , n = n ℓ | µ = θ σ + γ } = 1 − P s ℓ =1 Pr { Reject H 0 , n = n ℓ | µ = θ σ + γ } > 1 − P ( θ , ζ , α, β ) for any θ ∈ ( −∞ , − ε ]. By symmetry , we have Pr { Accept H 0 , n = n ℓ | µ = θ σ + γ } < P ( − θ , ζ , β , α ) for any θ ∈ [ ε, ∞ ). This completes the pro of of the theorem. C Pro of of Theorem 5 Without loss of a ny gener alit y , we ca n assume that A ⊆ [0 , 2 π ] for any conv ex do ma in D which contains the origin (0 , 0). Let A ∗ = [0 , 2 π ] \ A . Since Pr { ( U, V ) ∈ D } = 1 2 π R R ( u,v ) ∈ D exp  − u 2 + v 2 2  dudv , using po lar co ordinates, we have 2 π Pr { ( r, φ ) ∈ D } = Z A " Z B ( φ ) r =0 exp  − r 2 2  rdr # dφ + Z A ∗  Z ∞ r =0 exp  − r 2 2  rdr  dφ = Z A  1 − exp  − B 2 ( φ ) 2  dφ + Z A ∗ dφ = Z A ∪ A ∗ dφ − Z A exp  − B 2 ( φ ) 2  dφ = 2 π − Z A exp  − B 2 ( φ ) 2  dφ, from which the theorem immediately follows. D Pro of of Theorem 6 Without loss o f any g e ne r alit y , w e can a ssume that A i ⊆ A v for any conv ex domain D which do es not contain the orig in (0 , 0). Hence, we ca n write D = D ′ ∪ D ′′ with D ′ = { ( r, φ ) : B v ( φ ) ≤ r ≤ B i ( φ ) , φ ∈ A i } and D ′′ = { ( r, φ ) : r ≥ B v ( φ ) , φ ∈ A v \ A i } , where ( r, φ ) r epresen ts po lar co ordinates. 16 Since Pr { ( U, V ) ∈ D } = 1 2 π R R ( u,v ) ∈ D exp  − u 2 + v 2 2  dudv , using p olar co ordinates, we hav e 2 π Pr { ( r, φ ) ∈ D } = Z Z ( r,φ ) ∈ D exp  − r 2 2  rdr dφ = Z Z ( r,φ ) ∈ D ′ exp  − r 2 2  rdr dφ + Z Z ( r,φ ) ∈ D ′′ exp  − r 2 2  rdr dφ = Z A i " Z B i ( φ ) r = B v ( φ ) exp  − r 2 2  rdr # dφ + Z A v \ A i " Z ∞ r = B v ( φ ) exp  − r 2 2  rdr # dφ = Z A i  exp  − B 2 v ( φ ) 2  − exp  − B 2 i ( φ ) 2  dφ + Z A v \ A i exp  − B 2 v ( φ ) 2  dφ = Z A v exp  − B 2 v ( φ ) 2  dφ − Z A i exp  − B 2 i ( φ ) 2  dφ, from which the theorem immediately follows. E Pro of of Theorem 7 W e use a geometrical a pproac h fo r proving the theorem. Le t the horizon tal axis b e the u -axis a nd the vertical ax is be the v -axis. Note that line u = k v + g in tercepts line u = h at p oin t R =  h, h − g k  . Line u = h in tercepts the u -axis a t P = ( h, 0). Line u = k v + g intercepts the u - axis at Q = ( g , 0). The theorem can b e shown by considering 6 cases : (i) h ≤ g < 0; (ii) h ≤ 0 ≤ g ; (iii) 0 < h ≤ g ; (iv) 0 < g < h ; (v) g ≤ 0 ≤ h ; (vi) g < h < 0. In the ca se of h ≤ g < 0 , R is b elo w the u -axis , P is on the left side of Q , and O is o n the right side of Q . As can b e seen from Figure 1, the visible a nd invisible parts o f the b oundary can be express ed, re s pectively , as B v = n g √ 1+ k 2 cos( φ + φ k ) , φ  : π 2 − φ k < φ ≤ π + φ R o and B i = n h cos φ , φ  : π 2 < φ < π + φ R o . By Theorem 6 and making use o f a change of v ariable in the integration, we ha ve P r { h ≤ U ≤ k V + g } = R π + φ k + φ R π / 2 Ψ g,k ( φ ) dφ − R π + φ R π / 2 Ψ h ( φ ) dφ . 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 R P Q O Figure 1: Confi guratio n of h ≤ g < 0 In the case of h ≤ 0 ≤ g , R is b elo w the u -a xis, P is on the left side of Q , and O is lo cated in b et ween 17 P and Q . As can b e seen from Figure 2, the b o undary c a n b e expressed as B =  h cos φ , φ  : π 2 < φ ≤ π + φ R  [  g √ 1 + k 2 cos( φ + φ k ) , φ  : π + φ R ≤ φ < 2 π + π 2 − φ k  . By T he o rem 5 and making use of a change of v ariable in the in tegr ation, we hav e Pr { h ≤ U ≤ k V + g } = 1 − R π + φ R π / 2 Ψ h ( φ ) dφ − R 3 π / 2 φ k + φ R Ψ g,k ( φ ) dφ . 