On Estimation of Finite Population Proportion

On Estimation of Finite P opulation Prop ortion ∗ Xinjia Chen No v ember 20, 2018 Abstract In this pap er, we study the classical problem of estimating the prop ortion of a finite po pulation. First, w e consider a fixed sample size metho d and derive an explicit s a mple size formula which ensures a mixe d criterion of absolute and relativ e er rors. Second, w e co nsider an inv erse sampling sc heme such that the s ampling is contin ue until the num b er of units having a cer tain attribute reaches a thres ho ld v alue o r the whole p opula tio n is examined. W e hav e established a simple metho d to determine the threshold so that a prescrib ed relative precision is guara nt ee d. Finally , we dev elop a multistage sampling scheme for c onstructing fixed-width confidence in terv al for the prop or tion of a finite p opula tio n. Pow erful computational techniques are in tro duced to make it p ossible that the fixed-width confidence int er v al e ns ures prescrib ed level of c overage pro bability . 1 Fixed Sample Size Metho d The estimation of the pr op ortion of a fi nite p opulation is a basic and very imp ortan t p r oblem in probabilit y and statistics [6, 8]. Suc h p roblem find s applications sp anning man y areas of sciences and engineering. The problem is formulated as follo ws. Consider a fi nite p opulation of N units, among which there are M units ha vin g a certain attribute. The ob j ectiv e is to estimate the prop ortion p = M N based on sampling without replace- men t. One p opu lar method of samp ling is to draw n units without replacemen t f r om the p opu lation and count the n umb er, k , of un its having the attribute. Then, the estimate of the pr op ortion is tak en as b p = k n . In this pro cess, the sample size n is fi xed. Clearly , th e random v ariable k p ossesses a hyp ergeometric distr ibution. Th e reliabilit y of the estimator b p = k n dep ends on n . F or error cont r ol pu rp ose, we are interested in a cru cial q u estion as follo ws: ∗ The aut hor is curren tly w ith Department of Elec t rical Engineering, Louisiana S tate U niversi ty at Ba ton Rouge, LA 70803, USA, and Department of Electrical Engineering, Southern Universit y and A&M Col lege, Baton Rouge, LA 70813, USA; Email: c henx injia@gmail. com 1 F or pr esc rib e d mar gin of absolute err or ε a ∈ (0 , 1) , mar gin of r elative err or ε r ∈ (0 , 1 ) , and c onfidenc e p ar ameter δ ∈ (0 , 1) , how lar ge the sample size n should b e to guar ante e Pr  | b p − p | < ε a or     b p − p p     < ε r  > 1 − δ ? (1) In this regard, w e hav e Theorem 1 L et ε a ∈ (0 , 1) and ε r ∈ (0 , 1) b e r e al numb ers such that ε a ε r + ε a ≤ 1 2 . Then, (1) is guar ante e d pr ovide d that n > ε r ln 2 δ ( ε a + ε a ε r ) ln(1 + ε r ) + ( ε r − ε a − ε a ε r ) ln  1 − ε a ε r ε r − ε a  . (2) The p r o of of Theorem 1 is giv en in App endix A. It should b e noted that con ven tional meth- o ds for determining sample sizes are based on normal app r o ximation, see [6] and the referen ces therein. In contrast, Theorem 1 offers a rigorous metho d for d etermining samp le sizes. T o re- duce co n serv ativ eness, a numerical