Estimation of a quantile in a mixture model of exponential distributions ...

Estimation of a quantile in a mixture model of exponential distributions with unknown location and scale parameter

Constantinos Petropoulos Department of Mathematics University of the Aegean 83 200 Karlovassi, Samos, Greece e-mail: [email protected]

Abstract Estimation of a quantile in a mixture model of exponential distributions is considered. For quadratic loss and specified extreme quantiles, better estimators than the best affine equivariant procedure are established. In particular, improved estimators for a quantile of an Exponential-Inverse Gaussian distribution and the multivariate Lomax distribution with unknown location and scale parameters are derived.

AMS 2000 subject classifications: 62C99, 62F10, 62H12 Key words and phrases: decision theory, Stein’s estimator of a scale parameter, quantile estimation, mixture of exponential distributions, Exponential-Inverse Gaussian distribution, multivariate Lomax distribution.

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1. Introduction In this paper we consider the data set X = (X1 , X2 , . . . , Xn ), n ≥ 2, where, given

˜

τ > 0, X1 , X2 , . . . , Xn are i.i.d. with common exponential distribution E(µ, σ/τ ). Thus, the unconditional joint density of X1 , X2 , . . . , Xn is ( ) Z ∞ n n τ X τ exp − (xi − µ) I[µ,+∞) (x(1) ) dG(τ ), (1.1) f (x1 , x2 , . . . , xn ; µ, σ) = σn σ i=1 0 where x(1) = min{x1 , x2 , . . . , xn }, I(a,b) (·) is the indicator function and µ ∈ R, σ > 0 are unknown. The distribution of the mixing parameter τ , G(·), is assumed to be known. The model (1.1) is called a mixture model of exponential distributions. It was first proposed by Lindley and Singpurwalla (1986) in assessing the reliability of a system of components and was further studied by Nayak (1987). In the mixture model (1.1) we are adopting the idea that the Xi ’s, i.e. the lifetimes of the components, are dependent which, as argued in Lindley and Singpurwalla (1986), is quite plausible because of the influence of the common environment shared by all components. The uncertainity of the common enviroment is described by the random variable τ > 0 with a distribution function G(·). Petropoulos and Kourouklis (2005) investigated estimation of σ in the model (1.1). They showed that the best affine equivariant (b.a.e.) estimator of σ is inadmissible under squared error loss and Stein’s loss. For specific forms of G(·) in (1.1), special distributions are derived. Indeed, two of these are of particular interest in actuarial applications and survival analysis. First, when G(·) is an Inverse Gaussian IG(γ, λ) distribution with density √ ½ ¾ λ (x − γ)2 λ √ exp − 2 I(0,+∞) (x), λ, γ > 0, 2γ x 2πx3 then (1.1) becomes

2

 λ

 v u n  u1 2 X t exp −λ + (xi − µ) γ  γ 2 σλ i=1 f2 (x1 , x2 , . . . , xn ; µ, σ) = × Ã !n/2 n X 1 2 σn + (xi − µ) 2 γ σλ i=1  v −i u n−1 n X X 1 2 (n − 1 + i)!  u 2λt 2 + × (xi − µ) I[µ,+∞) (x(1) ) i!(n − 1 − i)! γ σλ i=1 i=1

(1.2)

which is called Exponential - Inverse Gaussian (E-IG) distribution with location parameter µ and scale parameter σ. The E-IG distribution has been derived by Bhattacharya and Kumar (1986) in reliability modelling. Also, bivariate and multivariate versions of this distribution have been studied by Whitmore and Lee (1991) and Al-Mutairi (1997). This model can be used in actuarial applications, because of some useful properties of the density (see, Hesselager et al. 1998) and can be seen as a standard random effect exponential regression and hence it may have useful application to survival analysis, too. See, for example, the paper by Frangos and Karlis (2004) where the E-IG model is fitted to data from a large Greek insurance company, concerning the size of car accident claims. Another useful case is when G(·) is a Gamma G(a, 1) distribution, then the joint density of X1 , X2 , . . . , Xn in (1.1) is f1 (x1 , x2 , . . . , xn ; µ, σ) =

