Estimating a function of scale parameter of an exponential population with unknown location under general loss function

Abstract

In the present study, we consider the problem of estimating a function of scale parameter $\ln \sigma$ under an arbitrary location invariant bowl-shaped loss function, when location parameter $\mu$ is unknown. Various improved estimators are proposed. Inadmissibility of the best affine equivariant estimator (BAEE) of $\ln \sigma$ is established by deriving a Stein-type estimator. This improved estimator is not smooth. We derive a smooth estimator improving upon the BAEE. Further, the integral expression of risk difference (IERD) approach of Kubokawa is used to derive a class of improved estimators. To illustrate these results, we consider two specific loss functions: squared error and linex loss functions, and derive various estimators improving upon the BAEE. Finally, a simulation study has been carried out to numerically compare the risk performance of the improved estimators.

Appendix

Lemma 6.1

Let a, b, r and $\eta$ be positive real numbers with $0<a<b$ and $a_0\in (0,1]$. Then, the function

$$\begin{aligned} M(z)=\frac{a_0-e^{-az}}{a_0-e^{-bz}}I(az>\eta ), \end{aligned}$$

is increasing in $z>0$.

Proof

Differentiating M(z) with respect to z we have

$$\begin{aligned} M^{\prime }(z)=\frac{\left( a_0-e^{-bz}\right) ae^{-az}-\left( a_0-e^{-az}\right) be^{-bz}}{\left( a_0-e^{-bz}\right) ^2}I(az>\eta ). \end{aligned}$$

Now, $M^{\prime }(z)>0$ if

$$\begin{aligned} \left( a_0-e^{-bz}\right) ae^{-az}-\left( a_0-e^{-az}\right) be^{-bz}>0, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \left( a_0-e^{-bz}\right) ae^{-az}>\left( a_0-e^{-az}\right) be^{-bz}. \end{aligned}$$

Further, $te^{-zt}\left( a_0-e^{-tz}\right) ^{-1}$ is decreasing in t. Hence, under the assumption made, we obtain $M^{\prime }(z)>0.$ $\square$

Lemma 6.2

Let r and $r_1$ be positive real numbers such that $0<r_1<r$. Then, the function

$$\begin{aligned} R(x)=\frac{1-e^{-rx}}{1-e^{-r_1 x}}I(x>0), \end{aligned}$$

is nonincreasing in $x>0$.

Proof

The proof is similar to that of Lemma 6.1. $\square$

Lemma 6.3

Let r and $\lambda$ be positive real numbers. Then, the function

$$\begin{aligned} G(y)=\frac{e^{-\lambda }-e^{-re^y}}{1-e^{-re^y}}I(e^y>\lambda /r), \end{aligned}$$

is increasing in $y\in {\mathbb {R}}$.

Proof

Differentiating G(y) with respect to y, and then further simplification leads to $G^{\prime }(y)>0$. $\square$