Improved Sequential Stopping Rule for Monte Carlo Simulation

Impro v ed Sequen tial Stopping Rule for Mon te Carlo Sim ulation ∗ Luis Mendo and Jos ´ e M. Hernando † Septem b er 2008 Abstract This pap er presen ts an improv ed result on the negativ e-binomial Monte Carlo technique analyzed in a prev ious pap er 1 for t h e estimation of an un- known probabilit y p . Sp ecifically , the confidence lev el associated to a relative interv al [ p /µ 2 , pµ 1 ], with µ 1 , µ 2 > 1, is p roved to exceed its asymptotic val ue for a broader range of in terv als than that giv en in the referred pap er, and for any v alue of p . This ex tends the applicabili ty of the estima tor, relaxing the conditions that gu aran tee a giv en confidence lev el. Keywor ds: Simulati on, Monte Carlo metho ds, sequentia l stopping rule. 1 In tro duction Monte Ca rlo (MC) metho ds a re widely used for estimating an unknown parameter by means of rep eated trials or rea lizations of a random exp eriment. An imp ortant particular ca s e is that in which the para meter to b e estimated is the pro bability p of a certain even t H , a nd realizations are indep endent . In this setting, the technique of negative-binomial MC (NBMC) [1] can b e us ed. This technique applies a sequential stopping rule, which consis ts in carrying out as ma ny realizations a s necessary to obtain a giv en n umber N of o ccurr ences o f H . Based on this rule, an estimator is int ro duced in [1], and it is shown to have a num b er of interesting pro pe rties, in the form o f resp ective b ounds for its bias, r elative pr ecision, a nd co nfidence level for a relative interv al [ p/ µ 2 , pµ 1 ]; µ 1 , µ 2 > 1. Spec ific a lly , reg arding the latter, it is derived in [1] that the confidence level c = Pr[ p/µ 2 ≤ ˆ p ≤ pµ 1 ] has an asymptotic v a lue ¯ c as p → 0, g iven by 2 ¯ c = γ ( N , ( N − 1) µ 2 ) − γ  N , N − 1 µ 1  . (1) F urthermore, the co nfidence level c is assured to exc e ed ¯ c fo r µ 2 ≥ N + √ N N − 1 , µ 1 ≥ N − 1 N − q 3 2 N , (2) ∗ Pa p er accepted in IEEE T r ansactions on Communic ations. † E.T.S. Ingenieros de T elecom unicaci´ on, Polytec hnic Universit y of M adr i d, 28040 Madrid, Spain. E-mail: lmendo@grc.ssr.upm.es. 1 L. M endo a nd J. M. Hernando, “A si mple sequential stopping rule for Monte Carlo sim ulation,” IEEE T r ans. Commun. , vol. 54, no. 2, pp. 231–241, F eb. 2006. 2 γ ( r, x ) denotes the incomplete gamma function, defined as γ ( r, x ) = 1 / Γ( r ) · R x 0 e − t t r − 1 dt . 1 provided that 3 p < N − 1  7 2 N − 1  µ 1 . (3) In this pap e r, the sufficient conditions that assur e a confidence level c > ¯ c for the NBMC estimator are rela xed in tw o wa ys: • The restriction on p given by (3) is eliminated, i.e. p can be an unconstrained v a lue b etw een 0 and 1. • The condition for µ 1 given by (2) is weakened, while maintaining the condition for µ 2 . Thu s µ 1 can b e further decrea s ed while having the same guar a nteed confidence le vel g iven by (1). The re s ult is presented in Section 2, and conclusio ns are g iven in Section 3. 