1 Γ(n + a) Ã !n+a I[µ,+∞) (x(1) ) n n Γ(a)σ 1X 1+ (xi − µ) σ i=1

(1.3)

which is called multivariate Lomax distribution with location parameter µ and scale parameter σ, or Mardia’s multivariate Pareto II distribution, see Arnold (1983) and Kotz et al. (2000) for details. Several probabilistic properties of this distribution are reported and its usefulness in reliability theory is indicated in Arnold (1983) and Nayak (1987). The model (1.3) is a multivariate generalization of the univariate Lomax (Pareto µ ¶−1−a a x−µ II) distribution with density , x > µ, which is a useful model in 1+ σ σ a wide variety of socioeconomic as well as lifetime contexts and has been used in the analysis of income data and business failure data (Lomax 1954). The univariate 3

Lomax has also been recommended by Bryson (1974) as a heavy tail alternative to the exponential distribution. Based on observations X1 , X2 , . . . , Xn in the framework (1.1), our aim is the estimation of the linear function θ = µ + κσ, for given κ > 0, from a decision theoretic point of view. When κ = −mτ (p), 0 < p 6 1, where mτ (·) is the moment generating function of the random variable τ > 0, then θ is the 100(1 − p)th quantile of the marginal distribution of Xi . Note that for p = 1, the problem reduces to the estimation of the location parameter µ whereas the case p = mτ (−Eτ −1 ) corresponds to the estimation of the common mean of Xi ’s. Previous work on quantile estimation concerns the normal distribution (Zidek 1971) and the exponential distribution E(µ, σ). In the latter case Rukhin and Strawderman (1982) and Rukhin and Zidek (1985) showed the inadmissibility of the best 1 1 affine equivariant estimator for 0 6 κ < and κ > 1 + , whereas Rukhin (1986) n n 1 1 6 κ 6 1 + . Analogous inadmissibility was able to establish admissibility for n n results were obtained by Elfessi (1997) based on a doubly censored sample from an exponential distribution, while Petropoulos and Kourouklis (2001) generalized results in Rukhin and Strawderman (1982) to a strictly convex loss. Moreover, they proposed a new minimax estimator for µ + κσ under quadratic loss. Recently, Petropoulos and Kourouklis (2004) considered estimation of µ + κσ in the multivariate Lomax distribution (1.3). Under quadratic loss and for specified “small” and “large” values of κ, they established that the b.a.e. procedure is inadmissible by constructing a better estimator. In the present work and in Section 2, under quadratic loss, we construct better estimators than the b.a.e. estimator of θ = µ + κσ in the mixture µ model ¶ (1.1), 1 1 Eτ −2 Eτ −2 for extreme values of κ, namely 0 6 κ < and κ > 1 + . n(n + 1) Eτ −1 n Eτ −1 As a special case, for κ = 0, it is shown that the b.a.e. estimator of the location parameter µ is inadmissible. In Sections 3 and 4, we give some examples for specific mixture models, i.e. the EIG distribution and the multivariate Lomax distribution. In each case, we improve upon the b.a.e. estimator of the quantile θ = µ + κσ, for the above mentioned

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“small” and “large” values of κ. In the case of the multivariate Lomax distribution, for “small” values of κ, we reproduce the improved estimator in Petropoulos and Kourouklis (2004). Interestingly, however, in the case of “large” κ, our approach is setting off a new improved estimator, i.e., different than that in Petropoulos and Kourouklis (2004).

2. Mixture of Exponential Distributions The (minimal) sufficient statistic in the model (1.1) is (X, S), where X = X(1) = n X min{X1 , X2 , . . . , Xn } and S = (Xi − X(1) ). Conditionally on τ , X and S are i=1

independent, with τX |τ ∼ E σ

µ

τµ 1 , σ n

¶ and

τS |τ ∼ Gamma(n − 1, 1). σ

(2.1)

The problem of estimating θ = µ + κσ under the loss (d − µ − κσ)2 /σ 2 is invariant under the affine group of transformations (X, S) → (cX + b, cS) and the equivariant estimators are of the form δ = X + cS, where c is a real constant. A particular member of this class of estimators is the minimum variance unbiased estimator corresponding to c = (κ(Eτ −1 )−1 − 1/n)/(n − 1). The risk of δ, as a function of c, is minimized at c0 = −

E[(X − µ − κσ)S] . ES 2

Upon conditioning on τ and using (2.1) it can be showed that µ ¶ 1 1 Eτ −1 . c0 = κ −2 − Eτ n n

(2.2)

So, the best affine equivariant estimator of θ is δ0 = X + c0 S.

