Optimal rules and robust Bayes estimation of a Gamma scale parameter

Leila Golparver; Ali Karimnezhad; Ahmad Parsian

doi:10.1007/s00184-012-0407-7

Metrika

July 2013, Volume 76, Issue 5, pp 595–622 | Cite as

Optimal rules and robust Bayes estimation of a Gamma scale parameter

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Authors and affiliations

Leila Golparver
Ali Karimnezhad
Ahmad ParsianEmail author

Article

First Online: 21 August 2012

Received: 15 February 2011

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4 Citations

Abstract

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $cX\,{+}\,d$, when $X\sim \varGamma (\alpha ,\eta )$. In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.

Keywords

Admissible estimator Bayes estimator Minimax estimator Precautionary loss function Robust Bayes estimator

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Appendix

Proof of Theorem 1

(i)

When $c=0$ and $d>0$, the estimator $cX+d$ is admissible since it is the only estimator for which $R(\eta ,d)=0$, when $\eta =d$.
(ii)

In (10), we showed that the unique Bayes estimator of $\eta $ w.r.t. the proper prior (9) under the loss function (3) is
$$\begin{aligned} \delta _{\gamma ,g}(X)&= \frac{1}{\sqrt{(\alpha +\gamma )(\alpha +\gamma -1)}}\,X+\frac{g}{\sqrt{(\alpha +\gamma )(\alpha +\gamma -1)}}\\&= cX+d, \end{aligned}$$
provided $\alpha >1$.

When $\gamma >0,\,g>0$ and $\alpha >1$, $c$ varies between $0$ and $c^*$ and the constant $d$ is positive. This proves that $c\,X+d$ is admissible when $0 <c <c^{*}$ and $d>0$.

(iii)

Consider the sequence of conjugate prior distributions $\pi _{m,g}$, i.e., $I\varGamma (\frac{1}{m},\frac{1}{g})$-distributions. Using (10), the unique Bayes estimator of $\eta $ w.r.t. $\pi _{m,g}$ under the loss (3) is given by
$$\begin{aligned} \delta _{m,g}(X)=\frac{X+g}{\sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}} \end{aligned}$$
provided $\alpha +\frac{1}{m}>1$. Let $\delta (X)=\frac{X+g}{\sqrt{\alpha (\alpha -1)}},\,\alpha >1$. It is easy to show that the risk difference is
$$\begin{aligned} D_m&= R(\eta ,\delta )-R(\eta ,\delta _{m,g})\\&= E\left[\frac{\left(X+g\right)}{\eta \sqrt{\alpha (\alpha -1)}}\right]+E\left[\frac{\eta \sqrt{\alpha (\alpha -1)}}{\left(X+g\right)}\right]\\&-E\left[\frac{\left(X+g\right)}{\eta \sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}\right]-E\left[\frac{\eta \sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}{\left(X+g\right)}\right]\\&= \left(\alpha +\frac{g}{\eta }\right)\left\{ \frac{1}{\sqrt{\alpha (\alpha -1)}}-\frac{1}{\sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}\right\} \\&+\eta \left\{ \sqrt{\alpha \alpha -1)}-\sqrt{\left(\alpha +\frac{1}{m}\right)\left(\alpha +\frac{1}{m}-1\right)}\right\} \,E\left[\frac{1}{X+g}\right]\\&= \eta \,c_{\alpha ,m}\int \limits _{0}^{\infty }\frac{\eta ^{-\alpha }}{\varGamma (\alpha )}\frac{t^{\alpha -1}}{t+g}\,e^{-t/\eta }\,dt+\left(\alpha +\frac{g}{\eta }\right)d_{\alpha ,m} \end{aligned}$$
where
$$\begin{aligned} c_{\alpha ,m}&= \sqrt{\alpha (\alpha -1)}-\sqrt{\left(\alpha +\frac{1}{m}\right)\left(\alpha +\frac{1}{m}-1\right)}\,,\\ d_{\alpha ,m}&= \frac{1}{\sqrt{\alpha (\alpha -1)}}-\frac{1}{\sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}.\\ \end{aligned}$$
Hence
$$\begin{aligned} \varDelta _m&= r(\pi _{m,g},\delta )-r(\pi _{m,g},\delta _{m,g})\\&= \int \limits _{0}^{\infty }D_m\pi _{m,g}(\eta )d\eta \\&= c_{\alpha ,m}\int \limits _{0}^{\infty }\eta \left[\int \limits _{0}^{\infty }\frac{\eta ^{-\alpha }}{\varGamma (\alpha )}\frac{t^{\alpha -1}}{t+g}\,e^{-t/\eta }\,dt\right]\frac{g^\frac{1}{m}}{\varGamma (\frac{1}{m})}\eta ^{-\frac{1}{m}-1}\,e^{-g/\eta }\,d\eta \\&+\left(\alpha +g\int \limits _{0}^{\infty }\frac{1}{\eta }\,g^{\frac{1}{m}}\eta ^{-\frac{1}{m}-1}e^{-g/\eta }\frac{1}{\varGamma (\frac{1}{m})}\,d\eta \right)d_{\alpha ,m}\\&= c_{\alpha ,m}\,\frac{g^\frac{1}{m}}{\varGamma (\frac{1}{m})}\frac{1}{\varGamma (\alpha )}\int \limits _{0}^{\infty }\frac{t^{\alpha -1}}{t\!+\!g}\left[\int \limits _{0}^{\infty }\eta ^{-\alpha -\frac{1}{m}}\,e^{-(g+t)/\eta }\,d\eta \right]\,dt\!+\!\left(\alpha \!+\!\frac{1}{m}\right)d_{\alpha ,m}\\&= c_{\alpha ,m}\,g^\frac{1}{m}\frac{\varGamma (\alpha +\frac{1}{m}-1)}{\varGamma (\frac{1}{m})\varGamma (\alpha )}\int \limits _{0}^{\infty }\frac{t^{\alpha -1}}{\left(t+g\right)^{\alpha +\frac{1}{m}}}\,dt+\left(\alpha +\frac{1}{m}\right)d_{\alpha ,m}\\&= c_{\alpha ,m}\,\frac{\varGamma (\alpha \!+\!\frac{1}{m}\!-\!1)}{\varGamma (\frac{1}{m})\varGamma (\alpha )}\int \limits _{0}^{1}z^{\alpha -1}\left(1\!-\!z\right)^{\frac{1}{m}-1}dz\!+\!\left(\alpha \!+\!\frac{1}{m}\right)d_{\alpha ,m}\quad \left(z=\frac{t}{t+g}\right)\\&= \frac{1}{\alpha +\frac{1}{m}-1}\,c_{\alpha ,m}+\left(\alpha +\frac{1}{m}\right)d_{\alpha ,m}. \end{aligned}$$
Now since $c_{\alpha ,m}$ and $d_{\alpha ,m}$ tend to zero as $m \longrightarrow \infty $, we reach in $\lim _{m\longrightarrow \infty }\varDelta _m=0$. This guarantees the admissibility of $\delta (X)=c^*X +c^* g$, when $\alpha >1$.
(iv)

