A Simple Sample Size Formula for Estimating Means of Poisson Random Variables

A Simple Sample Size F orm ula for Estimating Means of P oisson Random V ariables ∗ Xinjia Chen Submitted in April, 200 8 Abstract In this pap er, we derive an explicit sample size formula based a mixed criterio n of absolute and relative err ors for estimating means o f Poisson random v ariables. 1 Sample Size F orm ula It is a frequen t problem to estimate the mean v alue of a P oisson random v ariable based o n sampling. Sp ecifically , let X b e a P oisson random v ariable with mean E [ X ] = λ > 0, one wishes to estimate λ as b λ = P n i =1 X i n where X 1 , · · · , X n are i.i.d. random samples of X . Since b λ is of random nature, it is imp ortant to control th e statistic al error of the estimate . F or this purp ose, we h a ve Theorem 1 L et ε a > 0 , ε r ∈ (0 , 1) and δ ∈ (0 , 1) . Then Pr n    b λ − λ    < ε a or    b λ − λ    < ε r λ o > 1 − δ pr o vide d that n > ε r ε a × ln 2 δ (1 + ε r ) ln(1 + ε r ) − ε r . (1) It sh ould b e noted that con v entio n al metho ds for determining sample sizes are based on normal appro ximation, see [3] and the r eferences therein. In con trast, Th eorem 1 offers a r igorous metho d for determining sample sizes. T o redu ce c onserv atism, a n umerical approac h has been dev elop ed b y Chen [1] whic h p ermits exa ct computation of the minim um sample size. ∗ The auth or is cu rrently with Department of Electrical Engineering, Louisiana State Universit y at Baton Rouge, LA 70803, US A, and Department of Electrical Engineering, South ern Universit y and A&M College, Baton Rouge, LA 70813, US A; Email: c henxinjia@gmail.com 1 2 Pro of of Theorem 1 W e need some preliminary results. Lemma 1 L et K b e a Poisson r andom variable with me an θ > 0 . Then, Pr { K ≥ r } ≤ e − θ  θ e r  r for any r e a l numb er r > θ and Pr { K ≤ r } ≤ e − θ  θ e r  r for any p ositive r e a l nu mb er r < θ . Pro of . F or an y real n umber r > θ , using the Chernoff b ound [2], w e h a v e Pr { K ≥ r } ≤ inf t> 0 E h e t ( K − r ) i = inf t> 0 ∞ X i =0 e t ( i − r ) θ i i ! e − θ = inf t> 0 e θ e t e − θ e − r t ∞ X i =0 ( θ e t ) i i ! e − θ e t = inf t> 0 e − θ e θ e t − r t , where the infi m um is achiev ed at t = ln  r θ  > 0. F or this v alue of t , we ha v e e − θ e θ e t − tr = e − θ  θ e r  r . It follo ws that Pr { K ≥ r } ≤ e − θ  θ e r  r for any real n umber r > θ . Similarly , for any real n umber r < θ , w e ha v e Pr { K ≤ r } ≤ e − θ  θ e r  r . ✷ In the sequel, w e s hall in tro duce the follo wing function g ( ε, λ ) = ε + ( λ + ε ) ln λ λ + ε . Lemma 2 L et λ > ε > 0 . Then, Pr n b λ ≤ λ − ε o ≤ exp ( n g ( − ε, λ )) and g ( − ε, λ ) is monotoni- c al ly incr e asing with r esp e ct to λ ∈ ( ε, ∞ ) . Pro of . Letting K = P n i =1 X i , θ = nλ and r = n ( λ − ε ) and applying Lemma 1, for λ > ε > 0, w e ha v e Pr n b λ ≤ λ − ε o = Pr { K ≤ r } ≤ e − θ  θ e r  r = exp ( n g ( − ε, λ )) , where g ( − ε, λ ) is m onotonically increasing with resp ect to λ ∈ ( ε, ∞ ) b ecause ∂ g ( − ε, λ ) ∂ λ = − ln  1 − ε λ  − ε λ > 0 for λ > ε > 0. ✷ Lemma 3 L et ε > 0 . Then, Pr n b λ ≥ λ + ε o ≤ exp ( n g ( ε, λ )) and g ( ε, λ ) is monotonic al ly incr e a sing with r esp e ct to λ ∈ (0 , ∞ ) . 