(2.3)

Our aim is to provide Stein(1964)-type estimators which improve upon δ0 in (2.3). To this end we study the more general class of scale equivariant estimators of the form δ = φ(W )S, 5

(2.4)

X and φ(·) is a measurable function. Notice that the b.a.e. estimator S belongs in the above class, corresponding to where W =

φ0 (W ) = W + c0 . For “large” values of κ, Theorem 2.1, establishes the inadmissibility of δ0 by deriving a Stein(1964)-type estimator of the form (2.4) which is better than δ0 . Theorem 2.1. Assume that the following conditions hold. Eτ r−1 Eτ −1 (A1) − 6 c∗ r, for some c∗ > 0, ∀ r ∈ {0, 1, 2, . . .} Eτ r−2 Eτ −2 and R ∞ n −[(1+nw)s−nµ]τ τ e dG(τ ) (A2) 0 R ∞ n −(1+nw)sτ is decreasing in s. τ e dG(τ ) 0 µ ¶ n+1 1 Eτ −2 κ Eτ −1 Eτ −3 ∗ n(n + 2) Eτ Let κ > 1 + and φ (W ) = W + (1+nW )+κc W. 1 n Eτ −1 n + 1 Eτ −2 n + 1 Eτ n Eτ −2 Then the risk of the estimator   min{φ1 (W ), φ0 (W )}S , W>0 δ1 =  δ0 = φ0 (W )S , otherwise is strictly smaller than that of δ0 in (2.3). Proof. The risk of δ in (2.4) depends on (µ, σ) only through

µ , so without loss σ

of generality one can take σ = 1 and write R(δ; µ) = Eµ [Eµ {(φ(W )S − µ − κ)2 |W = w}].

(2.5)

The conditional expectation Eµ {(cS − µ − κ)2 |W = w} is minimized with respect to c, at c(µ; w) = (µ + κ)

Eµ (S|W = w) = (µ + κ)ψ(µ; w). Eµ (S 2 |W = w)

(2.6)

The key in Stein’s (1964) technique is to find an upper bound for c(µ; w), as a function of µ for each w > 0. Using (2.1), we observe, given τ > 0, that the (conditional) density of S|W = w is propotional to n µo τ n sn−1 enτ µ e−(1+nw)τ s , s > max 0, . w

6

(2.7)

Consider, first, µ 6 0 and fix w > 0, then from (2.6) and (2.7) we get R ∞ R ∞ n n −[(1+nw)s−nµ]τ s τ e dsdG(τ ) Eµ (S|W = w) ψ(µ; w) = . = R ∞0 R ∞0 n+1 n −[(1+nw)s−nµ]τ 2 Eµ (S |W = w) s τ e dsdG(τ ) 0 0 In the above expression, we make a change of variable from τ s to s and conclude that

R ∞ −1 nτ µ τ e dG(τ ) 1 + nw ψ(µ; w) = R0∞ −2 nτ µ . τ e dG(τ ) n + 1 0

(2.8)

Using Lemma 5.1 (see Appendix), one has that R ∞ −1 nτ µ R ∞ −1 τ e dG(τ ) τ dG(τ ) 0 R∞ R0∞ < . τ −2 enτ µ dG(τ ) τ −2 dG(τ ) 0 0

(2.9)

So, because of (2.9), (2.8) becomes Eτ −1 1 + nw . Eτ −2 n + 1

ψ(µ; w)
0 and fix, again, w > 0. Then, c(µ; w) = µψ(µ; w) + κψ(µ; w).

