MATHEMATICS

A. M. KAGAN, V. N. SUDAKOV

ON THE STRUCTURE OF THE COMPLETE CLASS OF UNBIASED ESTIMATORS FOR FAMILIES OF DISTRIBUTIONS OF A SPECIAL FORM

(Presented by Academician Yu. V. Linnik on 17 II 1965)

Let a family of probability measures be given on the space \(\{\mathfrak X,\mathfrak A\}\),

\[ P_\theta=\sum_{i=1}^{m} c_i(\theta) P_i, \tag{1} \]

depending on an abstract parameter \(\theta\in\Theta\). In all that follows we shall assume that \(P_i,\ i=1,\ldots,m,\) are mutually absolutely continuous probability measures, and that the functions \(c_i(\theta),\ i=1,\ldots,m,\) are linearly independent and \(c_i(\theta)\ge 0\).

Below we shall be concerned with unbiased estimators \((^1)\) of functions of the parameter for the family (1). It is obvious that a function \(g(\theta)\) can be unbiasedly estimated if and only if

\[ g(\theta)=\sum_{i=1}^{m} a_i c_i(\theta), \tag{2} \]

where \(a_1,\ldots,a_m\) are certain real numbers. In this case every unbiased estimator \(\tilde g(x)\) of the function \(g(\theta)\) satisfies the condition

\[ \int_{\mathfrak X} \tilde g(x)\,dP_i=a_i,\qquad i=1,\ldots,m. \tag{3} \]

The usual measure of quality of an unbiased estimator \(\tilde g(x)\) of the function \(g(\theta)\) is

\[ E_\theta[\tilde g(x)-g(\theta)]^2 = \int_{\mathfrak X}[\tilde g(x)-g(\theta)]^2\,dP_\theta. \tag{4} \]

The main result of the present paper consists in a description of the complete class \((^2,\ \text{p. }31)\) of unbiased estimators of functions of the form (2) for families of distributions (1), when the loss function (4) is used.

Since

\[ E_\theta[\tilde g(x)-g(\theta)]^2 = \sum_{i=1}^{m} c_i(\theta)\int_{\mathfrak X}\tilde g(x)^2\,dP_i - g(\theta)^2 \]

and \(c_i(\theta)\ge 0\), it is clear that a necessary condition for the admissibility of \(\tilde g_1(x)\) as an unbiased estimator of \(g(\theta)\) is the nonexistence of a function \(\tilde g(x)\) such that

\[ \int_{\mathfrak X}\tilde g(x)\,dP_i=a_i,\quad i=1,\ldots,m;\qquad \int_{\mathfrak X}\tilde g(x)^2\,dP_i \le \int_{\mathfrak X}\tilde g_1(x)^2\,dP_i,\quad i=1,\ldots,m, \tag{5} \]

with strict inequality for at least one \(i\). Fix \(m\) numbers \(a_1,\ldots,a_m\), and denote by \(L\) the set of functions \(\tilde g(x)\) on \(\{\mathfrak X,\mathfrak A\}\) satisfying...

satisfying conditions (3) and

\[ \int_{\mathfrak X}\tilde g(x)^2\,dP_i<\infty,\qquad i=1,\ldots,m. \tag{6} \]

We now proceed to describe, for a nonempty subset \(L_1\subset L\), all such functions \(\tilde g_1(x)\) that, if \(\tilde g(x)\in L\) and \(\tilde g(x)\ne \tilde g_1(x)\), then the inequalities

\[ \int_{\mathfrak X}\tilde g_1(x)^2\,dP_i \ge \int_{\mathfrak X}\tilde g(x)^2\,dP_i,\qquad i=1,\ldots,m, \tag{7} \]

cannot hold simultaneously.

Consider the Hilbert space \(H\) of all functions square-summable with respect to the measure

\[ P=\frac1m\sum_{i=1}^m P_i; \]

it consists precisely of the functions satisfying condition (6). Introduce the operators

\[ A_k h(x)=p_k(x)h(x), \tag{8} \]

where \(p_k(x)=dP_k/dP,\ h\in H\). Obviously,

\[ 0\le p_k(x)\le m,\quad \sum_{k=1}^m p_k=m,\quad \|A_k\|\le m,\quad \int_{\mathfrak X}h(x)^2\,dP_k=(A_kh,h), \]

\[ \int_{\mathfrak X}h(x)\,dP_k=(h,p_k),\qquad L=\bigcap_{k=1}^m\{h\in H:\ (h,p_k)=a_k\}. \]

Now let \(A_1,\ldots,A_m\) be some bounded self-adjoint positive operators, and \(p_1,\ldots,p_m\) some functionals (we ignore the connection, due to the condition, between \(A_k\) and \(p_k\)). By mutual absolute continuity of the measures \(P_k\), the value \(\lambda=0\) is not an eigenvalue of any \(A_k\). Let

\[ S_k(\tilde g_1)=L\cap\{h:\ (A_kh,h)\le (A_k\tilde g_1,\tilde g_1)\}. \tag{9} \]

Since \(\lambda=0\) is not an eigenvalue of the operators \(A_k\), the ellipsoids \(S_k(\tilde g_1)\) are strictly convex. Therefore either their intersection consists of the single element \(\tilde g_1\), or the intersection of the interiors of these ellipsoids is nonempty, and then the estimator \(\tilde g_1(x)\) is not an admissible estimator of \(g(\theta)\).