approac h h as b een dev elop ed b y Chen [4] w hic h p erm its exact computation of the minimum s ample size. 2 In v erse Sampling of Finite P opulation T o estimate the prop ortion p , a frequently- u sed sampling metho d is the inverse sampling sc h eme describ ed as follo ws: Con tinuing sampling from the p opulation (without r eplacemen t) until r u nits found to carry the attribu te or the num b er of sample size n reac hes the p opu lation size N . The estimator of the pr op ortion p is tak en as the r atio e p = k n , where k is the n u m b er of units h a ving the attribu te among the n u nits. Clearly , the reliabilit y of the estimator e p dep ends on the threshold v alue r . Hence, we are in terested in a crucial question as follo ws: F or pr esc rib e d mar gin of r elative err or ε ∈ (0 , 1) and c onfidenc e p ar ameter δ ∈ (0 , 1 ) , how lar ge the thr eshold r should b e to guar ante e Pr {| e p − p | < εp } > 1 − δ ? F or this pu rp ose, we hav e Theorem 2 F or any ε ∈ (0 , 1) , Pr {| e p − p | ≥ εp } ≤ Q ( ε, r ) 2 wher e Q ( ε, r ) = (1 + ε ) − r exp  εr 1 + ε  + (1 − ε ) − r exp  − εr 1 − ε  , which is monotonic al ly de c r e asing with r esp e ct to r . Mor e over, for any δ ∈ (0 , 1) , ther e exists a unique numb er r ∗ such that Q ( ε, r ∗ ) = δ and max  (1 + ε ) ln 1 δ (1 + ε ) ln(1 + ε ) − ε , (1 − ε ) ln 2 δ (1 − ε ) ln(1 − ε ) + ε  < r ∗ < (1 + ε ) ln 2 δ (1 + ε ) ln(1 + ε ) − ε . The pro of of Theorem 2 is giv en in App end ix B. As an immediate consequ ence of Th eorem 2, w e ha ve Corollary 1 L et ε, δ ∈ (0 , 1) . Then, Pr {| e p − p | < εp } > 1 − δ pr ovide d that r > (1 + ε ) ln 2 δ (1 + ε ) ln (1 + ε ) − ε (3) 3 Multistage Fixed-width Confidenc e In terv als So far we ha ve only considered p oint estimation for the p r op ortion p . Interv al estimation is also an imp ortant metho d f or estimating p . Motiv ated b y the fact that a confiden ce in terv al must b e sufficien tly n arr o w to b e u seful, w e sh all dev elop a multistag e sampling sc h eme for constructing a fixed-width confid ence interv al for the p rop ortion, p , of the finite pop u lation discussed in previous sections. Note that the pro cedur e of sampling without r eplacemen t can b e p recisely described as follo ws: Eac h time a single un it is drawn without rep lacement from the remaining p opulation so that ev ery u nit of the remaining p opulation has equal chance of b eing selected. Suc h a sampling pr o cess ca n b e exactly c haracterized by random v ariables X 1 , · · · , X N defined in a probability s pace (Ω , F , Pr) such that X i denotes the charact eristics of the i -th sample in the sense that X i = 1 if the i -th sample h as the attribute and X i = 0 otherw ise. By the nature of the sampling pr o cedure, it can b e shown that Pr { X i = x i , i = 1 , · · · , n } =  M P n i =1 x i  N − M n − P n i =1 x i   n P n i =1 x i  N n   for an y n ∈ { 1 , · · · , N } and a ny x i ∈ { 0 , 1 } , i = 1 , · · · , n . Based on random v ariables X 1 , · · · , X N , w e can defin e a m ultistage sampling sc heme of the follo win g basic structure. The sampling pro cess is d ivided into s stages with sample sizes n 1 < n 2 < · · · < n s . The con