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 R P Q O Figure 2: Confi guratio n of h ≤ 0 ≤ g In the case o f 0 < h ≤ g , O is on the left side of P , P is on the left s ide of Q , and R is b elo w the u -axis. As can be seen fro m Figure 3 , the visible and in visible par ts o f the b oundary can be expressed as B v = n h cos φ , φ  : φ R ≤ φ < π 2 o and B i = n g √ 1+ k 2 cos( φ + φ k ) , φ  : φ R < φ < π 2 − φ k o resp ectiv ely . B y Theorem 6 and making use o f a change of v ariable in the integration, we ha ve P r { h ≤ U ≤ k V + g } = R π / 2 φ R Ψ h ( φ ) dφ − R π / 2 φ k + φ R Ψ g,k ( φ ) dφ . 0.6 0.8 1 1.2 1.4 1.6 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 R P Q O Figure 3: Confi guratio n of 0 < h ≤ g In the ca se o f 0 < g < h , R is ab o ve the u -a xis, Q is on the left side o f P , and O is on the left side of Q . As can b e seen from Fig ure 4, the vis ible and in visible parts of the b oundary can b e express ed as B v = n h cos φ , φ  : φ R ≤ φ < π 2 o and B i = n g √ 1+ k 2 cos( φ + φ k ) , φ  : φ R < φ < π 2 − φ k o resp ectiv ely . By Theorem 6 and making use o f a change of v ariable in the integration, we ha ve P r { h ≤ U ≤ k V + g } = R π / 2 φ R Ψ h ( φ ) dφ − R π / 2 φ k + φ R Ψ g,k ( φ ) dφ . 18 0.8 0.9 1 1.1 1.2 1.3 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R P Q O Figure 4: Confi guratio n of 0 < g < h In the case of g ≤ 0 ≤ h , R is ab o ve the u -a xis, Q is on the left side of P , and O is lo cated in b et ween Q and P . As can be seen from Figur e 5, the bo unda ry is completely visible a nd can b e expressed as B v = n h cos φ , φ  : φ R ≤ φ < π 2 o S n g √ 1+ k 2 cos( φ + φ k ) , φ  : π 2 − φ k < φ < φ R o . By Theo r em 6 and making use o f a change of v ariable in the integration, we have Pr { h ≤ U ≤ k V + g } = R π / 2 φ R Ψ h ( φ ) dφ − R π / 2 φ k + φ R Ψ g,k ( φ ) dφ . 0.8 0.9 1 1.1 1.2 1.3 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R P Q O Figure 5: Confi guratio n of g ≤ 0 ≤ h In the c ase of g < h < 0, R is ab ov e the u -axis, Q is on the left s ide of P , and P is on the left side of O . A s can be seen from Figure 6, the visible and invisible pa rts of the b oundary c an b e expr essed, resp ectively , as B v = n g √ 1+ k 2 cos( φ + φ k ) , φ  : π 2 − φ k < φ ≤ π + φ R o and B i = n h cos φ , φ  : π 2 < φ < π + φ R o . By Theorem 6 and making use o f a change of v ariable in the integration, we ha ve P r { h ≤ U ≤ k V + g } = R π + φ k + φ R π / 2 Ψ g,k ( φ ) dφ − R π + φ R π / 2 Ψ h ( φ ) dφ . This concludes the pro of of the theor em. F Pro of of Theorem 8 W e sha ll take a g eometrical a pproac h to pr o ve Theorem 8. Befo re pr oceeding to the details of pro of, we shall introduce some notations. F o r tw o p oin ts P 1 , P 2 on the u -axis, when P 1 is on the left side o f P 2 , we write P 1 < P 2 . Similarly , when P 1 is o n the right side of P 2 , we write P 1 > P 2 . W e use [ P 1 P 2 to denote the hyperb olic ar c with end p oin ts P 1 and P 2 . W e define so me sp ecial p oin ts O = (0 , 0 ) , A = ( u A , v A ) , B = 19 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R P Q O Figure 6: Confi guratio n of g < h < 0 ( u B , v B ) , C = ( ϑ + √ h, 0 ) , D = ( ϑ − √ h, 0 ) a nd M = ( ϑ, 0) that will b e frequently referr ed in the pro of. The domain D is shade d for all configura tio ns. The pro of of Theo rem 8 can b e accomplished by showing Lemmas 2 to 9 in the sequel. Lemma 2 F o r Pr { ( U, V ) ∈ D } t o b e non-zer o, ϑ, λ, g , h, k must satisfy one of t h e fol lowing four c onditio ns: (i) k 2 < λ, g > √ h, ∆ ≥ 0 ; (ii) k 2 < λ, 0 < g ≤ √ h, ∆ ≥ 0 ; (iii) k 2 > λ, g k > √ ∆ ; (iv) k 2 > λ, g k ≤ √ ∆ . Pro o f . Clearly , for Pr { ( U, V ) ∈ D } to b e non-zero , a necessary condition is that there exists a t leas t one tuple ( u, v ) satisfying equa tions √ λv 2 + h = u − ϑ = k v + g . By letting z = u − ϑ , w e can write the equations as z − k v = g and ( k 2 − λ ) z 2 + 2 λg z − λg 2 − k 2 h = 0 with z ≥ 0, wher e the discriminant for the quadratic equa tion