Consider the sequence of conjugate prior distribution $\pi _{m,m}\equiv \pi _m$, i.e., $I\varGamma (\frac{1}{m},\frac{1}{m})$-distributions. Using (10), the unique Bayes estimator of $\eta $ w.r.t. $\pi _m$ under the loss (3) is given by
$$\begin{aligned} \delta _{m}(X)=\frac{X+\frac{1}{m}}{\sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}},\quad \alpha +\frac{1}{m}>1. \end{aligned}$$
Let $\delta ^*(X)=\frac{X}{\sqrt{\alpha (\alpha -1)}},\,\alpha >1$, then the risk difference is
$$\begin{aligned} D_m^*&= R(\eta ,\delta ^*)-R(\eta ,\delta _{m})\\&= E\left[\frac{X}{\eta \sqrt{\alpha (\alpha -1)}}\right]+E\left[\frac{\eta \sqrt{\alpha (\alpha -1)}}{X}\right]\\&-E\left[\frac{\left(X+\frac{1}{m}\right)}{\eta \sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}\right]-E\left[\frac{\eta \sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}{\left(X+\frac{1}{m}\right)}\right]\\&= 2\,\sqrt{\frac{\alpha }{\alpha -1}}-\left(\alpha +\frac{1}{m\eta }\right)\left(\frac{1}{\sqrt{(\alpha +\frac{1}{m})(\alpha +\frac{1}{m}-1)}}\right)\\&-\eta \,\sqrt{\left(\alpha +\frac{1}{m}\right)\left(\alpha +\frac{1}{m}-1\right)}\,E\left[\frac{1}{X+\frac{1}{m}}\right]\\&= 2\,\sqrt{\frac{\alpha }{\alpha -1}}-\left(\alpha +\frac{1}{m\eta }\right)c_{\alpha ,m}^{*^{-1}}-\eta \,c_{\alpha ,m}^*\,\int \limits _{0}^{\infty }\frac{\eta ^{-\alpha }}{\varGamma (\alpha )}\frac{t^{\alpha -1}}{t+\frac{1}{m}}\,e^{-t/\eta }\,dt, \end{aligned}$$
where
$$\begin{aligned} c_{\alpha ,m}^*=\sqrt{\left(\alpha +\frac{1}{m}\right)\left(\alpha +\frac{1}{m}-1\right)}. \end{aligned}$$
Hence,
$$\begin{aligned} \varDelta _m&= r(\pi _{m},\delta ^*)-r(\pi _{m},\delta _{m})\\&= \int \limits _{0}^{\infty }D_m^*\pi _{m}(\eta )d\eta \\&= 2\,\sqrt{\frac{\alpha }{\alpha -1}}-c_{\alpha ,m}^{*^{-1}}\left(\alpha +\frac{1}{m}\int \limits _{0}^{\infty }\frac{1}{\eta }\left(\frac{1}{m}\right)^{\frac{1}{m}}\eta ^{-\frac{1}{m}-1}e^{-\eta /m}\frac{1}{\varGamma (\frac{1}{m})}\,d\eta \right)d_{\alpha ,m}\\&-\,c_{\alpha ,m}^*\int \limits _{0}^{\infty }\eta \left[\int \limits _{0}^{\infty }\frac{\eta ^{-\alpha }}{\varGamma (\alpha )}\frac{t^{\alpha -1}}{t+\frac{1}{m}}\,e^{-t/\eta }\,dt\right]\frac{\frac{1}{m}^\frac{1}{m}}{\varGamma (\frac{1}{m})}\eta ^{-\frac{1}{m}-1}\,e^{-\frac{1}{m}\lambda }\,d\eta \\&= 2\,\sqrt{\frac{\alpha }{\alpha -1}}-c_{\alpha ,m}^{*^{-1}}\,\left(\alpha +\frac{1}{m}\right)\\&-\,c_{\alpha ,m}^*\,\frac{\left(\frac{1}{m}\right)^\frac{1}{m}\,\varGamma (\alpha +\frac{1}{m}-1)}{\varGamma (\frac{1}{m})\varGamma (\alpha )}\int \limits _{0}^{\infty }\frac{t^{\alpha -1}}{\left(t+\frac{1}{m}\right)^{\alpha +\frac{1}{m}}}\,dt\\&= 2\,\sqrt{\frac{\alpha }{\alpha -1}}-c_{\alpha ,m}^{*^{-1}}\,\left(\alpha +\frac{1}{m}\right)\\&-\,c_{\alpha ,m}^*\,\frac{\varGamma (\alpha +\frac{1}{m}-1)}{\varGamma (\frac{1}{m})\varGamma (\alpha )}\int \limits _{0}^{1}z^{\alpha -1}\left(1-z\right)^{\frac{1}{m}-1}dz\quad \left(z=\frac{t}{t+\frac{1}{m}}\right)\\&= 2\,\sqrt{\frac{\alpha }{\alpha -1}}-c_{\alpha ,m}^{*^{-1}}\,\left(\alpha +\frac{1}{m}\right)-\frac{1}{\alpha +\frac{1}{m}-1}\,c_{\alpha ,m}^*. \end{aligned}$$
Now since $c_{\alpha ,m}^*\longrightarrow \sqrt{\alpha (\alpha -1)}$ as $m\longrightarrow \infty $, we reach in $\lim _{m\rightarrow \infty }\varDelta _m^*=0$. This guarantees the admissibility of $\delta ^*(X)=c^*X$, when $\alpha >1$. $\square $