tinuation or termination of samp ling is determin ed by decision v ariables. F or eac h s tage with index ℓ , a decision v ariable D ℓ = D ℓ ( X 1 , · · · , X n ℓ ) is defi ned b ased on random v ariables X 1 , · · · , X n ℓ . The decision v ariable D ℓ assumes only t w o p ossible v alues 0 , 1 w ith the notion that the sampling is con tinued until D ℓ = 1 f or some ℓ ∈ { 1 , · · · , s } . Since th e sampling m us t b e terminated at or b efore the s -th stage, it is requir ed that D s = 1. F or simplicit y of notations, w e also define D ℓ = 0 for ℓ = 0. 3 Our goal is to construct a fix ed -width confidence in terv al ( L , U ) suc h that U − L ≤ 2 ε and that Pr { L < p < U | p } > 1 − δ for any p ∈ { i N : 0 ≤ i ≤ N } with p r escrib ed ε ∈ (0 , 1 2 ) and δ ∈ (0 , 1). T o w ard this goal, we need to d efi ne some multi v ariate f u nctions as follo ws. F or α ∈ (0 , 1) and integ ers 0 ≤ k ≤ n ≤ N , let L ( N , n, k , α ) b e the smallest int eger M l suc h that P n i = k  M l i  N − M l n − i  /  N n  > α 2 . Let U ( N , n, k , α ) b e the largest in teger M u suc h that P k i =0  M u i  N − M u n − i  /  N n  > α 2 . Let n max ( N , α ) b e the smallest n umb er n su ch th at U ( N , n, k , α ) − L ( N , n, k , α ) ≤ 2 εN for 0 ≤ k ≤ n . Let n min ( N , α ) b e the largest num b er n suc h that U ( N , n, k , α ) − L ( N , n, k , α ) > 2 εN for 0 ≤ k ≤ n . Theorem 3 L et ζ > 0 and ρ > 0 . L et n 1 < n 2 < · · · < n s b e the asc e nding arr angement of al l dis- tinct elements of  h n max ( N ,ζ δ ) n min ( N ,ζ δ ) i i τ n min ( N , ζ δ )  : i = 0 , 1 , · · · , τ  with τ = l 1 ln(1+ ρ ) ln n max ( N ,ζ δ ) n min ( N ,ζ δ ) m . F or ℓ = 1 , · · · , s , define K ℓ = P n ℓ i =1 X i and D ℓ such that D ℓ = 1 if U ( N , n ℓ , K ℓ , ζ δ ) −L ( N , n ℓ , K ℓ , ζ δ ) ≤ 2 εN ; and D ℓ = 0 otherwise. Supp ose the stopping rule is that sampling is c ontinue d until D ℓ = 1 for some ℓ ∈ { 1 , · · · , s } . Define L = 1 N × L  N , n , P n i =1 X i , ζ δ  and U = 1 N × U  N , n , P n i =1 X i , ζ δ  , wher e n i s the sample size when the sampling is terminate d. Then, a suffic i ent c ondition to guar ante e Pr { L < p < U | p } > 1 − δ for any p ∈ { i N : 0 ≤ i ≤ N } is that s X ℓ =1 [Pr {L ( N , n ℓ , K ℓ , ζ δ ) ≥ M , D ℓ − 1 = 0 , D ℓ = 1 | M } + Pr {U ( N , n ℓ , K ℓ , ζ δ ) ≤ M , D ℓ − 1 = 0 , D ℓ = 1 | M } ] < δ (4) for al l M ∈ { 0 , 1 , · · · , N } , wher e (4) is satisfie d if ζ > 0 is su ffic i ently smal l. It should be noted that Th eorem 3 has emplo y ed the double-decision-v ariable metho d recen tly prop osed b y Chen in [1]. T o further reduce compu tational complexit y , the tec hniques of bisection confidence tuning and d omain tru ncation dev elop ed in [1, 2] can b e v ery useful. A Pro of of Theorem 1 T o pro ve the theorem, we shall introdu ce fu nction g ( ε, p ) = ( p + ε ) ln p p + ε + (1 − p − ε ) ln 1 − p 1 − p − ε where 0 < ε < 1 − p . W e need some p reliminary r esu lts. The follo wing lemma is du e to Ho effd ing [7]. Lemma 1 Pr { b p ≥ p + ε } ≤ exp( n g ( ε, p )) for 0 < ε < 1 − p < 1 , Pr { b p ≤ p − ε } ≤ exp( n g ( − ε, p )) for 0 < ε < p < 1 . The follo wing Lemmas 2–4 hav e b een established in [3]. 