Coverage Probability of Wald Interval for Binomial Parameters

Co v erage Probabilit y o f W ald In terv al for Binomial P arameters ∗ Xinjia Chen Submitted in April, 200 8 Abstract In this paper, we develop an exact metho d for computing the minimum cov erage proba bilit y of W ald in terv al for estimation of binomial parameters. Simila r approach can be used for other t yp e of confidence int erv als. 1 W ald In terv al Let X b e a Bernoulli random v ariable with distribution Pr { X = 1 } = 1 − Pr { X = 0 } = p ∈ (0 , 1) . Let X 1 , · · · , X n b e i.i.d. random samp les of X . Let K = P n i =1 X i . The widely-used W ald in terv a l is [ L, U ] with low er confidence limit L = K n − Z δ/ 2 s K n (1 − K n ) n and up p er confidence limit U = K n + Z δ/ 2 s K n (1 − K n ) n where Z δ/ 2 is the cr itical v alue suc h that R ∞ Z δ/ 2 1 √ 2 π e − x 2 2 = δ 2 . It h as b e en disco v ered by Bro wn et al. [1] that the co v erag e probabilit y of W ald int erv al is surprisingly p o or. 2 Co v e rage Probabilit y The co v erage probability of W ald in terv al for binomial parameters has b een in ve stigated by [1] and other researc hers by Edgew orth exp ansion method and n umerical metho ds based on d iscretizing the binomial parameter. Here, we ha ve obtained expression of the minim um cov erag e probabilit y of W ald in terv a l, whic h r equires only finite many ev aluations of co ve rage p robabilit y . ∗ The aut hor is currently with Department of Electrical Engineering, Louisiana State Universit y at Baton Rouge, LA 708 03, USA , and Department of Electrical Engineering, Southern Univ ersity and A&M Col lege, Baton Rouge, LA 708 13, USA ; Email: c henxinjia@gmail.com 1 Theorem 1 Define T − ( p ) = 2 p + θ − p θ 2 + 4 θ p (1 − p ) 2(1 + θ ) , T + ( p ) = 2 p + θ + p θ 2 + 4 θ p (1 − p ) 2(1 + θ ) with θ = Z 2 δ/ 2 n . Define L ( k ) = k n − Z δ/ 2 s k n (1 − k n ) n , U ( k ) = k n + Z δ/ 2 s k n (1 − k n ) n for k = 0 , 1 , · · · , n . Define C l ( k ) = Pr {⌈ T − ( L ( k )) ⌉ ≤ K ≤ k − 1 | L ( k ) } , C ′ l ( k ) = Pr {⌊ T − ( L ( k )) ⌋ + 1 ≤ K ≤ k − 1 | L ( k ) } for k ∈ { 0 , 1 , · · · , n } such that 0 < L ( k ) < 1 . Define C u ( k ) = Pr { k + 1 ≤ K ≤ ⌊ T + ( U ( k )) ⌋ | U ( k ) } , C ′ u ( k ) = Pr { k + 1 ≤ K ≤ ⌈ T + ( U ( k )) ⌉ − 1 | U ( k ) } for k ∈ { 0 , 1 , · · · , n } such that 0 < U ( k ) < 1 . Supp ose θ < 3 . Then, the fol lowing statements hold true: (I): inf p ∈ (0 , 1) Pr { L ≤ p ≤ U | p } e quals to the minimum of { C l ( k ) : 0 ≤ k ≤ n ; 0 < L ( k ) < 1 } ∪ { C u ( k ) : 0 ≤ k ≤ n ; 0 < U ( k ) < 1 } . (II): min p ∈ (0 , 1) Pr { L < p < U | p } e quals to the minimum of { C ′ l ( k ) : 0 ≤ k ≤ n ; 0 < L ( k ) < 1 } ∪ { C ′ u ( k ) : 0 ≤ k ≤ n ; 0 < U ( k ) < 1 } . The pr o of of Theorem 1 is giv en in the next section. 