Proof of Theorem 2

(i)

If $c=0$ and $d=0$, then $cX+d$ has infinite risk for all $\eta $ and is dominated by $\delta _0(X)\equiv k$ for some $k>0$.
(ii)

If $c^*\ne c>0$ and $d=0$, then $R(\eta ,cX)$ takes its minimum at $c=c^*$ and hence $c^*X$ dominates $cX$.
(iii)

If $c>c^*$ and $d>0$, then $cX+d$ is dominated by $\delta ^{**}(X)=c^*X+\frac{c^*d}{c}$. To see this, note that the risk difference is
$$\begin{aligned} D&= R(\eta ,cX+d)-R(\eta ,\delta ^{**}(X))\\&= E\left[\frac{cX+d}{\eta }\right]+E\left[\frac{\eta }{cX+d}\right]\\&-E\left[\frac{c^*X+\frac{c^*d}{c}}{\eta }\right]-E\left[\frac{\eta }{c^*X+\frac{c^*d}{c}}\right]\\&> \left(c\alpha +\frac{d}{\eta }\right)\left(1-\frac{c^*}{c}\right)+E\left[\frac{\eta }{cX}\left(1-\frac{c}{c^*}\right)\right]\\&> \left(1-\frac{c^*}{c}\right)\left\{ c^*\,\alpha +\frac{d}{\eta }-\frac{1}{(\alpha -1)}\frac{1}{c^*}\right\} \\&= \left(1-\frac{c^*}{c}\right)\frac{d}{\eta }>0 \end{aligned}$$
and this completes the proof. $\square $

Proof of Proposition 2

We need to verify the conditions of the Meczarski’s theorem. Note that when $\pi \in \varGamma _g$ then from (11),

$$\begin{aligned} \rho (\gamma _0,g,\delta )=\rho _{\gamma _0}(g,\delta ) =\frac{\delta (\alpha +\gamma _0)}{g+x}+\frac{g+x}{\delta (\alpha +\gamma _0-1)}-2. \end{aligned}$$

Obviously, the function $\rho _{\gamma _0}(g,.)$ is strictly convex since,

$$\begin{aligned} \frac{\partial }{\partial \delta }\rho _{\gamma _0}(g,\delta )&= \frac{(\alpha +\gamma _0)}{g+x}-\frac{g+x}{\delta ^2(\alpha +\gamma _0-1)}\\ \frac{\partial ^2}{\partial \delta ^2}\rho _{\gamma _0}(g,\delta )&= 2\frac{g+x}{\delta ^3(\alpha +\gamma _0-1)}>0. \end{aligned}$$

On the other hand,

$$\begin{aligned} \frac{\partial }{\partial g}\rho _{\gamma _0}(g,\delta )&= -\frac{\delta (\alpha +\gamma _0)}{(g+x)^2}+\frac{1}{\delta (\alpha +\gamma _0-1)}\\ \frac{\partial ^2}{\partial g^2}\rho _{\gamma _0}(g,\delta )&= 2\frac{\delta (\alpha +\gamma _0)}{(g+x)^3}>0, \end{aligned}$$

thus $\rho _{\gamma _0}(g,\delta )$ takes its unique minimum at $g_{min}(\delta )\!\equiv \!\hat{g}_{\delta }\!=\!\delta \sqrt{(\alpha \!+\!\gamma _0)(\alpha +\gamma _0-1)}-x$. Also, when $\eta _1<\eta _2\le \delta $,

$$\begin{aligned} \rho _{\gamma _0}(\hat{g}_{\eta _2},\eta _2)-\rho _{\gamma _0}(\hat{g}_{\eta _1},\eta _1)=0 \end{aligned}$$

and

$$\begin{aligned} \rho _{\gamma _0}(\hat{g}_{\delta },\eta _2)-\rho _{\gamma _0}(\hat{g}_{\delta },\eta _1) =\sqrt{\frac{\alpha +\gamma _0}{\alpha +\gamma _0-1}}\left(\frac{1}{\delta }(\eta _2-\eta _1)+\delta \left(\frac{1}{\eta _2}-\frac{1}{\eta _1}\right)\right)<0. \end{aligned}$$

Thus, $\forall \eta _1<\eta _2\le \delta $,

$$\begin{aligned} \rho _{\gamma _0}(\hat{g}_{\delta },\eta _2)-\rho _{\gamma _0}(\hat{g}_{\delta },\eta _1)<\rho _{\gamma _0}(\hat{g}_{\eta _2},\eta _2)-\rho _{\gamma _0}(\hat{g}_{\eta _1},\eta _1). \end{aligned}$$

Similarly, we can show that $\forall \eta _2>\eta _1\ge \delta $,

$$\begin{aligned} \rho _{\gamma _0}(\hat{g}_{\eta _2},\eta _2)-\rho _{\gamma _0}(\hat{g}_{\eta _1},\eta _1)<\rho _{\gamma _0}(\hat{g}_{\delta },\eta _2)-\rho _{\gamma _0}(\hat{g}_{\delta },\eta _1). \end{aligned}$$