4 Lemma 2 L et 0 < ε < 1 2 . Then, g ( ε, p ) i s monot onic al ly incr e asing with r esp e ctive to p ∈ (0 , 1 2 − ε ) and monotonic al ly de cr e asing with r esp e ctive to p ∈ ( 1 2 , 1 − ε ) . Similarly, g ( − ε, p ) is monotonic al ly incr e asing with r esp e ctive to p ∈ ( ε, 1 2 ) and monot onic al ly de cr e asing with r esp e ctive to p ∈ ( 1 2 + ε, 1) . Lemma 3 L et 0 < ε < 1 2 . Then, g ( ε, p ) > g ( − ε, p ) ∀ p ∈  ε, 1 2  , g ( ε, p ) < g ( − ε, p ) ∀ p ∈  1 2 , 1 − ε  . Lemma 4 L et 0 < ε < 1 . Then, g ( εp, p ) is monotonic al ly de cr e asing with r esp e ct to p ∈  0 , 1 1+ ε  . Similarly, g ( − εp, p ) i s monotonic al ly de c r e asing with r esp e ct to p ∈ (0 , 1) . Lemma 5 Supp ose 0 < ε r < 1 and 0 < ε a ε r + ε a ≤ 1 2 . Then, Pr { b p ≤ p − ε a } ≤ exp  n g  − ε a , ε a ε r  (5) for 0 < p ≤ ε a ε r . Pro of . W e shall sho w (5) by inv estiga ting three cases as follo ws. In the case of p < ε a , it is clear that Pr { b p ≤ p − ε a } = 0 < exp  n g  − ε a , ε a ε r  . In the case of p = ε a , we hav e Pr { b p ≤ p − ε a } = Pr { b p = 0 } = Pr { k = 0 } =  N − M n   N n  ≤  N − M N  n = (1 − p ) n = (1 − ε a ) n = lim p → ε a exp( n g ( − ε a , p )) < exp  n g  − ε a , ε a ε r  , where the last in equalit y follo ws fr om L emm a 2 and the fact that ε a < ε a ε r ≤ 1 2 − ε a . In the case of ε a < p ≤ ε a ε r , we ha ve Pr { b p ≤ p − ε a } ≤ exp ( n g ( − ε a , p )) < exp  n g  − ε a , ε a ε r  , where the firs t inequalit y follo w s from Lemma 1 and the second inequalit y f ollo w s from Lemma 2 and the fact th at ε a < ε a ε r ≤ 1 2 − ε a . S o, (5) is established. ✷ 5 Lemma 6 Supp ose 0 < ε r < 1 and 0 < ε a ε r + ε a ≤ 1 2 . Then, Pr { b p ≥ (1 + ε r ) p } ≤ exp  n g  ε a , ε a ε r  (6) for ε a ε r < p < 1 . Pro of . W e s h all sh ow (6 ) by in vesti gating three cases as follo ws . In th e case of p > 1 1+ ε r , it is clear that Pr { b p ≥ (1 + ε r ) p } = 0 < exp  n g  ε a , ε a ε r  . In the case of p = 1 1+ ε r , we ha ve Pr { b p ≥ (1 + ε r ) p } = Pr { b p = 1 } = Pr { k = n } =  M n   N n  ≤  M N  n = p n =  1 1 + ε r  n = lim p → 1 1+ ε r exp( n g ( ε r p, p )) < exp  n g  ε a , ε a ε r  , where the last inequ alit y follo ws f rom Lemma 4 and th e fact that ε a ε r ≤ 1 2 1 1+ ε r < 1 1+ ε r as a result of 0 < ε a ε r + ε a ≤ 1 2 . In the case of ε a ε r < p < 1 1+ ε r , we ha ve Pr { b p ≤ (1 + ε r ) p } ≤ exp( n g ( ε r p, p )) < exp  n g  ε a , ε a ε r  , where the fi rst inequalit y f ollo w s fr om Lemma 1 and the second inequalit y follo ws from Lemma 4. So, (6) is established. ✷ W e are now in a position to pro v e the theorem. W e shall assu me (2 ) is satisfied and s ho w that (1) is true. It suffi ces to sh o w that Pr {| b p − p | ≥ ε a , | b p − p | ≥ ε r p } < δ . F or 0 < p ≤ ε a ε r , we hav e Pr {| b p − p | ≥ ε a , | b p − p | ≥ ε r p } = P r {| b p − p | ≥ ε a } = Pr { b p ≥ p + ε a } + Pr { b p ≤ p − ε a } . (7) Noting that 0 < p + ε a ≤ ε a ε r + ε a ≤ 1 2 , we hav e Pr { b p ≥ p + ε a } ≤ exp( n g ( ε a , p )) ≤ exp  n g  ε a , ε a ε r  , 6 where the fi rst inequalit y f ollo w s fr om Lemma 1 and the second inequalit y follo ws from Lemma 2. It can b e c h eck ed that (2) is equiv alen t to exp  n g  ε a , ε a ε r  < δ 2 . Therefore, Pr { b p ≥ p + ε a } < δ 2 for 0 < p ≤ ε a ε r . On the other h and, sin ce ε a < ε a ε r < 1 2 , by Lemma 5 and L emm a 3 , we ha ve Pr { b p ≤ p − ε a } ≤ exp  n g  − ε a , ε a ε r  ≤ exp  n g  ε a , ε a ε r  < δ 2 for 0 < p ≤ ε a ε r . Hence, b y (7), Pr {| b p − p | ≥ ε a , | b p − p | ≥ ε r p } < δ 2 + δ 2 = δ. This prov es (1) for 0 < p ≤ ε a ε r . F or ε a ε r < p < 1, we hav e Pr {| b p − p | ≥ ε a , | b p − p | ≥ ε r p } = P r {| b p − p | ≥ ε r p } = Pr { b p ≥ p + ε r p } + Pr { b p ≤ p − ε r p } . In voking Lemma 6, we hav e Pr { b p ≥ p + ε r p } ≤ exp  n g  ε a , ε a ε r  . On the other h and, Pr { b p ≤ p − ε r p } ≤ exp( n g ( − ε r p, p )) ≤ exp  n g  − ε a , ε a ε r  ≤ exp  n g  ε a , ε a ε r  where the first inequalit y follo ws f rom Lemma 1, the second inequ ality follo ws fr om Lemma 4, and the last in equalit y follo ws fr om L emm a 3 . Hence, Pr {| b p − p | ≥ ε a , | b p − p | ≥ ε r p } ≤ 2 exp  n g  ε a , ε a ε r  < δ. This prov es (1) for ε a ε r < p < 1. The pro of of Theorem 1 is th us completed. 7 B Pro of T heorem 2 W e need some preliminary r esu lts. W e shall in tro d uce f unctions M ( z , p ) = ln  p z  +  1 z − 1  ln  1 − p 1 − z  and H ( z , p ) = z M ( z , p ) for 0 < z < 1 and 0 < p < 1. Lemma 7 Supp ose 1 ≤ r ≤ M < N . Then, Pr n r n ≤ (1 − ε ) p o ≤ (1 − ε ) − r exp  − εr 1 − ε  . Pro of . Clearly , Pr n r n ≤ (1 − ε ) p o = Pr  n ≥ r (1 − ε ) p  = Pr { n ≥ m } where m =  r (1 − ε ) p  . It can b e seen that there exists a r eal n u m b er ε ∗ ∈ (0 , 1) su c h that ε ∗ ≥ ε an d r (1 − ε ∗ ) p =  r (1 − ε ) p  . No w let K m b e the n umber of u nits ha ving a certa in attribute among m units dra wn b y a sampling without replacemen t from a finite p opulation of size N w ith M u nits ha ving the attribute. Then, Pr { n ≥ m } = Pr { K m ≤ r } = Pr  K m m ≤ r m  = Pr  K m m ≤ (1 − ε ∗ ) p  . Applying the well-kno wn Ho effding inequalit y [7 ] f or th e case of fin ite p opulation, we ha ve Pr  K m m ≤ (1 − ε ∗ ) p  ≤ exp ( m H ( p − ε ∗ p, p )) = exp  r (1 − ε ∗ ) p H ( p − ε ∗ p, p )  = exp ( r M ( p − ε ∗ p, p )) ≤ exp ( r M ( p − εp, p )) 8 where the last inequ alit y follo ws from ε ∗ ≥ ε and the m onotone prop ert y of M ( p − εp, p ) with resp ect to ε , wh ich has b een established as Lemma 5 in [5]. F rom the pr o of of Lemma 6 of [5], we know that M ( p − εp, p ) is monotonically decreasing with resp ect to p ∈ (0 , 1). Hence, Pr n r n ≤ (1 − ε ) p o ≤ exp ( r M ( p − εp, p )) ≤ lim p → 0 exp ( r M ( p − εp, p )) = (1 − ε ) − r exp  − εr 1 − ε  . The pro of of the lemma is thus completed. ✷ Lemma 8 Supp ose 1 ≤ r ≤ M < N and p + εp < 1 . Then, Pr n r n ≥ (1 + ε ) p o ≤ (1 + ε ) − r exp  εr 1 + ε  . Pro of . It is clear that Pr n r n ≥ (1 + ε ) p o = Pr  n ≤ r (1 + ε ) p  = Pr { n ≤ m } where m =  r (1 + ε ) p  . It can b e seen that there exists a r eal n u m b er ε ∗ ∈ (0 , 1) su c h that ε ∗ ≥ ε an d r (1 + ε ∗ ) p =  r (1 + ε ) p  . No w let K m b e the n umber of u nits ha ving a certa in attribute among m units dra wn b y a sampling without replacemen t from a finite p opulation of size N w ith M u nits ha ving the attribute. Then, Pr { n ≤ m } = Pr { K m ≥ r } = Pr  K