2 Pro of . Letting K = P n i =1 X i , θ = nλ and r = n ( λ + ε ) and applying Lemma 1, for λ > 0, we ha ve Pr n b λ ≥ λ + ε o = Pr { K ≥ r } ≤ e − θ  θ e r  r ≤ exp ( n g ( ε, λ )) , where g ( ε, λ ) is monoto n ically increasing with resp ect to λ ∈ (0 , ∞ ) b ecause ∂ g ( ε, λ ) ∂ λ = − ln  1 + ε λ  + ε λ > 0 . ✷ Lemma 4 g ( ε, λ ) > g ( − ε, λ ) for λ > ε > 0 . Pro of . Since g ( ε, λ ) − g ( − ε, λ ) = 0 for ε = 0 and ∂ [ g ( ε, λ ) − g ( − ε, λ )] ∂ ε = ln λ 2 λ 2 − ε 2 > 0 for λ > ε > 0, w e ha v e g ( ε, λ ) − g ( − ε, λ ) > 0 for any ε ∈ (0 , λ ). S ince su c h argumen ts hold for arbitrary λ > 0, w e can conclude that g ( ε, λ ) > g ( − ε, λ ) for λ > ε > 0. ✷ Lemma 5 L et 0 < ε < 1 . Then, Pr n b λ ≤ λ (1 − ε ) o ≤ exp ( n g ( − ελ, λ )) and g ( − ελ, λ ) is mono- tonic al ly de cr e asing with r esp e ct to λ > 0 . Pro of . Letting K = P n i =1 X i , θ = nλ and r = nλ (1 − ε ) and making u se of Lemma 1, for 0 < ε < 1, w e ha v e Pr n b λ ≤ λ (1 − ε ) o = Pr { K ≤ r } ≤ e − θ  θ e r  r ≤ exp ( n g ( − ελ, λ )) , where g ( − ελ, λ ) = [ − ε − (1 − ε ) ln(1 − ε ) ] λ, whic h i s monotonica lly decreasing with resp ec t to λ > 0, since − ε − (1 − ε ) ln(1 − ε ) < 0 f or 0 < ε < 1. ✷ Lemma 6 L et ε > 0 . Th en, Pr n b λ ≥ λ (1 + ε ) o ≤ exp ( n g ( ελ, λ )) and g ( ελ, λ ) is monotonic al ly de cr e asing with r esp e ct to λ > 0 . 3 Pro of . Letting K = P n i =1 X i , θ = n λ and r = nλ (1 + ε ) and m aking us e of Lemma 1, for ε > 0, w e ha v e Pr n b λ ≥ λ (1 + ε ) o ≤ exp ( n g ( ελ, λ )) where g ( ελ, λ ) = [ ε − (1 + ε ) ln(1 + ε ) ] λ, whic h is monotonically decreasing with r esp ect to λ > 0, since ε − (1 + ε ) ln (1 + ε ) < 0 for ε > 0. ✷ W e are no w in a p osition to pro v e the theorem. It suffices to sho w Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o < δ for n satisfying (1). It can sho wn that (1) is equiv alent to exp( n g ( ε a , ε a )) < δ 2 . (2) W e shall consider f our cases as follo ws. Case (i) : 0 < λ < ε a ; Case (ii ): λ = ε a ; Case (ii i): ε a < λ ≤ ε a ε r ; Case (iv) : λ > ε a ε r . In Case (i), w e ha v e Pr { b λ ≤ λ − ε a } = 0 and Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o = Pr n    b λ − λ    ≥ ε a o = Pr { b λ ≤ λ − ε a } + Pr { b λ ≥ λ + ε a } = Pr { b λ ≥ λ + ε a } . By Lemma (3), Pr { b λ ≥ λ + ε a } ≤ exp( n g ( ε a , λ )) ≤ exp( n g ( ε a , ε a )) < δ 2 . Hence, Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o < δ 2 < δ. In Case (ii), w e h a v e Pr { b λ ≤ λ − ε a } = Pr { b λ = 0 } and Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o = Pr n    b λ − λ    ≥ ε a o = Pr { b λ ≤ λ − ε a } + Pr { b λ ≥ λ + ε a } = Pr { b λ = 0 } + Pr { b λ ≥ λ + ε a } . 4 Noting that ln 2 < 1, w e can sh o w that − ε a < g ( ε a , ε a ) and hence Pr { b λ = 0 } = Pr { X i = 0 , i = 1 , · · · , n } = [Pr { X = 0 } ] n = e − nλ = e − n ε a < exp( n g ( ε a , ε a )) < exp  n g  ε a , ε a ε r  < δ 2 where the second inequalit y follo ws from Lemma (3 ). Hence, Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o < δ 2 < δ. In Case (iii), b y Lemma (2), Lemma (3) and Lemma (4), we hav e Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o = Pr { b λ ≤ λ − ε a } + Pr { b λ ≥ λ + ε a } ≤ exp( n g ( − ε a , λ )) + exp( n g ( ε a , λ )) < exp  n g  − ε a , ε a ε r  + exp  n g  ε a , ε a ε r  < 2 exp  n g  ε a , ε a ε r  < δ. In Case (iv), b y L emm a (5), L emma (6) and Lemma (4), w e ha v e Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o = Pr n    b λ − λ    ≥ ε r λ o = Pr { b λ ≤ (1 − ε r ) λ } + Pr { b λ ≥ (1 + ε r ) λ } ≤ exp( n g ( − ε r λ, λ )) + exp( n g ( ε r λ, λ )) < exp  n g  − ε a , ε a ε r  + exp  n g  ε a , ε a ε r  < 2 exp  n g  ε a , ε a ε r  < δ. Therefore, w e hav e s h o wn Pr n    b λ − λ    ≥ ε a &    b λ − λ    ≥ ε r λ o < δ for all ca ses. This completes the pro of of Theorem 1. References [1] X. Ch en , “Exact co mp utation of minim um sample size for estimat ion of P oisson parameters,” arXiv:0707 .2116 v1 [math.ST], July 2007. 5 [2] Chern off, H. (1952). A measur e of asymptotic efficiency for tests of a hyp othesis based on the su m of observ atio ns . An n. Math. Statist. 23 493–507. [3] M. M. Desu and D. Ragha v arao, Samp le Size Metho dolo gy , Academic Press, 19 90. 6