Some properties of the moment estimator of shape parameter for the gamma distribution

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1 Some properties of the moment estimator of shape parameter for the gamma distribution arxiv: v [math.st] 7 Aug 0 Piotr Nowak Mathematical Institute, University of Wrocław Pl. Grunwaldzki /4, Wrocław, Poland nowak@math.uni.wroc.pl Abstract Eact distribution of the moment estimator of shape parameter for the gamma distribution for small samples is derived. Order preserving properties of this estimator are presented. Keywords: moment estimator, stochastic ordering, student s ratio, dispersive ordering. 00 MSC: 60E5, 6E5. Introduction and preliminaries It is well known that the gamma distribution has wide application, for eample in meteorology to describe the distribution of rainfall. The moment estimators ˆα and ˆλ of the parameters α and λ of the gamma distribution with density f(;α,λ) = ep( /λ), > 0, α > 0,λ > 0 Γ(α)λ αα for a random sample X,...,X n are (.) ˆα = X and S ˆλ = S X, where X and S are sample mean and sample biased variance respectively. In this paper we are mainly interested in order preserving property of the estimator ˆα. In general, we say that the estimator ˆθ based on the samplex,...,x n from population with density f( ;θ) is increasing in θ with respect to the order if ˆθ ˆθ, whenever θ < θ, where ˆθ (ˆθ ) is the estimator based on sample from density f( ;θ ) (f( ;θ )). We shall consider when is one of the following orders: st, disp, *. For details and definitions of these stochastic orders we refer the reader to Shaked and Shanthikumar [5]. Nowak [3] proved the following theorem.

2 Theorem. The estimators ˆα and ˆλ are stochastically increasing respectively in α and λ. Deriving eact distributions of moment estimators ˆα and ˆλ is very tedious for n >. Now we find the distribution of ˆα for n =. First we prove the more general fact. Lemma. Suppose that X G(λ,α ) and X G(λ,α ), X and X are independent random variables. Then Z = S has density of the form X h(z;α,α ) = /B(α,α ) (α +α ) z / [ ( z) α (+ z) α +(+ z) α ( z) α ]. Corollary. Let X,X G(λ,α), Y E(), X, X and Z are independent random variables. Then (.) X +X (X +X ) = st (X +Y) +X (X +Y +X ). Problem. Equation (.) yields some interesting property of the eponential distribution. There arises a question: Does this equation characterize the eponential distribution? Corollary. For n = the statistics /ˆα has the beta distribution B(/,α). Hence ˆα has monotone likelihood ratio. Now we eamine dispersive ordering of the ˆα estimator. First we notice that /ˆα is not dispersive monotone in α. We conclude it from Theorem 3.B.4 of Shaked and Shanthikumar [5] since the support of the estimator / ˆα is the finite interval (0,n ). Theorem. The estimator /ˆα is not dispersively monotone in α. We now consider dispersive ordering of the estimator ˆα. From previous theorem it does not follow that ˆα is not dispersive monotone in α, though the function / is decreasing and conve for > 0. We should notice that Theorem 3.B.0(b) of Shaked and Shanthikumar [5] is not valid here. For eample, in the following theorem we prove that for n = the estimator ˆα is dispersively increasing in α. Theorem 3. For n = the estimator ˆα is dispersively increasing in α. Proof. Let f α denotes density of the estimator ˆα. We know that /ˆα has the beta distribution B(/,α), so f α () = B(/,α) 3/ ( /) α, >.