To verify the condition ($d$), note that

$$\begin{aligned} \varDelta r_\delta&= \rho _{\gamma _0}(g_1,\delta )-\rho _{\gamma _0}(g_2,\delta )\\&= \delta (\alpha +\gamma _0)\left\{ \frac{1}{g_1+x}-\frac{1}{g_2+x}\right\} +\frac{g_1-g_2}{\delta (\alpha +\gamma _0-1)} \end{aligned}$$

and

$$\begin{aligned} \frac{d}{d \delta }\varDelta r_\delta&= (\alpha +\gamma _0)\left\{ \frac{1}{g_1+x}-\frac{1}{g_2+x}\right\} +\frac{g_2-g_1}{\delta ^2(\alpha +\gamma _0-1)}\\&= \left\{ (\alpha +\gamma _0)\frac{1}{(g_1+x)(g_2+x)}+\frac{1}{\delta ^2(\alpha +\gamma _0-1)}\right\} \left(g_2-g_1\right),\\&\left\{ \begin{array}{ll} >0&g_2>g_1\\ <0&g_2<g_1 \end{array} \right., \end{aligned}$$

thus, $\varDelta r_\delta $ is a monotone function of $\delta $. Accordingly $\delta _{S}$ will be the solution of the equation

$$\begin{aligned} \rho _{\gamma _0}(g_1,\delta _{S})=\rho _{\gamma _0}(g_2,\delta _{S}), \end{aligned}$$

which leads to (12).

To obtain $CGM$ estimator w.r.t. $\varGamma _g$, note that the Bayes estimate of $\eta $ w.r.t. $\pi _{\gamma _0,g}$ under the loss function (3) is given by (10) with $\gamma =\gamma _0$, and it can be easily seen that $\delta _{S}\in [\delta _{\pi _{\gamma _0,g_1}},\delta _{\pi _{\gamma _0,g_2}}]$. So in this estimation problem $\delta _{CGM}(x)=\delta _{S}(x)$. $\square $

Proof of Proposition 3

When $\pi \in \varGamma _\gamma $, the posterior risk is

$$\begin{aligned} \rho (\gamma ,g_0,\delta )=\rho _{g_0}(\gamma ,\delta ) =\frac{\delta (\alpha +\gamma )}{g_0+x}+\frac{g_0+x}{\delta (\alpha +\gamma -1)}-2. \end{aligned}$$

Now, it is easy to verify the conditions of Meczarski’s theorem briefly. Obviously

(i)

The function $\rho _{g_0}(\gamma ,.)$ is strictly convex
(ii)

The function $\rho _{g_0}(.,\delta )$ is convex and it takes its unique minimum at $\gamma _{min}(\delta )\equiv \hat{\gamma }_{\delta }=\frac{g_0+x}{\delta }-\alpha +1$.
(iii)

If $\eta _1<\eta _2\le \delta $, then
$$\begin{aligned} \rho _{g_0}(\hat{\gamma }_{\eta _2},\eta _2)-\rho _{g_0}(\hat{\gamma }_{\eta _1},\eta _1)=\frac{\eta _2-\eta _1}{g_0+x} \end{aligned}$$
and
$$\begin{aligned} \rho _{g_0}(\hat{\gamma }_{\delta },\eta _2)-\rho _{g_0}(\hat{\gamma }_{\delta },\eta _1)&= \frac{1}{\delta }(\eta _2-\eta _1)+\delta \left(\frac{1}{\eta _2}-\frac{1}{\eta _1}\right)+\frac{\eta _2-\eta _1}{g_0+x}\nonumber \\&< \frac{\eta _2-\eta _1}{g_0+x}.\nonumber \end{aligned}$$

(17)
Thus,
$$\begin{aligned} \forall \,\eta _1<\eta _2\le \delta ,\,\rho _{g_0}(\hat{\gamma }_{\delta },\eta _2)-\rho _{g_0}(\hat{\gamma }_{\delta },\eta _1)<\rho _{g_0}(\hat{\gamma }_{\eta _2},\eta _2)-\rho _{g_0}(\hat{\gamma }_{\eta _1},\eta _1). \end{aligned}$$
Similarly, we can show that
$$\begin{aligned} \forall \, \eta _2>\eta _1\ge \delta ,\,\rho _{g_0}(\hat{\gamma }_{\eta _2},\eta _2)-\rho _{g_0}(\hat{\gamma }_{\eta _1},\eta _1)<\rho _{g_0}(\hat{\gamma }_{\delta },\eta _2)-\rho _{g_0}(\hat{\gamma }_{\delta },\eta _1) \end{aligned}$$
(iv)

Finally,
$$\begin{aligned} \varDelta r_\delta&= \rho _{g_0}(\gamma _1,\delta )-\rho _{g_0}(\gamma _2,\delta )\\&= \frac{\delta (\gamma _1-\gamma _2)}{g_0+x}+\frac{g_0+x}{\delta }\left(\frac{1}{\alpha +\gamma _1-1}-\frac{1}{\alpha +\gamma _2-1}\right) \end{aligned}$$
and
$$\begin{aligned} \frac{d}{d\delta }\varDelta r_\delta&= \frac{\gamma _1-\gamma _2}{g_0+x}-\frac{g_0+x}{\delta ^2}\left(\frac{1}{\alpha +\gamma _1-1}-\frac{1}{\alpha +\gamma _2-1}\right)\\&\left\{ \begin{array}{ll} >0&\gamma _2<\gamma _1 \\ <0&\gamma _2>\gamma _1 \end{array} \right. \end{aligned}$$
Hence, $\varDelta r_\delta $ is a monotone function of $\delta $. Accordingly $\delta _{S}$ will be the solution of the equation
$$\begin{aligned} \rho _{g_0}(\gamma _1,\delta _{S})=\rho _{g_0}(\gamma _2,\delta _{S}) \end{aligned}$$
which leads to (13).