of z is 4 k 2 ∆ . Therefore, the necess ary condition for P r { ( U, V ) ∈ D } to b e non-zer o can b e divided as tw o conditio ns : (I) ∆ ≥ 0 , g ≥ 0 , k 2 < λ ; (I I) k 2 > λ . If co ndit ion (I) holds, then the quadra t ic equa t ion of z hav e tw o no n-negativ e ro ots: z A = λg − k √ ∆ λ − k 2 , z B = λg + k √ ∆ λ − k 2 . Accordingly , there are tw o tuples ( u A , v A ) and ( u B , v B ) satisfying equations √ λv 2 + h = u − ϑ = k v + g with u A = z A + ϑ, v A = z A − g k , u B = z B + ϑ, v B = z B − g k . Noting that v A , v B are the ro ots for equation ( k 2 − λ ) v 2 + 2 kg v + g 2 − h = 0 with resp ect to v , conditio n (I) can b e divided into conditions (i) and (ii) of the lemma suc h that (i) implies √ h + ϑ < u A < u B , v A < 0 < v b and tha t (ii) implies √ h + ϑ < u A < u B , 0 ≤ v A < v B . If c ondition (I I) holds, then the qua dratic eq uation of z hav e tw o ro ots z A and z B of opp osite sig ns. Observing that z A > z B , we hav e z A = λg − k √ ∆ λ − k 2 > 0 > z B . Since v A = z A − g k = gk − √ ∆ λ − k 2 ≥ 0 if and only if g k ≤ √ ∆ , condition (II) can b e divided into conditions (iii) and (iv) of the lemma such that (iii) implies √ h + ϑ < u A , v A < 0 and that (iv) implies √ h + ϑ < u A , v A ≥ 0. This completes the pro of of the lemma. ✷ Now w e attempt to e xpress the right branch h yp erbo la, H R = { ( u, v ) : √ λv 2 + h ≤ u − ϑ } in p olar co ordinates ( r , φ ), which is related to the Ca rtesian c o ordinates by u = r cos φ, v = r sin φ . Note that the po lar co ordinates, ( r, φ ), of any p oin t of H R m ust s a tisfy the equation ( r cos φ − ϑ ) 2 − λ ( r sin φ ) 2 = h with resp ect to r ≥ 0, which can be written as (cos 2 φ − λ sin 2 φ ) r 2 − 2 ϑ cos φ r + η = 0 with η = ϑ 2 − h . F or φ such that ( h − λη ) cos 2 φ + λη ≥ 0, we have tw o rea l r oots r ⋄ ( φ ) = η ϑ co s φ + p ( h − λη ) cos 2 φ + λη , r ⋆ ( φ ) = η ϑ co s φ − p ( h − λη ) cos 2 φ + λη = − r ⋄ ( φ + π ) . 20 These are p o ssible expres sions for the relationship of p olar co ordinates r a nd φ of the rig h t branch hyperb ola H R . How ever, it is not clear which express ion should b e taken. The sp ecific express io n and the visibility of H R are to b e determined in the sequel. Lemma 3 If O ≤ M , t h en the right hyp erb ola H R is visible and c an b e expr esse d as B v = { ( r ⋆ , φ ) : | φ | < φ λ } . Pro o f . T o show the lemma, we first need to show that r ⋆ > 0 > r ⋄ for 1 − λ tan 2 φ > 0 and D < O ≤ M . F or D < O ≤ M , w e hav e ϑ − √ h < 0 ≤ ϑ ⇒ η = ϑ 2 − h < 0. Thus, r ⋄ < 0 as a result of 1 − λ tan 2 φ > 0 ⇐ ⇒ | φ | < φ λ < π 2 . On the other hand, r ⋆ = − η − ϑ cos φ + √ ( h − λη ) c os 2 φ + λη . Obser v ing that ( ϑ co s φ ) 2 −  ( h − λη ) cos 2 φ + λη  = η cos 2 φ (1 − λ tan 2 φ ) < 0 as a consequence of η < 0 and 1 − λ tan 2 φ > 0, we hav e r ⋆ > 0. Next, we nee d to show that r ⋆ > r ⋄ ≥ 0 for 1 − λ ta n 2 φ > 0 and O ≤ D . F or O ≤ D , we have ϑ − √ h ≥ 0 ⇒ η = ϑ 2 − h ≥ 0 . Thus, r ⋄ ≥ 0. On the other hand, obs erving that ( ϑ cos φ ) 2 −  ( h − λη ) cos 2 φ + λη  = η c o s 2 φ (1 − λ tan 2 φ ) ≥ 0 as a consequence of η ≥ 0 and 1 − λ tan 2 φ > 0, we hav e r ⋆ ≥ 0. Since the denominator of r ⋆ is s m aller than that of r ⋄ , we hav e r ⋆ > r ⋄ ≥ 0. This completes the pro of o f the lemma. ✷ Lemma 4 If M < O ≤ C , then B v = { ( r ⋆ , φ ) : | φ | ≤ φ m } and B i = { ( r ⋄ , φ ) : φ λ < | φ | < φ m } . Pro o f . Since M < O ≤ C , we have ϑ < 0 ≤ ϑ + √ h ⇒ η = ϑ 2 − h ≤ 0. Hence, ( h − λη ) cos 2 φ + λη = h c o s 2 φ + λη s in 2 φ = − λη cos 2 φ  − h λη − tan 2 φ  , whic h implies that ( h − λη ) cos 2 φ + λη is nonneg ativ e for | φ | ≤ φ m and nega tiv e for φ m < | φ | < π 2 . T o show the lemma, w e first need to show that r ⋆ ≥ 0 ≥ r ⋄ if 1 − λ tan 2 φ > 0 . Since η ≤ 0 and ϑ < 0, w e hav e r ⋆ = − η − ϑ cos φ + √ ( h − λη ) c os 2 φ + λη ≥ 0 in view o f 1 − λ tan 2 φ > 0 ⇐ ⇒ | φ | < φ λ < π 2 . On the other hand, observing that r ⋄ = − η − ϑ cos φ − √ ( h − λη ) c os 2 φ + λη and ( ϑ cos φ ) 2 −  ( h − λη ) cos 2 φ + λη  = η c o s 2 φ (1 − λ tan 2 φ ) < 0 as a consequence of η ≤ 0 a nd 1 − λ tan 2 φ > 0, w e have r ⋄ ≤ 0. Next, w e need to show that 0 ≤ r ⋆ ≤ r ⋄ if φ λ < | φ | < φ m . By the same argument as ab o ve, we hav e r ⋆ ≥ 0 b ecause | φ | < π 2 . It r emains to show r ⋆ < r ⋄ . Note that ( ϑ cos φ ) 2 −  ( h − λη ) cos 2 φ + λη  = η c o s 2 φ (1 − λ tan 2 φ ) is p ositiv e a s a r esult of η ≤ 0 and φ λ < | φ | < φ m ⇒ 1 − λ tan 2 φ < 0. Since ϑ cos φ < 0 as a cons e q uence of ϑ < 0 a nd φ λ < | φ | < φ m , it follows tha t − ϑ cos φ − p ( h − λη ) cos 2 φ + λη > 0 and thus r ⋄ ≥ 0. Since the numerators o f r ⋆ and r ⋄ are equal to the same non-negative num b er and the denominator of r ⋄ is a p ositiv e num be r smaller than that o f r ⋆ , we have r ⋄ ≥ r ⋆ ≥ 0. This completes the pro of of the lemma. ✷ As can b e seen from the pro of of Lemma 4, the b oundary is divided into visible part B v and invisible part B i by the upp er cr itical point  η ϑ cos φ m , φ m  and the lower critica l p oin t  η ϑ cos φ m , − φ m  . The visible part is o n the left side o f the critical line, which is referred to as the v ertica l line connecting the low er and upper critical p oint s. The invisible part is on the right side of the c ritical line. Lemma 5 If O > C , then the right hyp erb ola H R c an b e r epr esente d as { ( r ⋄ , φ ) : φ λ < φ < 2 π − φ λ } . 21 Pro o f . T o show the lemma, w e first need to show that r ⋆ < 0 < r ⋄ for φ λ < φ < π − φ λ and π + φ λ < φ < 2 φ − φ λ . Since O > C , we hav e ϑ < − √ h a nd thus η = ϑ 2 − h > 0. Since 1 − λ tan 2 φ < 0 for φ λ < φ < π − φ λ and π + φ λ < φ < 2 π − φ λ , w e hav e | ϑ cos φ | − p ( h − λη ) cos 2 φ + λη < 0, lea din g to r ⋆ < 0. On the other hand, ϑ cos φ + p ( h − λη ) cos 2 φ + λη > −| ϑ cos φ | + p ( h − λη ) cos 2 φ + λη > 0, leading to r ⋄ > 0. Next, we need to s ho w that r ⋆ > r ⋄ > 0 for π − φ λ < φ < π + φ λ . F or π − φ λ < φ < π + φ λ , we have 1 − λ ta n 2 φ > 0. Since η > 0 and ϑ < 0, it must be true that ϑ cos φ > 0 and r ⋄ > 0. As a consequence of ϑ cos φ > 0 and 1 − λ tan 2 φ > 0, w e hav e that the denominator of r ⋆ is p ositiv e. Reca lling that the nu mera t or of r ⋆ is a p ositiv e num b er η , we hav e r ⋆ > 0. Since the num era tors of r ⋆ and r ⋄ are eq ual to the same po sitiv e n umber η and the denominato r of r ⋆ is a p ositiv e n umber smaller than that of r ⋄ , we hav e r ⋆ > r ⋄ > 0. This completes the pr oof of the lemma . ✷ Lemma 6 If k 2 < λ, g > √ h and ∆ ≥ 0 , then Pr { ( U, V ) ∈ D } = I np . Pro o f . As consequence of k 2 < λ, g > √ h and ∆ ≥ 0, we have √ h + ϑ < u A < u B , v A < 0 < v B . The tangent line at A in tercepts the u -axis at P = ( u P , 0 ) with u P satisfying √ ( u A − ϑ ) 2 − h √ λ ( u A − u P ) = u A − ϑ √ λ √ ( u A − ϑ ) 2 − h , from which we obtain u P = ϑ + h u A − ϑ > ϑ . Similarly , the tangen t line at B intercepts the u -axis at Q = ( u Q , 0 ) with u Q = ϑ + h u B − ϑ < u P < u C . Line AB int erce pt s the u -axis at R = ( u R , 0 ) with u R = g + ϑ . Clea rly , D < M < Q < P < C . The lemma can be sho wn by investigating five ca ses as follows. In the case of ϑ + h u B − ϑ ≥ 0, we hav e O ≤ Q . The situa tio n is s ho wn in Figure 7. If O ≤ M , then, b y Lemma 2, the right branc h h yp erbola H R is completely visible. Accordingly , the visible and invisible parts of the b oundary o f D can b e expr essed, res pectively , as B v = { ( r ⋆ , φ ) : − φ A ≤ φ ≤ φ B } and