(2.12)

Our goal now is to bound each term of the right hand side of (2.12) separately. First, we notice that R R ∞ ∞ ψ(µ; w) =

and hence

0 R∞ 0

µ w

R∞

sn τ n e−[(1+nw)s−nµ]τ ds dG(τ )

n+1 τ n e−[(1+nw)s−nµ]τ ds dG(τ ) µ s R w∞ R ∞ n n −[(1+nw)s−nµ]τ ds dG(τ ) µ s τ e w 0 < R ∞ R ∞w µ = µ sn τ n e−[(1+nw)s−nµ]τ ds dG(τ ) µ 0 w w

µψ(µ; w) < w.

(2.13)

Also, making a change of variable from τ s to s and expanding enµτ in a Taylor series, we can write, Z +∞ X (nµ)k ψ(µ; w) =

k=0 +∞ X k=0

k!

(nµ)k k!

∞

Z

0

Z

∞ 0

Z

∞ τµ w

∞ τµ w

7

sn τ k−1 e−(1+nw)s ds dG(τ ) .

sn+1 τ k−2 e−(1+nw)s ds dG(τ )

(2.14)

By Lemma 5.2 we see that R ∞ k−1 Z ∞Z ∞ Z ∞Z ∞ τ dG(τ ) 1 + nw n k−1 −(1+nw)s 0 s τ e ds dG(τ ) 6 R ∞ k−2 sn+1 τ k−2 e−(1+nw)s ds dG(τ ) τµ τµ n + 1 τ dG(τ ) 0 0 0 w w and hence (2.14) leads us to the conclusion that ½ R ∞ k−1 ¾Z ∞Z ∞ +∞ X τ dG(τ ) (nµ)k 0 R∞ sn+1 τ k−2 e−(1+nw)s ds dG(τ ) k−2 dG(τ ) τµ k! τ 0 1 + nw k=0 0 w ψ(µ; w) 6 . Z Z +∞ n+1 X (nµ)k ∞ ∞ sn+1 τ k−2 e−(1+nw)s ds dG(τ ) τµ k! 0 w k=0 (2.15) Because of condition (A1), (2.15) becomes R∞ 1 + nw 0 τ −1 dG(τ ) 1 + nw ∗ R∞ ψ(µ; w) 6 + c ψ1 (µ; w) n + 1 0 τ −2 dG(τ ) n+1 +∞ X (nµ)k

where, ψ1 (µ; w) =

k=0 +∞ X k=0

k! (nµ)k k! +∞ X

= nµ

k=0 +∞ X k=0

But,

+∞ X k=0

Z

∞

Z

k 0

Z

∞

Z

∞ τµ w

∞ τµ w

0

(nµ)k k! (nµ)k k!

Z

sn+1 τ k−2 e−(1+nw)s ds dG(τ )

sn+1 τ k−2 e−(1+nw)s ds dG(τ ) ∞

Z

∞

∞

Z

∞ τµ w

0

sn+1 τ k−1 e−(1+nw)s ds dG(τ )

τµ w

0

Z

(2.16)

. sn+1 τ k−2 e−(1+nw)s ds dG(τ )

(nµτ )k = enµτ and the above expression becomes, k! R∞ ψ1 (µ; w) = nµ

0 R∞ 0

τ −1 enµτ τ −2 enµτ

R∞ τµ w

R∞ τµ w

sn+1 e−(1+nw)s ds dG(τ ) sn+1 e−(1+nw)s ds dG(τ )

.

(2.17)

. From (2.12), (2.13), (2.16) and Lemma 5.3 (which in its proof, condition (A2) has been used), it follows that c(µ; w) < w +

κ Eτ −1 n(n + 2) ∗ Eτ n+1 Eτ −3 (1 + nw) + κc w = φ1 (w) n + 1 Eτ −2 n+1 Eτ n Eτ −2

for any µ and w > 0.

