UDC 519.281

MATHEMATICS

L. B. KLEBANOV

ADMISSIBILITY OF THE SAMPLE MEAN AS AN ESTIMATE OF A SHIFT PARAMETER UNDER POLYNOMIAL LOSSES

(Presented by Academician Yu. V. Linnik on 4 III 1970)

1. We consider the problem of estimating the parameter \(\theta \in R^1\) from the data of a repeated sample \((x_1,\ldots,x_n)\) from a population with distribution function (d.f.) \(F(x-\theta)\).

If \(\tilde\theta=\tilde\theta(x_1,\ldots,x_n)\) is an estimate of the parameter \(\theta\), then the losses that we incur in doing this are specified by a function \(r(\tilde\theta;\theta)\), called the loss function (loss). The mathematical expectation of the loss function
\[ R(\tilde\theta;\theta)=E_\theta r(\tilde\theta;\theta) \]
is called the risk of the estimate \(\tilde\theta\). For a given risk, the notion of admissibility is introduced in a natural way.

If the loss is quadratic, \(r(\tilde\theta;\theta)=(\tilde\theta-\theta)^2\), then for \(n \ge 3\) the admissibility of the sample mean \(\bar x=(x_1+\cdots+x_n)/n\) as an estimate of the parameter \(\theta\) is equivalent to the normality of \(F(x)\) \(\left({}^{1}\right)\) (see also \(\left({}^{2}\right)\)).

An analogous result (under rather severe restrictions on \(F(x)\)) was established in \(\left({}^{3}\right)\) for losses
\[ r(\tilde\theta;\theta)= \begin{cases} -\alpha(\tilde\theta-\theta), & \text{when } \tilde\theta \le \theta,\\ \beta(\tilde\theta-\theta), & \text{when } \tilde\theta \ge \theta,\ \alpha,\beta>0, \end{cases} \]
and losses of the form \(r(\tilde\theta;\theta)=|\tilde\theta-\theta|^{2m-1}\).

Here an analogous result will be formulated that pertains to polynomial losses.

1. Let \(P(u)\) be a polynomial in \(u\) of degree \(m\) with positive coefficients. Consider the loss function
   \[ r(\tilde\theta;\theta)=P\bigl((\tilde\theta-\theta)^2\bigr). \]

Theorem. Let \((x_1,\ldots,x_n)\) be a repeated sample from a population with d.f. \(F(x-\theta)\), having two continuous derivatives and satisfying the conditions
\[ \int x\,dF(x)=0, \tag{1} \]
\[ \int x^{2m}\,dF(x)<+\infty. \tag{2} \]
Then, if the sample mean \(\bar x\) is admissible in the class of unbiased estimates of \(\theta\), then \(F(x)\) is the d.f. of the normal law.

The proof is based on the same considerations that were used in \(\left({}^{1,3}\right)\).

From the admissibility of \(\bar x\) we obtain
\[ \sum_{\substack{k_1+\cdots+k_n=2m-1\\ k_i\ge 0}} A_{k_1\ldots k_n}\varphi_{(\tau_1)}^{(k_1)}\cdots \varphi_{(\tau_n)}^{(k_n)}=0 \tag{3} \]
for all \(\tau_1,\ldots,\tau_n\) connected by the relation \(\sum_{i=1}^n \tau_i=0\). Here
\[ \varphi(\tau)=\int_{-\infty}^{\infty} e^{i\tau x}\,dF(x). \]

By means of a change of the unknown function and subsequent differentiation, one can verify that only two cases are possible:

1) All derivatives of the function \(\varphi(x)\), evaluated at \(\tau=0\), coincide with the corresponding derivatives of the function \(\exp\{Q(\tau)\}\), where \(Q(\tau)\) is some polynomial. By the well-known theorem of Marcinkiewicz \((^4)\), this means that \(\varphi(\tau)\) is the characteristic function of a normal distribution.

2) \(\varphi(\tau)\) satisfies the equation

\[ \varphi_{\tau}^{(2m-1)}=(a_1\tau+b_1)\varphi(\tau), \tag{4} \]

where \(a_1,b_1\) are certain constants.

Applying the inverse Fourier transform to (4), we obtain that the probability density \(f(x)\) has the form

\[ f(x)=C\exp\{ax^{2m}+bx\}. \tag{5} \]

But the admissibility condition for \(\bar{x}\), written in terms of the density, has the form

\[ \int_{-\infty}^{\infty} P'(u)\prod_{i=1}^{n} f(u+z_i)\,du=0 \tag{6} \]

for all \(z_i:\ \sum_{i=1}^{n} z_i=0\).

By direct substitution we verify that the density (5), for \(m>1\), cannot satisfy condition (6), i.e., case 2) is impossible.

1. Let us note that we used the differentiability of \(F(x)\) only when establishing that (5) follows from (4). If \(m=1\), then (4) is integrated directly and, evidently, \(\varphi(x)\) is the characteristic function of the normal law. Thus, the result of \((^1)\) is a special case of ours.

In conclusion, I express my deep gratitude to Yu. V. Linnik and A. M. Kagan for their attention to this work.

Leningrad State University
named after A. A. Zhdanov

Received
9 I 1970

REFERENCES

\(^1\) A. M. Kagan, Y. V. Linnik, C. R. Rao, Sankhyā, A 27, 2–3–4 (1965).
\(^2\) A. M. Kagan, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 104 (1968).
\(^3\) A. A. Zinger, A. M. Kagan, L. B. Klebanov, DAN, 189, No. 1, 29 (1969).
\(^4\) J. Marcinkiewicz, Math. Zs., 44, No. 4–5, 612 (1938).