## Annals of Mathematical Statistics

### On a Theorem of Karlin Regarding Admissibility of Linear Estimates in Exponential Populations

#### Abstract

Let the random variable $X$ be distributed according to the probability density $p(x,\omega) = \beta(\omega) \exp (\omega x)$ with respect to a $\sigma$-finite measure $\mu$ defined on the real line, and $\omega$, the unknown state of nature, belongs to the natural parameter space:$\Omega = \{\omega\mid\int^\infty_{-\infty}\exp (\omega x) d\mu(x) < \infty\}$ which is an interval of the real line. Let $\bar\omega$ and $\underline{\omega}$ be the upper and lower end points of $\Omega$, respectively. $\bar\omega$ and $\underset{\bar}{\omega}$ may or may not belong to $\Omega$, and in some cases $\bar\omega = |\infty$ or $\underline{\omega} = -\infty$. The problem under consideration is the estimation of the quantity $\theta(\omega) = E_\omega(x) = -\beta'(\omega)/\beta(\omega)$ from a single observation $x$ on $X$. The loss-function is the standard squared-error loss function. Karlin [1] considered the admissibility of the linear estimates of the form $a_\gamma(x) = \gamma x = x/(\lambda + 1)$ for $0 \leqq \gamma \leqq 1$ and the following theorem of his gives sufficient conditions for the admissibility of $a_\gamma(x)$. Theorem 1.1 (Karlin). Let $p(x,\omega) = \beta(\omega) \exp (\omega x)$ describe the density of the exponential family wrt a measure $\mu$. If \begin{equation*}\tag{1.2)}\int^{\bar\omega}_c \beta^{-\lambda} (\omega) d\omega = +\infty\end{equation*} and \begin{equation*}\tag{1.2}\int^c_\omega \beta^{-\lambda}(\omega) d\omega = +\infty,\end{equation*} where $c$ is an interior point of $\Omega = (\underset{\bar}{\omega}, \bar\omega)$, then $\gamma x = x/(\lambda + 1)$ is an admissible estimate of $\theta(\omega) = E_\omega(X)$. In the sequel we shall refer to the integrals in (1.1) and (1.2) as Karlin's integrals. Karlin [1] conjectured that conditions (1.1) and (1.2) are also necessary for the admissibility of $x/(\lambda + 1)$. We shall, in the following, refer to this as Karlin's conjecture. Let us denote by $I^2(\omega)$ the square of the coefficient of variation given by $\lbrack(\beta'(\omega))^2 - \beta(\omega)\beta''(\omega)\rbrack/(\beta'(\omega))^2$. Suppose $I^2(\omega)$ ranges between $L$ and $L' (L < L')$ as $\omega$ traverses the interval $(\underset{\bar}{\omega}, \bar\omega)$. Karlin [1] showed that $x/(\lambda + 1)$ is inadmissible for all $\lambda < L$ and $\lambda > L'$. While criteria for inadmissibility of $x/(\lambda + 1)$ are given in terms of $L$ and $L'$, the conditions for admissibility are in terms of integrability of $\beta^{-\lambda}(\omega)$ near the end points of $\Omega$. The purpose of this paper is to link up these two criteria and characterize Karlin's integrability conditions in terms of the behavior of $I^2(\omega)$ near $\underline{\omega}$ and $\bar\omega$ (Theorem 2.2). It is also shown (Theorem 2.1) that the range of $\lambda$ for which both of Karlin's integrals are infinite form a sub-interval of $(0, \infty)$. While Karlin's conjecture still remains open, the characterizations we obtained serve to settle the conjecture for a class of cases, an example of which is given in Example 2.1. It is also interesting to regard $I^2(\omega)$ as the ratio of the risks of two linear estimates $x/(\lambda + 1)$ corresponding to $\lambda = 0$ and $\lambda = \infty$, the boundary points of the range of $\lambda$.

#### Article information

Source
Ann. Math. Statist., Volume 37, Number 6 (1966), 1809-1813.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177699172

Digital Object Identifier
doi:10.1214/aoms/1177699172

Mathematical Reviews number (MathSciNet)
MR207139

Zentralblatt MATH identifier
0147.18203

JSTOR