µ

(2.18)

¶ 1 Eτ −2 , it is easily verified that φ1 (w) < φ0 (w) on Furthermore, for κ > 1 + n Eτ −1 a set A of positive w-values having positive probability for all µ. Since Eµ {(cS − 8

µ − κ)2 |W = w} is strictly increasing in c for c > c(µ; w), by (2.18), we have Eµ {(φ1 (w)S − µ − κ)2 |W = w} < Eµ {(φ0 (w)S − µ − κ)2 |W = w}, w ∈ A, which in connection with (2.5) implies R(δ1 ; µ) < R(δ0 ; µ) for all µ. The next results establishes the inadmissibility of δ0 in (2.3) for “small” values of κ. 1 Eτ −2 . Then the risk of the estimator n(n + 1) Eτ −1  ½ ¾ 1 n + 2   max W, φ0 (W ) S , W < − n+1 n δ2 =   δ = φ (W )S , otherwise, 0 0

Theorem 2.2. Let 0 6 κ
0, we observe that the (conditional) density of S|W = w is propotional to τ n sn−1 enτ µ e−(1+nw)τ s , 0 < s
0 and hence from (2.6) and (2.24), we obtain that n+2 w. n+1 µ ¶ 1 Eτ −1 n+1 1 Now, for w < − , φ2 (w) > φ0 (w) holds for w > κ −2 − whereas, n n Eτ n 1 Eτ −2 provided that 0 6 κ < , we have n(n + 1) Eτ −1 c(µ; w) = (µ + κ)ψ(µ; w) > µψ(µ; w) >

c(µ; w) > φ2 (w) > φ0 (w), (2.25) µ µ ¶ ¶ n+1 Eτ −1 1 1 for w ∈ B = κ −2 − ,− . Finally, (2.25) and the fact that n Eτ n n Eµ {(cS − µ − κ)2 |W = w} is strictly decreasing for c < c(µ; w), ensure that Eµ {(φ2 (w)S − µ − κ)2 |W = w} < Eµ {(φ0 (w)S − µ − κ)2 |W = w}, w ∈ B, which in turn entails R(δ2 ; µ) < R(δ0 ; µ). This completes the proof of the theorem.

3. Exponential-Inverse Gaussian Model In this section we consider the framework in (1.1), taking the mixing distribution G(·) to be the Inverse Gaussian IG(γ, λ), in other words we study the problem of estimating θ = µ + κσ in the Exponential-Inverse Gaussian model. In this case 1 1 1 3 3 Eτ −1 = + and Eτ −2 = 2 + 2 + (see Seshadri 1993, p.47) so that the γ λ γ λ γλ b.a.e. estimator of θ in (2.2) is δ0 = X + c0 S, where µ ¶ 1 1 λγ(λ + γ) c0 = − κ 2 . (3.1) n λ + 3γ(λ + γ) n For “large” values of κ, an improved estimator of θ is given in the following result.

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µ

¶· 2 ¸ 1 λ + 3γ(λ + γ) κ λγ(λ + γ) Theorem 3.1. Let κ > 1 + , φ1 (W ) = W + 2 n λγ(λ + γ) n + 1 λ + 3γ(λ + γ) n ³ ´ X (n + i)! γ i · ¸ i!(n − i)! 2λ n(n + 2) 2γ 2 5 λ(λ + γ) i=0 (1+nW )+ κ γ n−1 + 2 W and 2 n+1 λ X (n − 1 + i)! ³ γ ´i λ λ γ + 3γ (λ + γ) i!(n − 1 − i)! 2λ i=0 φ0 (W ) = W + c0 with c0 as in (3.1). Then the risk of the estimator   min{φ (W ), φ (W )}S , W > 0 1 0 δ1 =  δ = φ (W )S , otherwise, 0 0 is strictly smaller than that of δ0 . Proof. We will prove the theorem provided (A1) and (A2), in Theorem 2.1, hold for the E-IG model. Because of the relation (2.12) in Seshadri (1993, p.47), we have that when a random variable X ∼ IG(γ, λ), then λ EX s = λEX s−2 + (2s − 3)EX s−1 . 2 γ

(3.2)

Setting, in (3.2), s = r − 1 we get λ r−1 Eτ = λEτ r−3 + (2r − 5)Eτ r−2 , so that γ2 λ Eτ r−1 Eτ r−3 = λ + 2r − 5. γ 2 Eτ r−2 Eτ r−2

(3.3)

Eτ −3 λ Eτ −1 = λ − 5. γ 2 Eτ −2 Eτ −2

(3.4)