2 Result Consider a rando m exper iment, and a n even t H ass o ciated to that e x per iment (m or e generally , there may b e a set of even ts asso c iated to the exp eriment, o ne o f whic h is o f interest). T he pro bability p of even t H is to b e es tima ted from indep endent realizations of the exp eriment, using the metho d describ ed in [1]. Sp ecifically , given 4 N ∈ N , with N ≥ 3, realizations are ca rried out until N o ccurrences of H are obtained. The n umber of realizations is th us a negativ e-binomia l random v a riable 5 n , from whic h p is estimated as [1 ] ˆ p = N − 1 n . (4) F or µ 1 , µ 2 > 1 given, c onsider the interv al [ p/µ 2 , pµ 1 ], and its ass o ciated con- fidence c = P r[ p/µ 2 ≤ ˆ p ≤ pµ 1 ]. As shown in [1], c tends to ¯ c g iven by (1) as p → 0. Prop ositio n 1. F or any p ∈ (0 , 1 ) , with ˆ p given by (4 ) , the lower b ound c > ¯ c holds if µ 2 ≥ N + √ N N − 1 , µ 1 ≥ N − 1 N − 1 2 − q N − 1 2 . (5) Pr o of. Cons ider N ≥ 3 , µ 1 , µ 2 > 1, and p ∈ (0 , 1). Let us define n 1 =  N − 1 pµ 1  (6) n 2 =  ( N − 1) µ 2 p  . (7) The co nfidence c is given by 1 − c 1 − c 2 with 6 c 1 = Pr[ n ≤ n 1 − 1] = p N ( N − 1)! n 1 − 1 X n = N ( n − 1) ( N − 1) (1 − p ) n − N , (8) 3 ⌊·⌋ and ⌈·⌉ resp ectiv ely denote rounding to the nearest integer tow ards −∞ and tow ards ∞ . 4 N denotes the set of natural num b ers, { 1, 2, . . . } . 5 Random v ariables are denoted i n b oldface throughout the pap er. 6 The following notation is used: k ( i ) = k ( k − 1) · · · ( k − i + 1), k (0) = 1. 2 c 2 = Pr[ n ≥ n 2 + 1] = p N ( N − 1)! ∞ X n = n 2 +1 ( n − 1) ( N − 1) (1 − p ) n − N . (9) Let ¯ c 1 and ¯ c 2 be re sp e c tively defined a s lim p → 0 c 1 and lim p → 0 c 2 . F rom [1, app endix C], ¯ c 1 = γ ( N , ( N − 1) /µ 1 ) and ¯ c 2 = 1 − γ ( N , ( N − 1) µ 2 ). W e will show that ¯ c 1 > c 1 and ¯ c 2 > c 2 for µ 1 , µ 2 as in (5). This will establish 7 that c > ¯ c . The inequality ¯ c 2 > c 2 is equiv alent to P r[ n ≤ n 2 ] > lim p → 0 Pr[ n ≤ n 2 ], whic h is established in [1, app endix C] for µ 2 as in (5) and n 2 given b y (7 ). In or der to show that ¯ c 1 > c 1 , we fir st note that ¯ c 1 = γ  N , N − 1 µ 1  = p N ( N − 1)! Z N − 1 pµ 1 0 t N − 1 e − pt dt > p N ( N − 1)! Z n 1 − 1 N − 1 t N − 1 e − pt dt. (10) Lemma 1 given in the Appendix implies that the right-hand side o f (10) will b e greater than or equa l to that of (8) if n 1 − 1 ≤ ( N − 1 / 2 − p N − 1 / 2) /p + 1 / 2. F rom (6), n 1 is upper bounded by ( N − 1) / ( pµ 1 ) + 1. Ther e fore, in order to assure that ¯ c 1 > c 1 it is sufficien t tha t N − 1 pµ 1 ≤ N − 1 2 − q N − 1 2 p + 1 2 , (11) or equiv alently µ 1 ≥ N − 1 N − 1 2 − q N − 1 2 + p 2 . (12) Since p > 0, (12) holds for µ 1 as in (5). This result remov es some of the restr ictions that ar e us e d in [1] to as sure that c > ¯ c . Sp ecifically , p can take any v alue, and the minim um requir e d v alue for µ 1 is low er. F or the pa rticular case that µ 1 = µ 2 = 1 + m , where m > 0 is a r e la tive er ror margin, it is easily seen that the limiting condition in (5) is that for µ 2 , i.e. m ≥ √ N + 1 N − 1 . (13) The das hed curves in Fig. 1 de pict the guaranteed co nfidence ¯ c (giv en by (1)) as a function of N and m , for m within the a llow ed r a nge (13). The solid line repr esents the minimum confidence ¯ c min that can b e guaranteed as a function o f m . This corres p o nds to the low est N p ermitted by (13 ) for a given m ; increa sing N g ives larger gua ranteed confidence lev els. The achiev able reg ion in the ( m, ¯ c ) pla ne is that ab ov e the solid curve, in the following sens e: for any ( m, ¯ c ) within this region, the confidence level a sso ciated to the error ma rgin m can b e a ssured to b e greater than ¯ c , irresp ective o f p ; this is accomplished b y s electing N a ccording to (1) (or, equiv ale ntly , using the curv es in 7 It should b e noted that, although [1, app endix C] considers n 2 ≥ a , the actual range of v alues for n 2 is n 2 > a − 1. Nonetheless, the pro ofs in [1, appendix C] can b e readily generalized to n 2 > a − 1. 