B i = { ( r l , φ ) : − φ A < φ < φ B } , where r l ( φ ) = g + ϑ √ 1+ k 2 cos( φ + φ k ) . Now consider the situatio n that M < O ≤ Q . Since the domain, H = { ( u, v ) : √ λv 2 + h ≤ u − ϑ } , corr e s ponding to the region included by the rig ht branch hyperb ola H R , is a c o n vex set, we have that H is divided by line O A into tw o sub- do mains of which one is below line OA and ab o ve the tang en t line P A , and the o ther is ab o ve both line O A and the tangent line P A . As can b e s e e n from Figure 7, the lower critical p oint  η ϑ cos φ m , − φ m  m ust b e b elo w line OA . It fo llo ws from Lemma 3 tha t arc d AC is visible. B y a simila r argument, we hav e that arc d C B is visible. There fo re, b y Lemma 3, the visible and invisible par ts o f the b oundary of D can b e express e d, resp ectiv ely , as B v and B i like the ca se of O ≤ M . Applying Theorem 6 yields Pr { ( U, V ) ∈ D } = I np , 1 . In the case of ϑ + h u B − ϑ < 0 ≤ ϑ + h u A − ϑ , w e have Q < O ≤ P . The situation is shown in Figure 8. Recall that arc d AC is visible as in the preceding case of O ≤ Q . Since the doma in H is a convex set, we have that H is divided by line OB in to t wo sub-domains o f which one is abov e line O B and b elo w the tangent line QB , and the other is below b oth line O B and the ta ng en t line Q B . As can b e se e n fr om Figure 8, the upp er critical p oint  η ϑ c os φ m , φ m  m ust b e ab o ve line OB . Hence, applying Lemma 3, the visible and invisible pa rts of the b oundary of D ca n be expr essed, resp ectiv ely , as B v = { ( r ⋆ , φ ) : − φ A ≤ φ ≤ φ m } and B i = { ( r l , φ ) : − φ A < φ < φ B } ∪ { ( r ⋄ , φ ) : φ B ≤ φ < φ m } . Applying Theorem 6 yields Pr { ( U, V ) ∈ D } = I np , 2 . In the case of ϑ + h u A − ϑ < 0 ≤ ϑ + √ h , w e have P < O ≤ C . The s it uation is shown in Figure 9. By a similar metho d a s that of the case of Q < O ≤ P , we hav e that the upp er cr itical point must be ab o ve line OB and in ar c d C B and that the low er critical p oin t m ust b e be low line OA and in ar c d AC . 22 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 A B C P Q R O Figure 7: Confi guratio n of O ≤ Q 0.2 0.3 0.4 0.5 0.6 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 A B C P Q R O Figure 8: Confi guratio n of Q < O ≤ P 23 Hence, b y Lemma 3 , the vis ible and invisible parts of the bo undary of D can b e expressed, resp ectiv ely , as B v = { ( r ⋆ , φ ) : − φ m ≤ φ ≤ φ m } and B i = { ( r l , φ ) : − φ A ≤ φ ≤ φ B } ∪ { ( r ⋄ , φ ) : − φ m < φ < − φ A } ∪ { ( r ⋄ , φ ) : φ B < φ < φ m } . By vir tue of Theor em 6, we hav e Pr { ( U, V ) ∈ D } = I np , 3 . 0.2 0.3 0.4 0.5 0.6 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 A B C P Q R O Figure 9: Confi guratio n of P < O ≤ C In the ca se of ϑ + √ h < 0 ≤ g + ϑ , we hav e C < O ≤ R . The situation is shown in Figure 10. By Lemma 4, the b oundary of D can b e express ed as B = { ( r l , φ ) : − φ A ≤ φ ≤ φ B } ∪ { ( r ⋄ , φ ) : φ B < φ < 2 π − φ A } . By virtue of Theorem 5, we hav e Pr { ( U, V ) ∈ D } = I np , 4 . 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 A B C R O Figure 10: Confi gu r a tion of C < O ≤ R In the case of g + ϑ < 0, we hav e O > R . The situation is shown in Figure 11. By L e m ma 4, the visible and invisible parts of the b oundary of D can be ex pressed, resp ectiv ely , a s B v = { ( r l , φ ) : φ B ≤ φ ≤ 2 π − φ A } and B i = { ( r ⋄ , φ ) : φ B < φ < 2 π − φ A } . By virtue of Theo rem 6, we hav e Pr { ( U, V ) ∈ D } = I np , 5 . ✷ Lemma 7 If k 2 < λ, 0 ≤ g ≤ √ h , then Pr { ( U, V ) ∈ D } = I pp . Pro o f . As a cons e q uence of k 2 < λ, 0 ≤ g ≤ √ h and ∆ ≥ 0, we have √ h + ϑ < u A < u B , 0 ≤ v A < v B . Clearly , D < M < Q < R < P < C . The le m ma ca n b e shown by inv estigating several ca ses as follows. In the case of ϑ + h u B − ϑ ≥ 0, we have O ≤ Q . The situation is shown in Figure 1 2 . By Lemmas 2 and 3 , and a similar argument as that of the first ca se o f Lemma 6, the visible and invisible pa rts of the b oundary of D can b e deter mined, r espectively , as B v = { ( r ⋆ , φ ) : φ A ≤ φ ≤ φ B } and B i = { ( r l , φ ) : φ A < φ < φ B } . By virtue of Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I pp , 1 . 