For r = 0 (3.3) gives

Combining (3.3) and (3.4), we have that µ ¶ µ r−3 ¶ λ Eτ r−1 Eτ −1 Eτ Eτ −3 − =λ − + 2r. γ 2 Eτ r−2 Eτ −2 Eτ r−2 Eτ −2

(3.5)

Using Lemma 5.1, we obtain Eτ −3 Eτ r−3 6 . Eτ r−2 Eτ −2 and thus (3.5) yields 2γ 2 2γ 2 Eτ r−1 Eτ −1 ∗ − 6 r. Hence (A1) holds with c = . Eτ r−2 Eτ −2 λ λ 11

(3.6)

The next step is to verify (A2), i.e., R ∞ n −[(1+nw)s−nµ]τ τ e dG(τ ) B(s) = 0 R ∞ n −(1+nw)sτ is decreasing in s. τ e dG(τ ) 0 Here, τ ∼ IG(γ, λ), so B(s) =

e

− aλ

µ

− aλ

e

0

Eµ Y n E0 Y n

−λ

=e

1 − a1 aµ 0

Eµ Y n , Y ∼ IG(aµ , λ), E0 Y n

·

2 1 2 where aµ = a(s; µ) = (1 + nw)s + 2 − nµ λ γ λ We proceed in showing that e

−λ

d ³ −λ( a1µ − a1 ) ´ −λ( 1 − 1 ) 0 e = e aµ a0 ds

(3.7)

¸−1/2 and a0 = a(s; 0).

1 − a1 aµ 0

is decreasing in s. We have, ¸ · 1 daµ 1 da0 (−λ) (−1) 2 − (−1) 2 . aµ ds a0 ds

(3.8)

Also,

daµ 1 = − (1 + nw) a3µ , ds λ da0 1 = − (1 + nw) a30 . ds λ From (3.8) and (3.9), we conclude that

(3.9)

d ³ −λ( a1µ − a1 ) ´ −λ( 1 − 1 ) 0 = −e aµ a0 (1 + nw)(aµ − a0 ) < 0, e ds that is −λ( a1 − a1 )

e

µ

0

↓ s.

(3.10)

On the other hand, n−1 X n

Eµ Y = E0 Y n

µ

aµ a0

¶n

k=0 n−1 X

Ak akµ ,

(3.11)

Ak ak0

k=0 n−1 X

where Ak =

(n − 1 + k)! aµ 1 k=0 . Clearly, is decreasing in s and so is n−1 k k!(n − 1 − k)! (2λ) a0 X

upon using Lemma 5.4. Hence from (3.11) it follows that Eµ Y n ↓ s. E0 Y n 12

Ak akµ Ak ak0

k=0

(3.12)

Finally, combining (3.7), (3.10) and (3.12) we establish that B(s) is a decreasing function of s, that is (A2) holds and the proof of the theorem is complete. For “small” values of κ application of Theorem 3.2 yields the following. · 2 ¸ λ + 3γ(λ + γ) 1 Theorem 3.2. Let 0 6 κ < and φ0 (W ) = W + c0 with n(n + 1) λγ(λ + γ) c0 as in (3.1). Then the risk of the estimator,  ½ ¾ n + 2 1   max W, φ0 (W ) S , W < − n+1 n δ2 =   δ , otherwise, 0 is nowhere larger than that of δ0 .

4. Multivariate Lomax Model In this section, we take the mixing distribution in the model (1.1) to be Gamma(a, 1), so we study the problem of estimating θ = µ + κσ in the multivariate Lomax model. 1 1 In this case, Eτ −1 = and Eτ −2 = , so that the b.a.e. estimator a−1 (a − 1)(a − 2) of θ is δ0 = X + c0 S, where ¶ µ 1 1 c0 = κ(a − 2) − . (4.1) n n Estimation of θ has been studied by Petropoulos and Kourouklis (2004), who µ ¶ pro1 1 posed better estimators than δ0 , mentioned above. Namely, for κ > 1 + , n a−2 they proved that the risk of the estimator  ½ µ ¶ ¾   min W + κ(1 + nW ) a − 2 + n(n + a) W, W + c0 S , W > 0 n+1 n+1 δ1 =   δ , otherwise, 0 (4.2) 1 , is strictly smaller than that of δ0 , where c0 as in (4.1), and for 0 6 κ < n(n + 1)(a − 2) they showed that the risk of the estimator  ½ ¾ n + 2 1   max W, φ0 (W ) S , W < − n+1 n δ2 = (4.3)   δ , otherwise, 0 13