3 10% 100% 65% 70% 75% 80% 85% 90% 95% 100% c m N=3 N=10 N=30 N=100 --- ¯ c — ¯ c min Figure 1 : Guar a nt eed confidence ¯ c a nd minimum confidence that ca n b e guar anteed ¯ c min [1, fig . 5(a)]). Compar ing with [1, fig. 4], it is se en that Prop osition 1 enlar ges the achiev able region, s pe c ially for low m ; b esides, it removes the restr ic tion on p . Fig . 1 th us replaces [1, fig. 4(a)]; and [1, figures 4(b) and 5(b)] are no long er necess a ry . As an example, consider the follo wing problem: g iven a n NBMC estimator ˆ p of an unknown p with N = 30, find the minimum m such that Pr[ p/ (1 + m ) ≤ ˆ p ≤ p (1 + m )] > 75%. F ro m Prop osition 1 and (1), m = 23 . 7%. The r esults in [1], on the con trary , c a n only give m = 24 . 5%, since (23 . 7% , 75%) is not in the ac hiev able region acco rding to [1, fig. 4(a)]; b esides, p < 0 . 22 4 is required. 3 Conclusions In this pa p er , the statistical characterization of the NBMC es tima tor introduced in [1] has b een improv ed b y relaxing the c o nditions that guar antee a certa in co nfidence level. It has b een es tablished that, for p ∈ (0 , 1 ) arbitrar y , the NBMC estimator has a confidence level better than (1) provided that µ 1 , µ 2 satisfy (5). This result extends the r ange o f a pplication of the NBMC es tima tio n technique. A App endix The following lemma, used in the pro o f of pro po sition 1, is no w establishe d. Lemma 1. Given N , n ∗ ∈ N with N ≥ 3 and n ∗ ≤ N − 1 2 − q N − 1 2 p + 1 2 (14) 4 the curve is convex p N N / ) 1 1 ( − − − np N e n − − 1 N n N p n − − − − ) 1 ( ) 1 ( ) 1 ( N n * n Figure 2: Gr aphical representation for the pr o of of Lemma 1 the fol lowing ine quality holds: Z n ∗ N − 1 n N − 1 e − pn dn ≥ n ∗ X n = N ( n − 1) ( N − 1) (1 − p ) n − N . (15) Pr o of. W e first note that the s ub- in tegr a l function is incre asing for n < ( N − 1) /p , and is conv ex for ( N − 1 − √ N − 1) /p < n < ( N − 1) /p . As figur e 2 illus tr ates, each term o f the sum in (1 5) ca n b e identified with the ar ea of a rectangle of unit w idth. Spec ific a lly , the term corre s po nding to a given n is a sso ciated to the rectangle that extends hor iz ontally from n − 1 to n in the figure. In addition, for the rectangles situated to the rig ht of ( N − 1 − √ N − 1) /p (where the s ub-int egr al function is conv ex), the flat tops are replaced by straig ht lines joining the centers, without altering the total area . The inequality (15) will hold if the area b elow the cur ve in the in terv al ( N − 1 , n ∗ ) is larger than the shaded a rea in the figure. W e divide this interv al in t wo: ( N − 1 , ( N − 1 − √ N − 1) /p ) and (( N − 1 − √ N − 1) /p, n ∗ ), and require tha t in each of these interv als the a rea of the curve b e larger than the part of the shade d area corr esp onding to that interv al. In the first interv al, since the curve is increa sing, it suffices that the curv e b e above the squa re ma r ks fo