To obtain $CGM$ estimator w.r.t. $\varGamma _\gamma $, note that the Bayes estimate of $\eta $ w.r.t. $\pi _{\gamma ,g_0}$ under the loss function (3) is given by (10) with $g=g_0$, and it can be easily seen that when $\gamma _2-\gamma _1>1$, $\delta _{S}\in [\delta _{\pi _{\gamma _2,g_0}},\delta _{\pi _{\gamma _1,g_0}}]$. So in this estimation problem $\delta _{CGM}(x)=\delta _{S}(x)$, provided $\gamma _2-\gamma _1>1$.

Proof of Theorem 5

Given $X=x$, the posterior regret risk $r_p(\delta ,\delta _{\pi })$ is equal to

$$\begin{aligned} r_p(\delta ,\delta _{\pi })&= \rho (\pi ,\delta )-\rho (\pi ,\delta _{\pi })\nonumber \\&= \frac{\delta (\alpha +\gamma )}{g+x}+\frac{g+x}{\delta (\alpha +\gamma -1)}-2\sqrt{\frac{\alpha +\gamma }{\alpha +\gamma -1} }\nonumber \\&= \sqrt{\frac{\alpha +\gamma }{\alpha +\gamma -1} }\,\left(\sqrt{\frac{\delta _\pi }{\delta }}-\sqrt{\frac{\delta }{\delta _\pi }}\right)^2\nonumber \\&= k_{\alpha ,\gamma }\left(\frac{\delta _{\pi }}{\delta }+\frac{\delta }{\delta _{\pi }}-2\right) \end{aligned}$$

(18)

where $k_{\alpha ,\gamma }=\sqrt{\frac{\alpha +\gamma }{\alpha +\gamma -1}}$.

(i)

Let $\varGamma _g$ be the class of priors, obviously $k_{\alpha ,\gamma _0}$ is a constant for any fixed $\alpha $ and
$$\begin{aligned} \frac{\partial }{\partial \delta _{\pi }}r_p(\delta ,\delta _{\pi })=k_{\alpha ,\gamma _0}\left(1/\delta -\delta /{\delta _\pi }^2\right) \end{aligned}$$
and
$$\begin{aligned} \frac{\partial ^2}{\partial {\delta _{\pi }}^2}r_p(\delta ,\delta _{\pi })=2k_{\alpha ,\gamma _0}\delta /{\delta _\pi }^3>0. \end{aligned}$$

So, $r_p(\delta ,\delta _{\pi })$ is strictly convex w.r.t. $\delta _{\pi }$. The following three steps leads us to the $PRGM$ estimate of $\eta $ under the loss function (3). Notice that $\underline{\delta }<\delta _{\pi }<\overline{\delta }$. $\square $

Step 1

When $\delta <\underline{\delta }$, differentiating $r_p(\delta ,\delta _{\pi })$ w.r.t. $\delta _\pi $, it is easy to verify that

$$\begin{aligned} \sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _{\pi })=r_p(\delta ,\overline{\delta }). \end{aligned}$$

Let $f_1(\delta )=r_p(\delta ,\overline{\delta })$, then $f_1^{^{\prime }}(\delta )=k_{\alpha ,\gamma _0}\left(1/\overline{\delta }-\overline{\delta }/{\delta }^2\right)$. So for all $\delta <\underline{\delta }$, $f_1(\delta )$ is decreasing in $\delta $ and $\inf _{\delta <\underline{\delta }}f_1(\delta )=f_1(\underline{\delta })$. It means that

$$\begin{aligned} \inf _{\delta <\underline{\delta }} \sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _\pi )=r_p(\overline{\delta },\underline{\delta }),\quad f_1(\overline{\delta })=0. \end{aligned}$$

Step 2

When $\delta >\overline{\delta }$, differentiating $r_p(\delta ,\delta _{\pi })$ w.r.t. $\delta _\pi $, it is easy to verify that

$$\begin{aligned} \sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _{\pi })=r_p(\delta ,\underline{\delta }). \end{aligned}$$

Let $f_2(\delta )=r_p(\delta ,\underline{\delta })$, then $f_2^{^{\prime }}(\delta )=k_{\alpha ,\gamma _0}\left(1/\underline{\delta }-\underline{\delta }/{\delta }^2\right)$. So for all $\delta >\overline{\delta }$, $f_2(\delta )$ is increasing in $\delta $ and $\inf _{\delta >\overline{\delta }}f_1(\delta )=f_1(\overline{\delta })$. It means that

$$\begin{aligned} \inf _{\delta >\overline{\delta }} \sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _\pi )=r_p(\underline{\delta },\overline{\delta }),\quad f_2(\underline{\delta })=0. \end{aligned}$$

Step 3

When $\underline{\delta }\,{<}\,\delta \,{<}\,\overline{\delta }$, there are two possibilities: either $\underline{\delta }\,{<}\,\delta \,{<}\,\delta _{\pi }\,{<}\,\overline{\delta }$ or $\underline{\delta }\,{<}\,\delta _{\pi }\,{<}\,\delta \,{<}\,\overline{\delta }$. So regarding these possibilities, $\sup r_p(\delta ,\delta _\pi )=\max \left\{ r_p(\delta ,\overline{\delta }),r_p(\delta ,\underline{\delta })\right\} $. Let

$$\begin{aligned} l(\delta )&= f_1(\delta )-f_2(\delta )\\&= k_{\alpha ,\gamma _0}\left\{ \frac{\overline{\delta }-\underline{\delta }}{\delta }+\delta \left(\frac{1}{\overline{\delta }}-\frac{1}{\underline{\delta }}\right)\right\} \end{aligned}$$