We shall indicate all \(\tilde g_1\) for which

\[ \bigcap_k^m S_k(\tilde g_1)=\{\tilde g\}. \]

Each of the ellipsoids \(S_k\), unless it degenerates into a point, contains, in view of the boundedness of the operator \(A_k\), some open set and therefore in the affine space \(L\) has a supporting hyperplane passing through the point \(\tilde g_1\). Let \(T_k\) be the open half-space of the space \(L\) lying on the same side of this hyperplane as the ellipsoid \(S_k\); in the case when \(S_k\) degenerates into a point, we agree to set \(T_k=L\).

Lemma 1. \(\displaystyle \bigcap_{k=1}^m S_k=\{\tilde g_1\}\) if and only if

\[ \bigcap_{k=1}^m T_k=\varnothing. \]

Indeed, the cone \(\displaystyle \bigcap_{k=1}^m T_k\) is the cone generated by the set

\[ \bigcap_{k=1}^m \operatorname{int} S_k = \operatorname{int}\bigcap_{k=1}^m S_k. \]

Each half-space \(T_k\) is characterized by an outer normal \(n_k\) (emanating from \(\tilde g_1\)).

Lemma 2. \(\displaystyle \bigcap_{k=1}^{m} T_k=\varnothing\) if and only if \(\Gamma\{l_k\}\) contains a subspace different from a point, where \(l_k\) are the rays generated by the vector \(n_k\); \(\Gamma\) is the convex hull.

The formulation of the lemma also covers the degenerate case, if one agrees to regard \(l_k=L\) when \(T_k=\varnothing\).

Proof. If \(\Gamma\{l_k\}\) contains some straight line \(l\), then the intersection of the sets \(\overline{T}_k\) is contained in the orthogonal complement to \(l\) and therefore contains no open subset; the same property in this case is possessed also by \(\displaystyle \bigcap_{k=1}^{m} T_k\), i.e., it is empty. Conversely, if \(\displaystyle \bigcap_{k=1}^{m} T_k=\varnothing\), then \(\displaystyle \bigcap_{k=1}^{m} \overline{T}_k\) is contained in a proper subspace of the space \(L\). In this case \(\Gamma\{l_k\}\) contains its orthogonal complement.

Let us now write the vectors \(n_k\) explicitly. The vector \(n_k\) is the vector of the (external) normal to the ellipsoid \(S_k\) in \(L\). Consider the subspace \(L_0\subset H,\ L_0=L-\tilde g_1\). Let \(Q_k\) be the projection operator onto \(L_0\), orthogonal in the sense of the quadratic form \((A_kh,h)\), i.e., such that if \((A_kh,\tilde g_0)=0\) for all \(\tilde g_0\in L_0\), then \(Q_kh=0\). Under projection by means of \(Q_k\), the vector \(\tilde g_1\) passes into \(Q_k\tilde g_1\in L_0\), while the vector \(n_k\) preserves its direction. Since the vector \(Q_k\tilde g_1\) is orthogonal to the supporting plane in \(L_0\) to the ellipsoid \(Q_kS_k\) in the sense of the quadratic form \((\Pi_kA_kh,h)\) (where \(\Pi_k\) is the ordinary orthogonal projection operator onto \(L_0\)), and \(n_k\) is orthogonal to this supporting plane in the ordinary sense, we obtain

\[ n_k=\Pi_kA_kQ_k\tilde g_1. \tag{10} \]

Thus, \(\tilde g_1\in L_1\) if and only if the convex cone spanned by the vectors \(\Pi_kA_kQ_k\tilde g_1\) contains some straight line. Here, to compute the operator \(Q_k\), one must solve a finite system of linear equations.

To the description of \(L_1\) one may also add the following. Let
\[ r_k(\tilde g)=(A_kQ_k\tilde g,Q_k\tilde g)^{1/2}; \]
then \(r(\tilde g)=\{r_1(\tilde g),\ldots,r_m(\tilde g)\}\) maps \(L\) continuously into \(R^m\), and it is not difficult to see that each \(\tilde g_1\in L_1\) is uniquely determined by its image and even by its \((n-1)\) coordinates. Thus, \(L_1\) is homeomorphic to a bounded subset of \(R^m\) of dimension not exceeding \((n-1)\).

In conclusion we make two remarks.

Remark 1. The results set out above can be transferred to the case when the measures \(P_k\) are not mutually absolutely continuous.

Remark 2. We intend to discuss separately the application of the results obtained to the analysis of estimates for certain families of distributions of the form (1).

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
5 II 1965

REFERENCES

\(^{1}\) A. N. Kolmogorov, Izv. AN SSSR, Ser. Mat., 14, 4 (1950).
\(^{2}\) E. Lehmann, Testing Statistical Hypotheses, “Nauka,” 1964.