r n ≤ ⌊ ( N − 1 − √ N − 1) /p ⌋ , as shown in the figure. In the second interv al, since the curve is increa sing and co n vex, it suffices that the curve b e ab ov e the square mark lo cated at ⌊ ( N − 1 − √ N − 1) /p ⌋ + 1 and above the triangle marks. As a re sult, to establish (15) it is sufficient that (i) the sub-integral function b e ab ov e the square mark s for N ≤ n ≤ ( N − 1 − √ N − 1) /p + 1; and (ii) the sub-integral function be a b ove the triangle marks for ( N − 1 − √ N − 1) /p + 1 < n < n ∗ . W e analyz e these conditions separ ately . 5 (i) With x defined as x = 1 p ln ( n − 1) N − 1 e − ( n − 1) p ( n − 1) ( N − 1) (1 − p ) n − N = − 1 p N − 1 X i =1 ln  1 − i − 1 n − 1  − n − N p ln(1 − p ) − ( n − 1) , (16) condition (i) is expresse d as x ≥ 0 for n ≤ ( N − 1 − √ N − 1) /p + 1. Defining M = N − 1 and ν = ( n − 1 ) p , x = − 1 p N − 1 X i =1 ln  1 − ( i − 1) p ν  − 1 p  ν p − M  ln(1 − p ) − ν p . (17) Using the T aylor expansio n ln(1 − t ) = − P ∞ j =1 t j /j , | t | < 1, (17) is tra nsformed int o x = P ∞ j =0 x j p j with x j = 1 ( j + 1) ν j +1 N − 1 X i =1 ( i − 1) j +1 + ν j + 2 − M j + 1 . (18) The term x 0 is easily seen to b e nonnega tive for ν ≤ M − √ M , i.e . for n ≤ ( N − 1 − √ N − 1) /p + 1. W e no w prove that the r emaining co efficients x j , j ≥ 1 are a lso nonnegative fo r ν ≤ M − √ M . W e b egin with the case N ≥ 5, j ≥ 2. Using the inequality N − 1 X i =1 ( i − 1) j +1 > Z N − 1 1 ( i − 1) j +1 di = ( N − 2) j +2 j + 2 = ( M − 1) j +2 j + 2 (19) in (18), w e can b ound x j > ( M − 1) j +2 + ( j + 1 ) ν j +2 − ( j + 2) M ν j +1 ( j + 1)( j + 2) ν j +1 . (20) The denominator in (20) is p ositive. Let y j denote the numerator. W e compute ∂ y j ∂ ν = ( j + 2)( j + 1)( ν − M ) ν j < 0 , (21) from which it suffices to consider ν = M − √ M . Expressing y j | ν = M − √ M = M j +2  1 − 1 √ M  j +1 · "  1 − 1 √ M   1 + 1 √ M  j +2 + ( j + 1 )  1 − 1 √ M  − ( j + 2 )  (22) 6 and denoting the brack eted term by Y j , w e now compute the following partial deriv a- tives a s if j were a con tinuous v ariable: ∂ Y j ∂ j =  1 − 1 √ M   1 + 1 √ M  j +2 · ln  1 + 1 √ M  − 1 √ M (23) ∂ 2 Y j ∂ j 2 =  1 − 1 √ M   1 + 1 √ M  j +2 ln 2  1 + 1 √ M  . (24) The right-hand side of (24) is positive, and using the inequality ln(1 + t ) > t − t 2 / 2 we ca n b ound (23 ) for j = 2 a s ∂ Y j ∂ j     j =2 =  1 − 1 √ M   1 + 1 √ M  4 1 √ M ·  1 − 1 2 √ M  − 1 √ M > 0 . (25) Therefore the right-hand side of (23) is po sitive for j ≥ 2. Consequently , in order to establish that y j ≥ 0, it suffices to show that Y 2 ≥ 0. Defining m = 1 / √ M , Y 2 can be expres s ed as − m 2 ( m 3 + 3 m 2 + 2 m − 2). According to Desc artes’ sign rule, this po lynomial has only one p ositive roo t. F or m → ∞ the polynomial takes negative v a lues, a nd for m = 1 / 2 it is p o sitive. Therefor e, it is pos itive for m ≤ 1 / 2, i.e. for M ≥ 4, o r N ≥ 5. F or N = 4, j ≥ 2, we hav e x j = ( j + 2)(1 + 2 j +1 ) + ( j + 1 ) ν j +2 − 3( j + 2) ν j +1 ( j + 2)( j + 1) ν j +1 (26) Let z j denote the numerator in (26). Since ∂ z j ∂ ν = ( j + 2)( j + 1) ν j ( ν − 3) < 0 , (27) it suffices to consider ν = 3 − √ 3. Bounding z j as z j > ( j + 2)(2 j +1 − 3 ν j +1 ) + ( j + 1) ν j +2 , (28) z j | ν =3 − √ 3 is see n to b e p os itive for j ≥ 2. F or N = 3, j ≥ 2, x j = j + 2 + ( j + 1) ν j +2 − 2( j + 2) ν j +1 ( j + 2)( j + 1) ν