Full Text
UDC 519.281
MATHEMATICS
A. I. SHALYT
SOME RESULTS ON SEQUENTIAL ESTIMATION OF A SHIFT PARAMETER
(Presented by Academician Yu. V. Linnik on 26 III 1969)
Let \((x_1, x_2, \ldots, x_n)\) be independent identically distributed random variables with distribution function (d.f.) \(F(x-\theta)\), depending on the shift parameter \(\theta \in R^1\). The problem of estimating the parameter \(\theta\) from a sample \((x_1, x_2, \ldots, x_n)\) of fixed size, after the fundamental work of Pitman \((^1)\), has been the subject of papers \((^{2-5})\), which substantially advanced the theory of estimation of a shift parameter. However, there are still many unsolved problems here, one of which is considered by us in § 1.
In our view, the problem of sequential estimation of a shift parameter (to which §§ 2, 3 are devoted) is very interesting, especially for d.f.’s \(F(x)\) concentrated on a bounded interval.
Probably, in this last case the efficiency of sequential estimation as compared with estimation from a sample of fixed size is the same as in problems of hypothesis testing. Although in \((^6)\) it is indicated that the existence of the Wolfowitz inequality does not allow sequential estimates to be noticeably superior in quality to the best estimates constructed from a sample of fixed size, our results show that, at least for distributions \(F(x-\theta)\), where \(F(x)\) is concentrated on a finite interval, this is not so. The point is that in this case the conditions required in deriving the Rao–Cramér and Wolfowitz inequalities, generally speaking, are not satisfied, and other considerations are required.
- Suppose that the d.f. \(F(x)\) is concentrated on a bounded interval. It may always be assumed that this interval is \((-a,a)\).
In estimating the shift parameter \(\theta\), it is natural to use equivariant estimates \(\bar\theta_n=\bar\theta_n(x_1,x_2,\ldots,x_n)\), satisfying, for all \(c\in R^1\), the condition
\[ \bar\theta_n(x_1+c,\ldots,x_n+c)=c+\bar\theta_n(x_1,\ldots,x_n). \]
Denote the class of equivariant estimates by \(\mathfrak R\). For \(\bar\theta_n\in\mathfrak R\), obviously, \(E_\theta(\bar\theta_n-\theta)^2\) does not depend on \(\theta\), and therefore there exists an optimal estimate \(\hat\theta_n\),
\[ E_\theta(\hat\theta_n-\theta)^2=\min_{\bar\theta_n\in\mathfrak R} E_\theta(\bar\theta_n-\theta)^2, \]
called the Pitman estimate.
Theorem 1. Let
\[ F(x)=\int_{-a}^{x} f(u)\,du \]
and let the density \(f(x)\), for some \(c_1,c_2,\alpha_1,\alpha_2,\ c_1>0,\ c_2>0,\ -1<\alpha_1<1,\ -1<\alpha_2<1\), satisfy the conditions \(f(x)\ge c_1(a+x)^{\alpha_1}\) in some neighborhood \(\Delta_1\) of the point \(x=-a\), and \(f(x)\ge c_2(a-x)^{\alpha_2}\) in some neighborhood \(\Delta_2\) of the point \(x=a\). Then, if \(\alpha=\max\{\alpha_1,\alpha_2\}\), then
\[ E_\theta(\hat\theta_n-\theta)^2 \le C n^{-2/(1+\alpha)}, \tag{1} \]
where the constant \(C\) depends only on \(a,c_1,c_2,\alpha_1,\alpha_2\) and \(\Delta_1,\Delta_2\).
This theorem gives, at least, a partial answer to the question posed by L. N. Bolshev about the behavior of the variance of the Pitman estimate depending on the form of the density \(f(x)\) at the endpoints of the interval \((-a,a)\).
Note that, whatever the d.f. \(F(x)\), the estimate
\[ E_\theta(\hat{\theta}_n-\theta)^2 \leq c/n . \]
Our estimate (1) is, obviously, better than this trivial one.
- We now turn to the scheme of sequential estimation of \(\theta\). Here the number of observations is determined by a stopping rule \(\tau\), and the estimates are statistics \(\bar{\theta}=\bar{\theta}(x_1,\ldots,x_\tau)\).