24 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 A B C R O Figure 11: Configuration of O > R −0.2 0 0.2 0.4 0.6 0.8 1 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 A B C Q O Figure 12: Confi gu r a tion of O ≤ Q 25 In the case of ϑ + h u B − ϑ < 0 ≤ g + ϑ , we ha ve Q < O ≤ R . The situation is shown in Figure 13. By Lemma 3 and a similar ar g umen t as that o f the second cas e o f Lemma 6, the visible and in visible pa rts of the b o undary of D ca n b e determined, r espectively , as B v = { ( r ⋆ , φ ) : φ A ≤ φ ≤ φ m } and B i = { ( r l , φ ) : φ A < φ ≤ φ B } ∪ { ( r ⋄ , φ ) : φ B < φ < φ m } . By virtue of Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I pp , 2 . 1.66 1.68 1.7 1.72 1.74 1.76 1.78 1.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 O A B Q R Figure 13: Confi gu r a tion of Q < O ≤ R In the case o f g + ϑ < 0 ≤ ϑ + h u A − ϑ , we hav e R < O ≤ P . The situatio n is shown in Figure 14. Obser ving that the upp er c r itical p o in t must b e above OA and thus must b e in arc c AS , by Lemma 3, we ha ve that the visible and in visible parts of the boundary of D can b e e x pressed, respe ctiv ely , as B v = { ( r ⋆ , φ ) : φ A ≤ φ ≤ φ m } ∪ { ( r l , φ ) : φ B ≤ φ < φ A } and B i = { ( r ⋄ , φ ) : φ B < φ < φ m } . By virtue of Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I pp , 2 . 1.72 1.74 1.76 1.78 1.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 O A B P R S Figure 14: Confi gu r a tion of R < O ≤ P In the cas e of ϑ + h u A − ϑ < 0 ≤ ϑ + √ h , w e hav e P < O ≤ C . The situa tio n is s hown in Figure 15. Observ ing that the upper critical p oin t must be in the part of ar c d C A that is ab o ve O A , by Lemma 3, we have that the visible and invisible par ts of the b o undary o f D can b e determined, respectively , as B v = { ( r l , φ ) : φ B ≤ φ ≤ φ A } and B i = { ( r ⋄ , φ ) : φ B < φ < φ A } . By virtue o f Theorem 6, we have Pr { ( U, V ) ∈ D } = I pp , 3 . In the case of ϑ + √ h < 0, we hav e O > C . The s itua t ion is shown in Figure 16. By Lemma 4, the visible and invisible parts o f the b oundary of D can b e determined, resp ectiv ely , a s B v = { ( r l , φ ) : φ B ≤ φ ≤ φ A } 26 1.73 1.74 1.75 1.76 1.77 0 0.05 0.1 0.15 0.2 0.25 0.3 O A B C P Figure 15: Configuration of P < O ≤ C and B i = { ( r ⋄ , φ ) : φ B < φ < φ A } . By virtue of Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I pp , 3 . 1.73 1.735 1.74 1.745 1.75 1.755 1.76 1.765 1.77 1.775 0 0.05 0.1 0.15 0.2 0.25 0.3 O A B C Figure 16: Configuration of O > C ✷ Lemma 8 If k 2 > λ and g k ≤ √ ∆ , then Pr { ( U, V ) ∈ D } = I p . Pro o f . Since k 2 > λ and g k ≤ √ ∆ , we hav e v A ≥ 0. Consider straight line AB describ ed b y equa tio n u − ϑ = k v + g , passing thro ugh A = ( u A , v A ). Supp ose that the tangent line at A int erce pts the u -a xis at P . Draw a line, denoted b y AF , fr om A with angle φ λ . Extend F A to int erce pt the u -ax is a t G . Then, u A − u G = √ λv A , leading to u G = u A − √ λ v A . The lemma can b e shown by considering several cases as follows. In the case of ϑ ≥ 0 a nd v A u A ≥ 1 k , we have that O ≤ M and AB is b elow OA . The situation is shown in Figure 17. Since O ≤ M , b y Lemma 2, the boundar y of D