It is easy to verify that $l(\delta )$ is continuous and decreasing in $\delta $ and $l(\underline{\delta })\,l(\overline{\delta })<0$. Thus, there exists $\delta ^*\in (\underline{\delta },\overline{\delta })$ such that $l(\delta ^*)=0$, i.e., $r_p(\delta ^*,\underline{\delta })=r_p(\delta ^*,\overline{\delta })$. Therefore for all $\underline{\delta }<\delta <\delta ^*$, $l(\delta )>0$ and $\max \left\{ r_p(\delta ,\overline{\delta }),r_p(\delta ,\underline{\delta })\right\} =r_p(\delta ,\overline{\delta })$. Furthermore for all $\underline{\delta }<\delta <\delta ^*$, $r_p(\delta ,\overline{\delta })$ is decreasing in $\delta $ and

$$\begin{aligned} \inf _{\underline{\delta }<\delta <\delta ^*}r_p(\delta ,\overline{\delta })=r_p(\delta ^*,\overline{\delta }). \end{aligned}$$

On the other hand, for all $\delta ^*<\delta <\overline{\delta }$, $l(\delta )<0$ and $\max \left\{ r_p(\delta ,\overline{\delta }),r_p(\delta ,\underline{\delta })\right\} =r_p(\delta ,\underline{\delta })$. Hence for all $\delta ^*<\delta <\overline{\delta }$, $r_p(\delta ,\underline{\delta })$ is increasing in $\delta $ and

$$\begin{aligned} \inf _{\delta ^*<\delta <\overline{\delta }}r_p(\delta ,\underline{\delta })=r_p(\delta ^*,\underline{\delta }). \end{aligned}$$

Combining these arguments leads us to

$$\begin{aligned} \inf _{\underline{\delta }<\delta <\overline{\delta }} \sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _\pi )=r_p(\delta ^*,\underline{\delta })=r_p(\delta ^*,\overline{\delta }). \end{aligned}$$

Using the above three steps, it is easy to see that

$$\begin{aligned} \inf _{\delta \in \mathcal{D}}\sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _\pi )&= \inf _{\underline{\delta }<\delta <\overline{\delta }} \sup _{\pi \in \varGamma _{g}}r_p(\delta ,\delta _\pi )\nonumber \\&= r_p(\delta ^*,\overline{\delta })\nonumber \\&= r_p(\delta ^*,\underline{\delta }).\nonumber \end{aligned}$$

Now, $\delta _{PRGM}(x)=\delta ^*\in \left(\underline{\delta },\overline{\delta }\right)$ is given by $l\left(\delta _{PRGM}(x)\right)=0$, which leads to $\delta _{PRGM}(x)=\sqrt{\underline{\delta }(x)\overline{\delta }(x)}$, i.e., the geometric mean of $\underline{\delta }(x)$ and $\overline{\delta }(x)$.

(ii) Let $\varGamma _\gamma $ be the class of priors. Notice that in this case $k_{\alpha ,\gamma }$ is a function of $\gamma $ and after awkward calculations we reach in

$$\begin{aligned} \frac{\partial }{\partial \delta _{\pi }}r_p(\delta ,\delta _{\pi })=AB\left\{ \frac{1}{2B^2-1}\left(\frac{\delta }{\delta _{\pi }^2}+\frac{1}{\delta }-\frac{2}{\delta _{\pi }}\right)+\left(\frac{1}{\delta }-\frac{\delta }{\delta _{\pi }^2}\right)\right\} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2}{\partial {\delta _{\pi }}^2}r_p(\delta ,\delta _{\pi })=\frac{2B}{\left(2B^2-1\right)}\left\{ \frac{A^{-1}}{\left(2B^2-1\right)^2}\frac{1}{\delta _{\pi }^2}\left(\frac{\delta }{\delta _{\pi }}+\frac{\delta _{\pi }}{\delta }-2\right)+\frac{2A^{-1}\delta }{\delta _{\pi }^3}+\frac{A}{\delta \delta _{\pi }}\right\} , \end{aligned}$$

where $A=(\alpha +\gamma -1)^{-\frac{1}{2}}$ and $B=(\alpha +\gamma )^{\frac{1}{2}}$. Obviously, $\frac{\partial ^2}{\partial {\delta _{\pi }}^2}r_p(\delta ,\delta _{\pi })>0$ when $\alpha +\gamma >1$, i.e., $r_p(\delta ,\delta _{\pi })$ is strictly convex w.r.t. to $\delta _\pi $.

Once again, following the above three steps in (i) with $\underline{\delta }(x)=\frac{x+g}{\sqrt{(\alpha +\gamma _2)(\alpha +\gamma _2-1)}}$ and $\overline{\delta }(x)=\frac{x+g}{\sqrt{(\alpha +\gamma _1)(\alpha +\gamma _1-1)}}$, we reach in $\delta _{PRGM}(x)=\frac{\sqrt{\overline{\delta }(x)}+q\sqrt{\underline{\delta }(x)}}{\sqrt{\underline{\delta }(x)}+q\sqrt{\overline{\delta }(x)}}\,\sqrt{\underline{\delta }(x)\overline{\delta }(x)}$, where $q=\root 4 \of {\frac{{(\alpha +\gamma _2)(\alpha +\gamma _1-1)}}{(\alpha +\gamma _1)(\alpha +\gamma _2-1)}}$.