It seems natural to consider—for estimating the location parameter—invariant stopping rules \(\tau\), i.e., such that the sets \(\{\tau=n\}\), for \(n\geq 2\), belong to the \(\sigma\)-algebra generated by the vector
\[
Y_n=(x_2-x_1,\ldots,x_n-x_1);
\]
we shall also assume that \(\{\tau=1\}=\varnothing\).
If, as before, we restrict ourselves to equivariant estimates, for which, for all \(c\in R^1\),
\[ \bar{\theta}(x_1+c,\ldots,x_\tau+c) = c+\bar{\theta}(x_1,\ldots,x_\tau), \]
then for a stopping rule \(\tau\) the optimal equivariant estimate \(\hat{\theta}_\tau\) (it may be called the sequential Pitman estimate corresponding to the rule \(\tau\)) has the following form: on the set \(\{\tau=n\}\), \(\hat{\theta}_\tau=\hat{\theta}_n\).
Put \(x_{(n)}=\min\{x_1,\ldots,x_n\}\) and \(x^{(n)}=\max\{x_1,\ldots,x_n\}\). Fix some small \(\varepsilon>0\) and consider the stopping rule \(\tau=\tau_\varepsilon\):
\[ \tau_\varepsilon=\min\{n:\ x^{(n)}-x_{(n)}>2a-\varepsilon\}. \tag{2} \]
It is easy to see that \(\tau_\varepsilon\) is an invariant rule and that, upon stopping at time \(\tau_\varepsilon\), we can determine the value of \(\theta\) with accuracy up to \(\varepsilon\).
Theorem 2. Under the conditions of Theorem 1, for the stopping rule (2) we have
\[ E_\theta(\hat{\theta}_\tau-\theta)^2 \leq C(E_\theta\tau)^{-2/(1+\alpha)}, \tag{3} \]
where \(C\) depends on the same quantities as in Theorem 1.
Although estimate (3) is of the same order as (1) (the role of \(n\) in the sequential case is played by \(E_\theta\tau\)), it should be remembered that both are merely upper bounds for the variances of Pitman statistics. For particular distributions, \(E_\theta(\hat{\theta}_\tau-\theta)^2\) turns out to be substantially smaller than \(E_\theta(\hat{\theta}_n-\theta)^2\).
- Suppose that \(F(x)\) is the d.f. of the uniform law on \((-a,a)\), and denote \(2a=\Delta\). In this case
\[ \hat{\theta}_n=(x_{(n)}+x^{(n)})/2, \qquad E_\theta(\hat{\theta}_n-\theta)^2=\Delta^2/2(n+1)(n+2). \]
For rule (2) we have
\[ E_\theta\tau=2\Delta/\varepsilon, \qquad E_\theta(\hat{\theta}_\tau-\theta)^2=\varepsilon^2/24=\Delta^2/6(E_\theta\tau)^2. \]
If we choose \(\varepsilon=2\Delta/n\), then the relative efficiency of the estimate \(\hat{\theta}_\tau\) in comparison with \(\hat{\theta}_n\) will be
\[ \operatorname{reff}(\hat{\theta}_\tau,\hat{\theta}_n) = E_\theta(\hat{\theta}_n-\theta)^2/E_\theta(\hat{\theta}_\tau-\theta)^2 = 3n^2/(n+1)(n+2)\xrightarrow[n\to\infty]{}3. \]
Apparently, sequential Pitman estimates for the location parameter of a d.f. \(F(x-\theta)\) concentrated on a bounded interval are, in many cases, appreciably better than estimates constructed from a sample of fixed size.
The author expresses sincere gratitude to A. M. Kagan for posing the problem and for assistance in the work.
Leningrad State University
named after A. A. Zhdanov
Received
28 II 1969
REFERENCES
- E. G. Pitman, Biometrika, 30, III—IV (1938).
- C. Stein, Ann. Math. Stat., 30, 4 (1969).
- A. M. Kagan, Yu. V. Linnik, C. R. Rao, Sankhya, Ser. A, 27, 2—3—4 (1965).
- A. M. Kagan, Sankhya, Ser. A, 28, 3—4 (1966).
- A. M. Kagan, Proc. Steklov Inst. Math. Acad. Sci. USSR, 104, 19 (1968).
- N. L. Johnson, J. Roy. Statist. Soc., A, 124, 3 (1961).