is completely visible and ca n b e expressed as B v = { ( r ⋆ , φ ) : φ A ≤ φ < φ λ } ∪  ( r l , φ ) : π 2 − φ k < φ < φ A  . By virtue o f Theor em 6, we hav e P r { ( U, V ) ∈ D } = I p , 1 . In the case of ϑ ≥ 0 a nd v A u A < 1 k , we have that O ≤ M and AB is a bov e OA . The situation is shown in Figure 18. By Lemma 2 , the visible and invisible par ts of the bo undary of D can b e determined, resp ectiv ely , as B v = { ( r ⋆ , φ ) : φ A ≤ φ < φ λ } and B i =  ( r l , φ ) : φ A < φ < π 2 − φ k  . By virtue of Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I p , 1 . 27 −0.2 0 0.2 0.4 0.6 0.8 1 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 A B C M O Figure 17: Confi gu r a tion for O ≤ M and AB b elo w O A −0.2 0 0.2 0.4 0.6 0.8 1 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 A B C M O Figure 18: Confi gu r a tion for O ≤ M and AB ab o v e O A 28 In the case of ϑ < 0 ≤ u A − √ λ v A and v A u A ≥ 1 k , we hav e that M < O ≤ G and AB is be low OA . The situation is shown in Figur e 1 9 . Ma king use o f Lemma 3 and the obser v ation that the upp er c r itical p oin t m ust b e ab o ve O A , the vis ible and invisible parts of the b oundary of D can be determined, res pectively , as B v = { ( r ⋆ , φ ) : φ A ≤ φ ≤ φ m } ∪  ( r l , φ ) : π 2 − φ k < φ < φ A  and B i = { ( r ⋄ , φ ) : φ λ < φ < φ m } . By virtue of Theo rem 6, we hav e Pr { ( U, V ) ∈ D } = I p , 2 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 A B C M O F G Figure 19: Configuration for M < O ≤ G and AB belo w O A In the case of ϑ < 0 ≤ u A − √ λ v A and v A u A < 1 k , we hav e that M < O ≤ G and AB is ab o ve O A . The situation is shown in Figur e 20. Since the upp er cr it ical po in t must b e ab o ve OA , by Lemma 3, the visible and invisible parts o f the b o undary of D can be determined, resp ectively , as B v = { ( r ⋆ , φ ) : φ A ≤ φ ≤ φ m } and B i = { ( r ⋄ , φ ) : φ λ < φ < φ m } ∪  ( r l , φ ) : φ A < φ < π 2 − φ k  . By virtue of Theorem 6, we have Pr { ( U, V ) ∈ D } = I p , 2 . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 A B C M O F G Figure 20: Configuration for M < O ≤ G and AB a b o v e O A In the case of u A − √ λ v A < 0 ≤ ϑ + h u A − ϑ , we have that G < O ≤ P . The s itu ation is shown in Figur e 21. Since k 2 > λ , the slo pe of line AB is s maller than that of line AF . As a consequence of G < O , the slop e of line AF must b e smaller than that of line OA . Hence, the s lo pe of line AB m ust b e smaller than that of line O A . Making use o f this observ ation and noting that the upp er critical p oint must be ab ov e OA , we c a n apply Lemma 3 to determine the v is ible and invisible pa r ts o f the bo undary of D , res pectively , as B v = { ( r ⋆ , φ ) : φ A ≤ φ ≤ φ m } ∪  ( r l , φ ) : π 2 − φ k < φ < φ A  and B i = { ( r ⋄ , φ ) : φ λ < φ < φ m } . By virtue of Theo rem 6, we hav e Pr { ( U, V ) ∈ D } = I p , 2 . In the cas e of ϑ + h u A − ϑ < 0 ≤ ϑ + √ h , w e hav e P < O ≤ C . The situa tio n is s hown in Figure 22. Observing that the upper critical p oin t must be in the part of ar c d C A tha t is ab o ve line O A , by 29 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.01 0.02 0.03 0.04 0.05 A B C P O F G S Figure 21: Configuration of G < O ≤ P Lemma 3, the visible and invisible pa rts of the bo un dar y of D can b e determined, resp ectiv ely , as B v =  ( r l , φ ) : π 2 − φ k < φ ≤ φ A  and B i = { ( r ⋄ , φ ) : φ λ < φ < φ A } . By virtue o f Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I p , 3 . 