3 Pro of of Theorem 1 W e n eed some pr eliminary results. Lemma 1 F or n > Z 2 δ/ 2 3 , b oth the lower and upp er c o nfidenc e limits of Wald interval ar e mono- tonic al ly incr e asing with r esp e c t to k . Pro of . F or simplicit y of notation, let z = k n . Then, the upp er confi dence limit can b e written as h ( z ) = z + Z δ/ 2 p z (1 − z ) /n. 2 Similarly , the lo wer confiden ce limit can b e written as g ( z ) = z − Z δ/ 2 p z (1 − z ) /n. T o show that the upp er confid en ce limit is monotonically increasing with resp ect to k , it su ffices to show th at ∂ h ( z ) ∂ z > 0 if 0 ≤ h ( z ) ≤ 1. Since ∂ h ( z ) ∂ z = 1 + √ θ 2 1 − 2 z p z (1 − z ) , whic h is clearly p ositiv e for 0 < z ≤ 1 2 , it remains to sho w √ θ 1 − 2 z p z (1 − z ) > − 2 , ∀ z ∈  1 2 , 1  or equiv alent ly , z (1 − z ) > θ 4 (2 z − 1) 2 , ∀ z ∈  1 2 , 1  . Note that h ( z ) < 1 for 0 < z < 1 1+ θ , and h ( z ) > 1 for 1 > z > 1 1+ θ . If 1 1+ θ ≤ 1 2 , i.e., θ ≥ 1, then we are done, since ∂ h ( z ) ∂ z > 0 for all 0 < z < 1 1+ θ ≤ 1 2 . Otherw ise, if θ < 1, it su ffices to sh o w w ( z ) = z (1 − z ) − θ 4 (2 z − 1) 2 > 0 for 1 2 < z < 1 1+ θ . Since z (1 − z ) is d ecreasing and θ 4 (2 z − 1) 2 is in creasing for 1 > z > 1 2 , w ( z ) is decreasing for 1 2 < z < 1 1+ θ . T herefore, it suffices to h a v e w  1 1 + θ  > 0 , i.e., 1 1 + θ  1 − 1 1 + θ  − θ 4  2 1 + θ − 1  2 > 0 , i.e., θ (1 + θ ) 2 − θ 4  1 − θ 1 + θ  2 > 0 , i.e., 4 > (1 − θ ) 2 , whic h is guaran teed since 0 < θ < 3. This sho ws that the up p er confidence limit is monotonically increasing with resp ect to k . O b serving that g ( z ) = 1 − h (1 − z ), w e h av e that the lo w er confidence limit is also monotonically increasing with resp ect to k . ✷ 3 No w we consid er the minim um co v erage pr obabilit y . By solving equation  p − k n  2 = θ k n  1 − k n  with resp ect to k , we can sh ow that Pr { L ≤ p < U | U ( k ) } = Pr { k < K ≤ T + ( p ) | U ( k ) } , 0 < U ( k ) < 1 Pr { L < p ≤ U | L ( k ) } = Pr { T − ( p ) ≤ K < k | L ( k ) } , 0 < L ( k ) < 1 Pr { L < p < U | U ( k ) } = Pr { k < K < T + ( p ) | U ( k ) } , 0 < U ( k ) < 1 Pr { L < p < U | L ( k ) } = Pr { T − ( p ) < K < k | L ( k ) } , 0 < L ( k ) < 1 Since b oth the lo w er and up p er confi dence limits of W ald in terv al are monotone as asserted by Lemma 1, the pro of of Theorem 1 can b e completed b y making us e of th e ab o v e r esults and applying the theory of co ve rage probability of r andom in terv als established by Chen in [2]. References [1] Bro wn, L. D., Cai, T. and DasGupta, A., “Interv al estimation for a binomial p rop ortion and asymptotic exp ansions,” The Annals of Statistics ,” V ol. 30, pp. 160 -201, 2002. [2] Chen X., “Co verag e Probability of Rand om Interv als,” arXiv:0707 .2814 , July 2007. 4