(iii) Let $\varGamma _{\gamma ,g}$ be the class of priors. Then from (ii), $r_p(\delta ,\delta _{\pi })$ is strictly convex for any fixed $g\in [g_1,g_2]$. Hence, it is convex for all $g\in [g_1,g_2]$. So, following the above three steps in (i) with $\underline{\delta }(x)=\frac{x+g_1}{\sqrt{(\alpha +\gamma _2)(\alpha +\gamma _2-1)}}$ and $\overline{\delta }(x)=\frac{x+g_2}{\sqrt{(\alpha +\gamma _1)(\alpha +\gamma _1-1)}}$, we reach in $\delta _{PRGM}(x)=\frac{\sqrt{\overline{\delta }(x)}+q\sqrt{\underline{\delta }(x)}}{\sqrt{\underline{\delta }(x)}+q\sqrt{\overline{\delta }(x)}}\,\sqrt{\underline{\delta }(x)\overline{\delta }(x)}$.

Remark 1

(i)

$0<q\le 1$ and when $q=1$, i.e., $\gamma _1=\gamma _2(=\gamma _0)$ which means the class of priors $\varGamma _g$, we have $\delta _{PRGM}(x)=\sqrt{\underline{\delta }(x)\overline{\delta }(x)}$.
(ii)

the solution of $l(\delta )=0$ in Step 3 of part (ii) and (iii) has only one acceptable solution.