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.01 0.02 0.03 0.04 0.05 A B C P O Figure 22: Configuration of P < O ≤ C In the case of ϑ + √ h < 0, we hav e C < O . The situation is shown in Fig ure 2 3 . By Lemma 4, the visible and invisible parts of the b oundary of D can b e expr e s sed, resp ectiv ely , a s B v =  ( r l , φ ) : π 2 − φ k < φ ≤ φ A  and B i = { ( r ⋄ , φ ) : φ λ < φ < φ A } . By vir tue of Theorem 6, we have Pr { ( U, V ) ∈ D } = I p , 3 . ✷ Lemma 9 If k 2 > λ and g k > √ ∆ , then Pr { ( U, V ) ∈ D } = I n . Pro o f . F or k 2 > λ and g k > √ ∆ . Then, v A < 0. The lemma c an b e shown by inv estigating five cas es as follows. In the case of ϑ ≥ 0, w e have O ≤ M . The situation is sho wn in Figure 24. Since O ≤ M , by Lemma 2, the rig h t branch hyperb ola H R is completely visible. There f or e , the visible and invisible parts of the bounda ry o f D can b e determined, r espectively , as B v = { ( r ⋆ , φ ) : − φ A ≤ φ < φ λ } and B i =  ( r l , φ ) : − φ A < φ < π 2 − φ k  . By vir tue o f Theo r em 6, we hav e P r { ( U, V ) ∈ D } = I n , 1 . In the ca se of ϑ < 0 ≤ ϑ + h u A − ϑ , we hav e M < O ≤ P . The situation is shown in Figure 25. Observing that the low er critical p oin t must be b elo w line O A , by L e m ma 3 , w e have that a rc d AC must be visible and that the visible and invisible pa rts o f the b oundary of D can b e determined, resp ectiv ely , as 30 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.01 0.02 0.03 0.04 0.05 A B C O Figure 23: Configuration of C < O −0.2 0 0.2 0.4 0.6 0.8 1 −0.05 0 0.05 0.1 A B C M O Figure 24: Confi gu r a tion of O ≤ M 31 B v = { ( r ⋆ , φ ) : − φ A ≤ φ ≤ φ m } a nd B i =  ( r l , φ ) : − φ A < φ < π 2 − φ k  ∪ { ( r ⋄ , φ ) : φ λ < φ < φ m } . By virtue of Theo rem 6, we hav e Pr { ( U, V ) ∈ D } = I n , 2 . 0 0.1 0.2 0.3 0.4 0.5 0.6 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 A B C M P O Figure 25: Confi gu r a tion of M < O ≤ P In the case of ϑ + h u A − ϑ < 0 ≤ ϑ + √ h , we have P < O ≤ C . The situation is shown in Figure 2 6. Observing that the low er critical p oin t must b e in the par t of ar c d AC that is b elo w line O A , by Lemma 3, we have that the visible and invisible pa rts of the b oundary o f D can b e deter min ed, resp ectiv ely , as B v = { ( r ⋆ , φ ) : − φ m ≤ φ ≤ φ m } and B i =  ( r l , φ ) : − φ A < φ < π 2 − φ k  ∪ { ( r ⋄ , φ ) : φ λ < φ < φ m } ∪ { ( r ⋄ , φ ) : − φ m < φ ≤ − φ A } . By vir tue of Theorem 6, we have Pr { ( U, V ) ∈ D } = I n , 3 . 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 A B C P O Figure 26: Confi gu r a tion of P < O ≤ C In the ca se of ϑ + √ h < 0 ≤ ϑ + g , we hav e C < O ≤ R . The situation is shown in Figure 27. By Lemma 4, the b oundary of D can b e expressed as B =  ( r l , φ ) : − φ A ≤ φ ≤ π 2 − φ k  ∪ { ( r ⋄ , φ ) : φ λ < φ < 2 π − φ A } . By virtue of Theorem 5, we hav e Pr { ( U, V ) ∈ D } = I n , 4 . In the case of ϑ + g < 0, we hav e R < O . The s it uation is shown in Figur e 2 8 . By L e m ma 4, the visible and invisible par ts of the b oundary o f D can b e determined, resp ectiv ely , as B v = { ( r l , φ ) : π 2 − φ k < φ ≤ 2 π − φ A } and B i = { ( r ⋄ , φ ) : φ λ < φ < 2 π − φ A } . By vir t ue o f Theorem 6, we hav e Pr { ( U, V ) ∈ D } = I n , 5 . This completes the pro of o f the theor em. ✷ 32 0.3 0.35 0.4 0.45 −0.02 −0.01 0 0.01 0.02 0.03 A B C O Figure 27: Confi gu r a tion of C < O ≤ R 0.3 0.35 0.4 0.45 0.5 0.55 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 A B C O R Figure 28: Configuration of R < O 33 References [1] X. Chen, “ A new framework of multistage estimation,” a rXiv:0809.124 1 [ma th .ST], Septem b er 2008 . [2] X. Chen, “A new framework of m ultistage hypothes is tests,” arXiv:080 9.3170 [math.ST], Septem b er 2008. [3] B. K. Gho sh a nd P . K. Sen (eds.), Handb o ok of S e quent ial Analysis , Dekker, New Y ork, 19 91. [4] A. W a ld, Se quential Analysis , Wiley , New Y ork, 1 9 47. 34