m m ≥ r m  = Pr  K m m ≥ (1 + ε ∗ ) p  . Applying the well-kno wn Ho effding inequalit y [7 ] f or th e case of fin ite p opulation, we ha ve Pr  K m m ≥ (1 + ε ∗ ) p  ≤ exp ( m H ( ε ∗ p, p )) = exp  r (1 + ε ∗ ) p H ( ε ∗ p, p )  = exp ( r M ( p + ε ∗ p, p )) ≤ exp ( r M ( p + εp, p )) 9 where the last inequ alit y follo ws from ε ∗ ≥ ε and the m onotone prop ert y of M ( p + εp, p ) with resp ect to ε , wh ich has b een established as Lemma 5 in [5]. F rom the pr o of of Lemma 6 of [5], we know that M ( p + εp, p ) is monotonically decreasing with resp ect to p ∈  0 , 1 1+ ε  . Hence, Pr n r n ≥ (1 + ε ) p o ≤ exp ( r M ( p + εp, p )) ≤ lim p → 0 exp ( r M ( p + εp, p )) = (1 + ε ) − r exp  εr 1 + ε  . The pro of of the lemma is thus completed. ✷ No w we are in a p osition to p ro ve Th eorem 2. W e sh all consider the follo win g cases: Case (i): M < r ; Case (ii): M = N ; Case (iii): r = N ; Case (iv): 1 ≤ r ≤ M < N and p < 1 1+ ε ; Case (v): 1 ≤ r ≤ M < N and p = 1 1+ ε ; Case (vi): 1 ≤ r ≤ M < N and p > 1 1+ ε . In Case (i), we h a v e n = N and k = M . Hence, e p = p and Pr {| e p − p | ≥ εp } = 0 ≤ Q ( ε, r ). In Case (ii), we h a v e e p = p and Pr {| e p − p | ≥ εp } = 0 ≤ Q ( ε, r ). In Case (iii), we hav e e p = p and Pr {| e p − p | ≥ εp } = 0 ≤ Q ( ε, r ). In Case (iv), we h a v e k = r and , by L emm a 7 and Lemma 8, Pr {| e p − p | ≥ εp } = Pr n r n ≤ (1 − ε ) p o + Pr n r n ≥ (1 + ε ) p o ≤ (1 − ε ) − r exp  − εr 1 − ε  + (1 + ε ) − r exp  εr 1 + ε  = Q ( ε, r ) . In Case (v), we ha ve k = r and Pr {| e p − p | ≥ εp } = Pr n r n ≤ (1 − ε ) p o + Pr n r n ≥ (1 + ε ) p o = Pr n r n ≤ (1 − ε ) p o + Pr { k = n = r } . Notice that Pr { k = n = r } =  M r   N r  <  M N  r = p r =  1 1 + ε  r < (1 + ε ) − r exp  εr 1 + ε  10 as a result of M ≤ N . Therefore, by Lemma 7, Pr {| e p − p | ≥ εp } ≤ (1 − ε ) − r exp  − εr 1 − ε  + (1 + ε ) − r exp  εr 1 + ε  = Q ( ε, r ) . In Case (vi), we h a v e k = r, Pr  r n ≥ (1 + ε ) p  = 0 and, b y Lemm a 7, Pr {| e p − p | ≥ εp } = Pr n r n ≤ (1 − ε ) p o + Pr n r n ≥ (1 + ε ) p o = Pr n r n ≤ (1 − ε ) p o ≤ (1 − ε ) − r exp  − εr 1 − ε  < Q ( ε, r ) . So, w e ha v e sho wn P r {| e p − p | ≥ εp } ≤ Q ( ε, r ). The other statemen ts of Theorem 2 ha ve b een established in [5]. This concludes the p ro of of Theorem 2. References [1] Chen , X. , “A new framewo rk of m u ltistage estimati on,” arXiv:0809 .1241 v5 [math.ST], Jan- uary 2009. [2] X. Chen, “A truncation appr oac h for fast computation of distribution fu nctions,” arXiv:0802 .3455 [math.ST], F ebruary 2008. [3] X. Chen, “On Estimation and Op timization of Probabilit y ,” arXiv:0804.13 99 , April 2008. [4] X. Chen, “Exact computation of min im um samp le size f or estimating prop ortion of finite p opulation,” arXiv:0707.21 15 , July 2007. [5] X. Chen, “In v ers e samplin g for n onasymptotic sequentia l estimation of b ounded v ariable means,” arXiv:0711.2 801 , No ve mb er 2007. [6] M. M. Desu and D. Raghav arao, Sample Size Metho dolo gy , Academic Press, 1990. [7] W. Ho effding, “Probabilit y inequalities for s ums of b oun ded v ariables,” J. Amer. Statist. Asso c . , vo l. 58, 13–29, 1963. [8] S . K . T hompson, Sampling , Wiley , 2002. 11