References

Ahmadi J, Doostparast M, Parsian A (2005) Estimation and prediction in a two-parameter exponential distribution based on $k$-records values under LINEX loss function. Comm Stat Theory Methods 34:795–805MathSciNetMATHCrossRefGoogle Scholar
Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, LondonMATHCrossRefGoogle Scholar
Bayati Eshkaftaki A, Parsian A (2011) Robust Bayes estimation. Commun Stat Theory Methods 40(5):929–941MATHCrossRefGoogle Scholar
Berger JO (1984) An overview of robust Bayesian analysis. Test 3(1):5–124CrossRefGoogle Scholar
Berger JO (1985) Statistical decision theory and Bayesian analysis, 2nd edn. Springer, New YorkMATHCrossRefGoogle Scholar
Berger JO (1990) Robust Bayesian analysis: sensitivity to the prior. J Stat Plan Infer 25:303–328MATHCrossRefGoogle Scholar
Betro B, Ruggeri F (1992) Conditional $\Gamma $-minimax actions under convex losses. Commun Stat Theory Methods 21(4):1051–1066MathSciNetMATHCrossRefGoogle Scholar
Boratynska A (1997) Stability of Bayesian inference in exponential families. Stat Prob Lett 36:173–178MathSciNetMATHCrossRefGoogle Scholar
Boratynska A (2002a) Posterior regret $\Gamma $-minimax estimation in a normal model with asymmetric loss function. Appl Math 29:7–13MathSciNetMATHGoogle Scholar
Boratynska A (2002b) Robust Bayesian estimation with asymmetric loss function. Appl Math 29:297–306MathSciNetMATHGoogle Scholar
Boratynska A (2004) Estymatory o $\Gamma $-minimaksowej utracie a posteriori dla pewnych rozładów dyskretnych. Annals of Collegium of Economics Analysis, Warsaw School of Economics, vol 13. pp 31–46 (in Polish) 13:31–46Google Scholar
Boratynska A (2008) Posterior regret $\Gamma $-minimax estimation of insurance premium in collective risk model. Astin Bull 38:277–291MathSciNetMATHCrossRefGoogle Scholar
Boratynska A, Meczarski M (1994) Robust Bayesian estimation in the one-dimensional normal model. Stat Decis 12:221–230Google Scholar
Brown LD (1968) Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. Ann Math Stat 39:29–48MATHCrossRefGoogle Scholar
Brown LD (1990) Comment on Matta and Casella’s paper. Stat Sci 5:103–106Google Scholar
Chung Y (1995) Estimation of scale parameter from rayleigh distribution under entropy loss. Korean J CAM 2(1):33–40Google Scholar
DasGupta A, Studden WJ (1989) Frequentists behavior of robus Bayes estimates of normal means. Stat Decis 7:333–361MathSciNetMATHGoogle Scholar
Dey DK, Srinivasan C (1985) Estimation of a covariance matrix under Stein’s loss. Ann Stat 13:1581–1591MathSciNetMATHCrossRefGoogle Scholar
DeGroot MH (2004) Optimal statistical decisions. Wiley Interscience, LondonMATHCrossRefGoogle Scholar
Ferguson TS (1967) Mathematical statistics, a decision theoric approach. Academic Press, New YorkGoogle Scholar
Ghitany ME (1990) Estimation of the scale parameter of gamma model in the presence of outlier observations. Int J Math Math Sci 13(1):121–127MathSciNetMATHCrossRefGoogle Scholar
Ghosh JK, Singh R (1966) Unbiased estimation of location and scale parameters. Ann Math Stat 37(6):1671–1675MathSciNetMATHCrossRefGoogle Scholar
Good IJ (1952) Rational decisions. J R Stat Soc Ser B (Methodolog) 14(1):107–114MathSciNetGoogle Scholar
Harris TJ (1992) Optimal controliers of nonsymmetric and nonquadratic loss functions. Technometrics 34(3):298–306Google Scholar
Jafari Jozani M, Parsian A (2008) Posterior regret $\Gamma $-minimax estimation and prediction with applications on $k$-records data under entropy loss function. Commun Stat Theory Methods 37(14):2202–2212Google Scholar
Jafari Jozani M, Nematollahi N, Shafie K (2002) An admissible minimax estimator of a bounded scale parameter in a subclass of the exponential family under scale-invariant squared error loss. Stat Prob Lett 60(4):437–444MathSciNetMATHCrossRefGoogle Scholar
James W, Stein C (1961) Estimation with quadratic loss. In: Proceedings of 4th Berekely symposium on mathematical statistics and probability, vol 1, University of California Press, Berekeley, CA, pp 361–379Google Scholar
Jokiel-Rokita A, Magiera R (2008) On invariant estimation of a continuous cumulative distribution function. International workshop on applied probability, IWAP 2008, Compiegne, France, 7–10 July, Iniversite de Technologie CompiegneGoogle Scholar
Jokiel-Rokita A, Magiera R (2010) Minimum risk invariant estimators of a continuous cumulative distribution function. Commun Stat Theory Methods 39(18):3323342MathSciNetGoogle Scholar
Kaminska A, Porosinski Z (2009) On robust Bayesian estimation under some asymmetric and bounded loss function. Statistics 43(3):253–265MathSciNetMATHCrossRefGoogle Scholar
Kiapour A, Nematollahi N (2011) Robust Bayesian prediction and estimation under a squared error loss function. Stat Prob Lett. doi: 10.16/j.spl.2011.07.002
Lehmann EL (1951) A general concept of unbiasedness. Ann Math Stat 22:587–592MATHCrossRefGoogle Scholar
Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. Springer, New YorkMATHGoogle Scholar
Lehmann EL, Romano JP (2005) Testing statistical hypothesis. Springer, New YorkGoogle Scholar
Meczarski M (1993) Stability and conditional $\Gamma $-minimax in Bayesian inference. Appl Math 22:117–122MathSciNetMATHGoogle Scholar
Meczarski M, Zeilinski R (1991) Stability of the Bayesian estimator of the Poisson mean under the inexactly specified gamma priors. Stat Prob Lett 52:77–91Google Scholar
Menon MV (1963) Estimation of the shape and scale parameters of the Weibull distribution. Technometrics 5(2):175–182MathSciNetMATHCrossRefGoogle Scholar
Nevzoro VB (2001) Records: mathematical theory. Translations of mathematical monographs, 194Google Scholar
Norstrom JG (1996) The use of precautionary loss function in risk analysis. IEEE Trans Reliab 3:400–403CrossRefGoogle Scholar
Pal N (1989) Estimation of generalized variance under entropy losses: admissibility results. Calcutta Stat Assoc Bull 38:147–156Google Scholar
Pal N, Ling C (1996) Estimation of a normal scale parameter under a balanced loss function. J Ind Stat Assoc Bull 34(1):21–38MathSciNetGoogle Scholar
Parsian A, Nematollahi N (1996) Estimation of scale parameter under entropy loss function. J Stat Plan Infer 12:329–333MathSciNetGoogle Scholar
Pitman EJG (1939) The estimation of the location and scale parameters of a continuous population of any given form. Biometrika 30:391–421MATHGoogle Scholar
Rahman MS, Gupta RP (1993) Family of transformed chi-square distributions. Commun Stat Theory Methods 22(1):135–146MathSciNetMATHGoogle Scholar
Rios Insua D, Ruggeri F, Vidakovic B (1995) Some results on posterior $\Gamma $-minimax estimation. Stat Decis 13:315–351MathSciNetMATHGoogle Scholar
Robinns H (1951) Asymptotically subminimax solutions of compound decision problems. In: Proceedings of the second Berkeley symposium on mathematical statistics and probability, vol 1950. University of California, Berkeley, pp 131–148Google Scholar
Rukhin AL, Ananda MMA (1992) Risk behavior of variance estimators in multivariate normal distribution. Stat Prob Lett 13:159–166MathSciNetMATHCrossRefGoogle Scholar
Samaniego JF (1975) On $\Gamma $-minimax estimation. Am Stat 29(4):168–169MathSciNetGoogle Scholar
Sanjari Farsipour N, Zakerzadeh H (2005) Estimation of a Gamma scale parameter under asymmetric squared log error loss. Commun Stat Theory Methods 34:1127–1135MathSciNetMATHCrossRefGoogle Scholar
Stacy EW, Mihram GA (1965) Parameter estimation for a generalized gamma distribution. Technometrics 7(3):349–358MathSciNetMATHCrossRefGoogle Scholar
Tate RF (1959) Unbiased estimation: functions of location and scale parameters. Ann Math Stat 30(2):341–366MathSciNetMATHCrossRefGoogle Scholar
Towhidi M, Behboodian J (1999) Estimation of scale parameter under a reflected gamma loss function. J Sci Islam Rep IR 10(4):265–269MathSciNetMATHGoogle Scholar
Towhidi M, Behboodian J (2002) Minimax estimation of a bounded parameter under some bounded loss functions. Far East J Theory Stat 6:39–48MathSciNetMATHGoogle Scholar
Vidakovic B (2000) $\Gamma $-minimax: a paradigm for conservative robust Bayesians. Lecture Notes in Statistics. Springer, New York, pp 241–259Google Scholar
Watson SR (1974) On Bayesian inference with incompletely specified prior distributions. Biometrika 61(1):193–196MathSciNetMATHCrossRefGoogle Scholar
Weke PGO (2007) Linear estimation of scale parameter for logistic distribution based on consecutive order statistics. Indian J Stat 69(4):870–884MathSciNetMATHGoogle Scholar
Zellner A (1986) Bayesian estimation and prediction using assymetric loss function. J Am Stat Assoc 81:446–451MathSciNetMATHCrossRefGoogle Scholar
Zellner A (1994) Bayesian and non-Bayesian estimation using balanced loss functions. In: Berger JO, Gupta SS (eds) Statistical decision theory and methods. Springer, New York, p 390Google Scholar
Zen M, DasGupta A (1993) Estimating a binomial parameter: is robust Bayes real Bayes? Stat Decis 11:37–60MathSciNetMATHGoogle Scholar
Zidek JV (1973) Estimating the scale parameter of the exponential with unknown location. Ann Math Stat 1(2):264–278Google Scholar

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Leila Golparver
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Ali Karimnezhad
- 1
Ahmad Parsian
- 1
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1.School of Mathematics, Statistics and Computer ScienceUniversity of TehranTehranIran

Optimal rules and robust Bayes estimation of a Gamma scale parameter

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