I. INTRODUCTION

Rukhin(1988) introduced a loss function given by,

Where, $\gamma$ is an estimator of the loss function $w(\theta ,\delta )$, which is non-negative. Guobing(2016) used this loss function and derived estimates of the loss and risk function of the parameter of Maxwell's distribution. Singh (2021) took various forms of $w(\theta ,\delta )$ and derived estimates of the loss and risk function of the parameter of a continuous distribution which gives Half-normal distribution, Rayleigh distribution and Maxwell's distribution as particular cases. Rukhin(1988) considered the Bayesian estimation of the unknown parameter $\theta$ of the binomial distribution by taking

In this paper, Bayes estimate of the unknown parameter $\theta$ of the binomial distribution has been obtained by replacing $w(\theta ,\delta )$ by $w_1(\theta ,\delta )$ given by

Where,

II. ESTIMATION OF LOSS AND RISK OF THE PARAMETER OF BINOMIAL DISTRIBUTION

Let the random variable $X$ follows binomial distribution with parameters $n$ and $\theta$. Where $\theta$ is unknown satisfying $0\le \theta \le 1$. The prior p.d.f of $\theta$, denoted by ${\pi }_1(\theta )$ is as follows:

Under the assumption of prior probability density function (p.d.f.) for $\theta$ as above, Bayes estimates of $\theta$ derived by Rukhin (1988) were as follows: For $\alpha \ge 0,\beta \ge 0$

and for $\alpha =0,\beta =0$

It was shown that

Under, $w_1(\theta ,\delta )$ as above, the corresponding Bayes estimate is given by, For $\alpha \ge 0,\beta \ge 0$

On simplification, provided, and,

on simplification, provided,

We, see that, in this case ${\gamma }_{1B}(X)$ does not depend upon $X$ and is function of $n,\alpha$ and $\beta$

Or,

In this case,

Or,

As mentioned by Keifer (1977), an estimator $\gamma (X)$ is said to be conservatively biased if,

In the light of this condition, ${\gamma }_0(X)$ as given by Rukhin (1988) is not conservatively biased. In this case,

Let ${\delta }_{0B}(X)$ and ${\gamma }_{0B}(X)$ be values of ${\delta }_{1B}(X)$ and ${\gamma }_{1B}(X)$, respectively when, $\alpha =\beta =0$. If possible let,

which holds if,

which is a contradiction, since and maximum value of $-2{\theta }^2+2\theta -1$ is $-\frac{1}{2}$ which corresponds to $\theta =\frac{1}{2}$. Moreover, $-2{\theta }^2+2\theta -1=-1$ for $\theta =1$ and $\theta =0$. Thus, ${\gamma }_{0B}(X)$ is not conservatively biased.

When $\alpha =\beta =1$, we have,

When, , $\theta =0.5$

When, , $\theta \ne 0.5$

which holds if

.Where,

$g(\theta )$ is a monotonically increasing function of $\theta$ over the set $S=(0,1)-\{0.5\}$. Hence, ${\gamma }_{1B}(X)$ as above, presents a valid 'frequentist report' as mentioned by Berger(1985).

The results are summarized in the following:

THEOREM. Let $({\delta }_{1B},{\gamma }_{1B})$ be Bayes estimators of the unknown parameter $\theta$ of the binomial distribution under the loss function $L(\theta ,\delta ,\gamma )=\frac{1}{\{\theta (1-\theta )\}}(\theta -\delta {)}^2{\gamma }^{-\frac{1}{2}}+{\gamma }^{\frac{1}{2}}$ and beta prior density with known parameters $\alpha$ and $\beta$. Then, the frequentist risk $E_{\theta }L(\theta ,{\delta }_{1B},{\gamma }_{1B})$ is finite for all values of $\alpha$ and $\beta$ provided . For $\alpha =\beta =0$, ${\gamma }_{1B}(X)$ is not conservatively biased. The estimator ${\gamma }_{1B}(X)$ is conservatively biased for $\alpha =\beta =1$ and for satisfying $\alpha \le 1+\frac{2\theta (1-\theta )}{(2\theta -1{)}^2},\theta \ne 0.5$. However, if , ${\gamma }_{1B}(X)$ is also conservatively biased.