j +1 , (29) and similar argument s to those for N = 4, j ≥ 2 s how that x j > 0. F or N ≥ 3, j = 1, us ing the identit y N − 1 X i =1 ( i − 1) 2 = ( N − 2) 3 3 + ( N − 2) 2 2 + N − 2 6 = ( M − 1) 3 3 + ( M − 1) 2 2 + M − 1 6 (30) 7 we o bta in from (18) x j = 4 ν 3 − 6 M ν 2 + 2( M − 1) 3 + 3( M − 1) 2 + M − 1 12 ν 2 . (31) F rom Descar tes’ sign r ule, the n umerator of (31) considered as a p olynomial in ν has t wo po sitive ro o ts at mos t. This p olynomial is po sitive for ν = 0 and for ν → ∞ , and ne g ative for ν = M . Thus, it will b e p o s itive for ν ≤ M − √ M if it is for ν = M − √ M . Substituting this ν and defining m = √ M , the numerator is expressed as m 2 (3 m 2 − 4 m + 1), which is positive for m > 1, or equiv alently M > 1, and thus for N ≥ 3 . (ii) With x ′ defined as x ′ = 1 p ln  n − 1 2  N − 1 e − ( n − 1 2 ) p ( n − 1) ( N − 1) (1 − p ) n − N = − 1 p N − 1 X i =1 ln  1 − i − 1 2 n − 1 2  − n − N p ln(1 − p ) −  n − 1 2  , (32) and taking into ac c ount (14), in orde r to fulfil condition (ii) it is sufficient that x ′ ≥ 0 for n ≤ ( N − 1 / 2 − p N − 1 / 2) /p + 1 / 2. Defining M ′ = N − 1 / 2 and ν ′ = ( n − 1 / 2 ) p , x ′ = − 1 p N − 1 X i =1 ln 1 −  i − 1 2  p ν ′ ! − 1 p  ν ′ p − M ′  ln(1 − p ) − ν ′ p . (33) Pro ceeding as with x , we can expr ess x ′ = P ∞ j =0 x ′ j p j with x ′ j = 1 ( j + 1) ν ′ j +1 N − 1 X i =1  i − 1 2  j +1 + ν ′ j + 2 − M ′ j + 1 . (34) x ′ 0 is seen to be nonnegative for ν ′ ≤ M ′ − p M ′ − 1 / 4, and thus for ν ′ ≤ M ′ − √ M ′ , i.e. for n ≤ ( N − 1 / 2 − p N − 1 / 2) /p + 1 / 2. W e now prov e that the remaining co efficients x ′ j , j ≥ 1 a re also no nnegative for ν ′ ≤ M ′ − √ M ′ . W e b egin with the case N ≥ 5, j ≥ 2. Using the inequality N − 1 X i =1  i − 1 2  j +1 > Z N − 1 1 / 2  i − 1 2  j +1 di =  N − 3 2  j +2 j + 2 = ( M ′ − 1) j +2 j + 2 (35) in (34), w e can b ound x ′ j > ( M ′ − 1) j +2 + ( j + 1 ) ν ′ j +2 − ( j + 2) M ′ ν ′ j +1 ( j + 1)( j + 2) ν ′ j +1 . (36) 8 The denominator in (36) is p ositive, and the numerator is as that in (20) with M replaced by M ′ and ν replac e d by ν ′ . It stems that x ′ j > 0 for M ′ ≥ 4, i.e. N ≥ 5, and j ≥ 2 . F or N = 4, j ≥ 2, (3 4) gives x ′ j = h ( j + 2)   1 2  j +1 +  3 2  j +1 +  5 2  j +1  + ( j + 1) ν ′ j +2 − 7 2 ( j + 2) ν ′ j +1 i.  ( j + 2)( j + 1) ν ′ j +1  , (37) which is shown to be positive with analo gous arguments as for x j . F or N = 3, j ≥ 2, x ′ j = h ( j + 2)   1 2  j +1 +  3 2  j +1  + ( j + 1 ) ν ′ j +2 − 5 2 ( j + 2) ν ′ j +1 i.  ( j + 2)( j + 1) ν ′ j +1  , (38) and similarly it is shown to b e positive. F or N ≥ 3, j = 1, us ing the identit y N − 1 X i =1  i − 1 2  2 =  N − 3 2  3 3 +  N − 3 2  2 2 + N − 3 2 6 = ( M ′ − 1) 3 3 + ( M ′ − 1) 2 2 + M ′ − 1 6 (39) we obtain an expressio n for x ′ j which co inc ides with (3 1) r eplacing M by M ′ and ν by ν ′ . Therefore, x ′ j is p ositive for M ′ > 1, and thus for N ≥ 3. According to the foreg o ing, conditions (i) and (ii) hold for N ≥ 3 . This estab- lishes the stated result (15). Ac kno wledgmen t The autho rs would like to thank the anonymous reviewers and the Editor for Wire- less Systems Performa nce, F. Santucci, for their helpful commen ts. References [1] L. Mendo and J. M. Hernando, “A simple seq uen tial s topping r ule for Mo n te Carlo simulation,” IEEE T r ans. Commun. , v ol. 54, no. 2, pp. 231–241 , F eb. 2006. 9