is nowhere larger than that of δ0 . In Theorem 4.3 a Stein (1964)-type estimator of θ = µ + κσ is produced which is better than that of δ0 , for “large” values of κ, but different from δ1 in (4.2). µ ¶ 1 1 κ Theorem 4.3. Let κ > 1 + , φ1 (W ) = W + (a − 2)(1 + nW ) + n a−2 n+1 ¶ µ nκ n+a 1 1 (n + 2) W and φ0 (W ) = W + c0 , c0 = κ(a − 2) − . Then the risk n+1 a−3 n n of the estimator   min{φ (W ), φ (W )}S , W > 0 1 0 δ3 =  δ = φ (W )S , otherwise, 0

0

is strictly smaller than that of δ0 . Proof. As in Theorem 3.1, we must show that the Conditions (A1) and (A2) in Theorem 2.1 hold for the mixing distribution G(τ ) ≡ Gamma(a, 1). R ∞ r−1 R ∞ −1 τ dG(τ ) τ dG(τ ) 0 It is straightforward that, R ∞ r−2 − R0∞ −2 = r, that is (A1) holds τ dG(τ ) τ dG(τ ) 0 0 as equality for c∗ = 1. In respect of (A2), R ∞ n −[(1+nw)s−nµ]τ ¸n+a · τ e dG(τ ) 1 + (1 + nw)s 0R , which, obviously, is a = ∞ n −(1+nw)sτ 1 + (1 + nw)s − nµ τ e dG(τ ) 0 decreasing function in s, i.e. (A2) also holds. For “small” values of κ, we give in Theorem 4.4 a better estimator of θ than δ0 , which coincides with δ2 in (4.3). The proof is an immediate consequence of Theorem 2.2. 1 1 1 and φ0 (W ) = W +c0 , c0 = Theorem 4.4. Let 0 6 κ < n(n + 1) a − 2 n Then the risk of the estimator  ½ ¾   max n + 2 W, φ0 (W ) S , W < − 1 n+1 n δ4 =   δ = φ (W )S , otherwise, 0 0 is nowhere larger than that of δ0 .

14

µ ¶ 1 κ(a − 2) − . n

5. Appendix Lemma 5.1. Suppose that b1 (x) and b2 (x) are densities supported on the same subset of the real line and b2 (x)/b1 (x) is strictly increasing. If X is a random variable with density b1 (x) or b2 (x) and h(x) is strictly increasing, then Eb2 h(X) > Eb1 h(X) (provided expectations exist). Proof. It is given in Lehmann(1986, p.85). Lemma 5.2. For a = a(τ ) > 0 and t > 0, R ∞ k−1 R ∞ R ∞ n k−1 −ts R ∞ R ∞ n+1 k−2 −ts τ dG(τ ) t s τ e ds dG(τ ) 6 R0∞ k−2 s τ e ds dG(τ ). 0 a τ dG(τ ) n + 1 0 a 0 Proof. By Lemma 5.1, putting R∞ R∞ sn−1 e−ts 0 τ k−1 I[a,∞) dG(τ ) sn−1 e−ts 0 τ k−1 I[0,∞) dG(τ ) b1 (s) = R R ∞ n+1 k−2 −ts , b2 (s) = R R ∞ n+1 k−2 −ts and h(s) = s, s τ e ds dG(τ ) s τ e ds dG(τ ) a a it follows that, R ∞ R ∞ n k−1 −ts R ∞ R ∞ n k−1 −ts R ∞ k−1 s τ e ds dG(τ ) s τ e ds dG(τ ) τ dG(τ ) t 0 a 0 0 R∞R∞ 6 R ∞ R ∞ n+1 k−2 −ts = R0∞ k−2 , ∀a > 0. n+1 k−2 −ts s τ e ds dG(τ ) s τ e ds dG(τ ) τ dG(τ ) n + 1 0 a 0 0 0 (5.1) n + 2 Eτ n+1 Eτ −3 Lemma 5.3. ψ1 (µ; w) < nw . 1 + nw Eτ n Eτ −2 Proof.