3 In order to prove that ˆα disp ˆα whenever α < α it suffices to prove that S (f α c (+) f α (+)) for every c > 0 with the sign sequence being,+, in the case of equality and ˆα st ˆα (see Theorem.6 and Remark. of Shaked [4]). Fi c > 0. When > c then f α c (+) f α (+) = B(/,α ) (+) 3/ ( /(+)) α [ A(+) 3/ ( c+) 3/ ( /( c+)) α /( /(+)) α ], where A = B(/,α )/B(/,α ). Thus S (f α c f α ) = S (h()), where h() = loga+3/log(+) 3/log( c+)+ +(α )log( /( c+)) (α )log( /(+)). Now we show that h has eactly one maimum when α > and the sign sequence is,+,. In the other case h is decreasing. After calculating we have that h () = 0 if and only if w()/((+)( c)( c+)) = 0, where w() = (α α +3c) +(α α +4c 4α c 3c )+c α c c +α c. So we must show that w has only one root in the interval (c, ) if α > and no roots when α (0,]. Define w() = w(+c). Therefore w() = ((α α )+3c) +((α α )+4c( α )+3c )+( α )(c+c ). It is easy to see that for α >, w has two different roots and that < 0 and for α (0,] w has no roots in (0, ). At the end we must show that the sign sequence is,+, when S (h) =. For α > lim c +h() =, lim h() = loga < 0. Combining these above facts with Theorem we end the proof. Remark. The above theorem does not hold in general for n 3, but direct calculating is impossible due to occurrence of hyper elliptic integrals. For eample, numerical calculations show, that for n = 3, ˆα disp ˆα if α = /5 and α = /4. Then G () F () has the local maimum at 0.7 and the local minimum at Deriving eact distribution when n = 3 is more tedious. We should add, that the estimator ˆα is closely related to the student statistic n X/S. The properties of this statistic was very intensively studied in the literature, especially under nonnormal conditions. First, we find joint distribution of the vector ( X,S). One can proof the following lemma, see for eample Craig []. 3

4 Lemma. If f is a density on (0, ), then the joint distribution ( X,S) for n = 3 is given by formula 8s +s s f( R )f( 3 +R )f( 3 R )d, 0 s /, [ F(,s) = 3 6s 8s 3 + ] +s s 3 f( R )f( 3 +R )f( 3 R )d, / s, where R = 6s 3( ). If we have the joint distribution ( X,S) it is easy to derive the distribution of the statistics T = X/S = ˆα from the formula f T (t) = uf(ut, u)du. Proposition. For n = 3 the cumulative distribution function of T for the eponential distribution with density f() = e, > 0 is given by { π 3, t, F(t) = ( ) t t + π +, t. 3 arcsin On the other hand it is not easy to derive in the general the distribution of T. For eample, if α (0,) above integrals can be calculated only by numerical methods. For n = 3, after a bit algebra we can also derive the distribution of T for the gamma distribution with shape parameter α =. Proposition. For n = 3 the cumulative distribution function oft for the gamma distribution with density g() = e, > 0 is given by 3) 0π(4t 7, t, G(t) = 4 ( ) t ( )+5πt(4t 3) 30t(4t 3)arcsin t 7 +, t. 5 From previous propositions it follows, that the statistic T is not monotone with respect to the star order, that is the function G F()/ is not monotone in > 0. To see it, let us calculate: ( ( )) ) ) G F() ) G F() 3) G F() = 0 = 6 = 0 0π+ 33 = = = 40π+ 0π(π 7 3) 7 3+8π π 89 ( π+600arccos 0 33 ( 79+00π 600arccos )

5 Since the star order is preserved under increasing function, we have the following corollary. Corollary 3. For n = 3 the estimator ˆα is not increasing in α with respect to the star order. The star order is closely related to the dispersive order, since X * Y iff logx disp logy. Using the same technique as in Theorem 3 we can proof the following proposition. Proposition 3. For n = the estimator ˆα is increasing in α with respect to the star order. Applying the property that for nonnegative random variables suchx st Y and X * Y implies the ordering X disp Y (see, for eample, Ahmed et al. []) we can also deduce from Theorem and Proposition 3 that ˆα is dispersively monotone for n =. References [] A. N Ahmed, A. Alzaid, J. Bartoszewicz, S. C. Kochar, Dispersive and superadditive ordering, Advances in Applied Probability 8, (986), [] Allen T. Craig, The simultaneous distribution of mean and standard deviation in small samples, The Annals of Mathematical statistics, Vol 3, No., 93, p [3] P. Nowak, Stochastic comparisons of moment estimators of gamma distribution parameters, to appear in Applicationes Mathematicae. [4] M. Shaked, Dispersive ordering of distributions, J. Appl. Prob. 9 (98), [5] M. Shaked, J. G. Shanthikumar, Stochastic Orders, Springer Verlag, New York,