R∞ ψ1 (µ; w) = nµ

0 R∞ 0

τ −1 enµτ τ −2 enµτ

R∞ τµ w

R∞ τµ w

sn+1 e−(1+nw)s ds dG(τ ) sn+1 e−(1+nw)s ds dG(τ )

Making a change of variable from s to τ s, (5.2) becomes R ∞ n+1 nµτ R ∞ n+1 −(1+nw)τ s e ds dG(τ ) τ e µ s 0 w < ψ1 (µ; w) = nµ R ∞ n nµτ R ∞ n+1 e−(1+nw)s ds dG(τ ) τ e µ s 0 w R ∞ n+2 R ∞ n+1 −[(1+nw)s−nµ]τ τ e dG(τ ) ds µ s 0 . < nw Rw ∞ n+1 R ∞ n −[(1+nw)s−nµ]τ τ e dG(τ ) ds µ s 0

.

(5.2)

(5.3)

w

Applying Lemma 5.1, it is seen Rthat R ∞ n+1 −[(1+nw)s−nµ]τ ∞ n+1 τ dG(τ ) τ e dG(τ ) 0 R0 ∞ R < and hence ∞ τ n e−[(1+nw)s−nµ]τ dG(τ ) τ n dG(τ ) 0 0 R ∞ n+1 Z ∞ Z ∞ τ dG(τ ) n+1 −[(1+nw)s−nµ]τ 0 R τ e dG(τ ) < ∞ n τ n e−[(1+nw)s−nµ]τ dG(τ ). (5.4) τ dG(τ ) 0 0 0 15

Combining (5.3) and (5.4), we deduce that R ∞ n+2 R ∞ n −(1+nw)τ s R ∞ n+1 τ e dG(τ ) ds τ dG(τ ) wµ s 0 R∞ R∞ ψ1 (µ; w) < nw R0 ∞ n . n+1 τ dG(τ ) τ n e−(1+nw)τ s dG(τ ) ds µ s 0 0

(5.5)

w

Another application of Lemma 5.1, (5.5) and Condition (A2) in Theorem 2.1 give the proof of the lemma. n X

Lemma 5.4. g(s) = ·

Ak akµ

k=0 n X

is decreasing in s, where Ak ak0

k=0

2 1 2 aµ = a(s; µ) = (1 + nw)s + 2 − nµ λ γ λ (n − 1 + k)! 1 and Ak = . k!(n − 1 − k)! (2λ)k

¸−1/2 , a0 = a(s; 0)

Proof. It suffices to show that g 0 (s) < 0. We have"that # # " n # " n # " n n X X X X 0 0 Ak kak−1 Ak akµ Ak ak0 − Ak kak−1 0 a0 µ aµ g 0 (s) =

k=0

k=1

" n X

k=0 #2

k=1

.

Ak ak0

k=0

The numerator of the above expression can be written as n n X n n X X X 0 k l−1 0 Ak akµ Al lal−1 Ak a0 Al laµ aµ − N (s) = 0 a0 , or

k=0 l=1

k=0 l=1

N (s) =

n n X X

£ ¤ 0 k l−1 0 lAk Al ak0 al−1 µ aµ − aµ a0 a0

(5.6)

k=0 l=1

Because of (3.9), (5.6) becomes N (s) = N1 (s) + N2 (s) where 2

N1 (s) = −c A0

n X

£ ¤ l+2 lAl al+2 µ − a0

(5.7)

¤ £ k l+2 a − a lAk Al ak0 al+2 µ 0 µ

(5.8)

l=1

and N2 (s) = −c

2

n X n X k=1 l=1

16

Obviously, N1 (s) in (5.7) is negative and if, every time we combine the term (l, k) with the term (k, l) in (5.8), we conclude that N2 (s) < 0. Hence N (s) < 0 and that completes the proof.

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