Mathematics

Academician Yu. V. LINNIK

TESTS, UNBIASED ESTIMATES, AND COTEST IDEALS

Let some space of elementary events \(\mathfrak X\) be endowed with a \(\sigma\)-algebra \(\mathfrak B\) and with a family of probability measures \(P=\{p_\theta,\ \theta\in\Omega\}\), where \(\theta\) belongs to some subset of the real Euclidean space \(E_s\), and the measures \(p_\theta\) are dominated by one measure. We shall consider problems of testing the null hypothesis \(H_0:\ \theta\in\omega\) (\(\omega\) a proper nonempty subset of \(\Omega\)) against the alternative \(H_1:\ \theta\in\Omega\setminus\omega\), and also the construction of unbiased estimates \(\chi\) for some projection \(\theta_i\) of the vector parameter \(\theta=(\theta_1,\ldots,\theta_s)\).

Tests for \(H_0\) will be regarded as randomized and similar of level \(\alpha\): a test \(\Phi=\Phi(x)\) will have the property

\[ E(\Phi\mid \theta_0)=\alpha \quad \text{for } \theta_0\in\omega,\ x\in\mathfrak X . \tag{1} \]

A cotest under these conditions will mean any function \(\psi\) of the form \(\psi=A(\Phi-\alpha)\), where \(A\) is a constant; obviously, then \(E(\psi\mid\theta_0)=0\) for \(\theta_0\in\omega\); a precotest is any statistic \(\xi(x)\), measurable with respect to all \(p_\theta\), under the condition \(E(\xi(x)\mid\theta_0)=0\) for \(\theta_0\in\omega\). An unbiased estimate \(\chi\) of some component of \(\theta\), say \(\theta_1\), behaves, obviously, analogously to a precotest: \(E(\chi-\theta_1)=0\) for all \(\theta\in\Omega\), so that one may say that a precotest is an “unbiased estimate of zero” for \(\theta_0\in\omega\).

Let the family of measures \(\{p_\theta,\ \theta\in\omega\}\) be complete in the sense, slightly strengthening the definition of Lehmann–Scheffé \((^{1,2})\): if for some measurable function \(f(x)\) there exist and are given, for all \(\theta\), its “images”:

\[ E(f(x)\mid\theta)=\varphi(\theta), \tag{2} \]

then from them one can reconstruct the “preimage” \(f(x)\) with probability 1 (in particular, in this case the unbiased estimate \(\chi\) is unique).

Among measurable functions \(f(x)\) we shall consider, when they exist, such classes of functions \(\mathfrak K\) for which the following requirements are fulfilled:

For \(f\in\mathfrak K\) the image \(\varphi(\theta)=E(f(x)\mid\theta)\) is a holomorphic function in some polycylinder \(Z\) of the complex Euclidean space \(E_{2s}\), and the images of the functions \(f\in\mathfrak K\) form a ring \(K_\theta\) with operators from the field of complex numbers. The totality of functions \(\varphi(\theta)\in K_\theta\) that vanish for \(\theta\in\omega\) forms an ideal \(I_{H_0}\). All of them, obviously, are precotests, and we shall for brevity call \(I_{H_0}\) a cotest ideal (although the more exact name would be precotest ideal). If the ring \(K_\theta\) is sufficiently broad (which will be the case, in particular, for exponential families of distributions), then one may try to describe the cotest ideal \(I_{H_0}\) by finding its basis, and then, with the aid of this description, seek a cotest \(\psi\) corresponding to an optimal, in a certain sense, test \(\Phi\) of level \(\alpha\). We may, for example, introduce a Bayes measure \(B(\theta)\) on the set \(\Omega\setminus\omega\) corresponding to the alternative \(H_1\), and seek a precotest under the conditions \(-\alpha\le \xi\le 1-\alpha\), which would give the greatest possible value of the functional

\[ \int_{(\Omega\setminus\omega)} E(\xi\mid\theta)\,dB(\theta). \]

The indicated conditions make the pre-cotest \(\xi\) a cotest. A cotest \(\psi_\varepsilon\) for which

\[ \int_{(\Omega\setminus\omega)} E(\psi_\varepsilon\mid\theta)\,dB(\theta)\geq \sup_{\psi}\int_{(\Omega\setminus\omega)} E(\psi\mid\theta)\,dB(\theta)-\varepsilon, \tag{3} \]

where the supremum is taken over all cotests of level \(\alpha\), will be called \(\varepsilon\)-optimal.

The description of ideals in rings of holomorphic functions requires the apparatus of the theory of coherent analytic sheaves, developed in \((^3)\). At present sufficiently complete results are available only for ideals of the ring of all functions holomorphic in a certain simply connected compact polycylinder of the space \(\bar E_{2s}\) \((^4,^5)\). In the aspect considered by us, rings of this type are applicable, for example, in considering exponential families of distributions in a natural parametrization \(((^6), p. 51)\). Here we have:

\[ p_\theta=C(\theta)\exp\left(\sum_{j=1}^{s}\theta_jT_j\right)h(T_1,\ldots,T_s), \tag{4} \]

where \(T_1,\ldots,T_s\) are scalar sufficient statistics for \(\theta\); \(h(T_1,\ldots,T_s)\) is a density with respect to Lebesgue measure.

If, for simplicity, we assume that the sufficient statistics \(T_1,\ldots,T_s\) are nonnegative and \(h(T_1,\ldots,T_s)>0\), then one may consider the ring of all functions holomorphic in the product \(R_1\times\cdots\times R_s\) of half-planes \(R_i:\operatorname{Re}\theta_i>0\).

Let now the set \(\omega\subset\Omega\), corresponding to the hypothesis \(H_0\), be described by \(r<s\) conditions:

\[ \Pi_1\left(\frac1{\theta_1},\ldots,\frac1{\theta_s}\right)=0;\ldots;\quad \Pi_2\left(\frac1{\theta_1},\ldots,\frac1{\theta_s}\right)=0, \tag{5} \]

where the functions \(\Pi_j\) are holomorphic for \(\theta_i\ne0\); the \(\theta_i\) are real and subject to the condition \(\theta_1>\varepsilon_0,\theta_s>\varepsilon_0\), where \(\varepsilon_0\) is a small positive number. Then, under certain additional conditions imposed on the relations (5), it is possible to describe a basis of the pre-cotest ideal \(I_H\), and to reduce the problem of the optimal test \(\Phi\) of level \(\alpha\) (for a given Bayesian measure \(B(\theta)\)) to problems of linear programming. The situation described occurs, for example, in the construction of tests of the “even type” for a linear hypothesis in the method of least squares with unknown weights—in particular, the test for the Behrens–Fisher problem with sufficient \(\bar x,\bar y,s_1^2,s_2^2\) (in the usual notation), depending only on \(|\bar x-\bar y|,s_1^2,s_2^2\). In order to formulate one of the theorems relating to this question, replace in condition (5) \(\theta_l\) by \(p_l^{-1}\), so that we obtain the conditions \(\Pi_j(p_1,\ldots,p_l)=0\) \((j=1,2,\ldots,r)\); here \(p_l=x_l+iy_l\) will lie in the disk \((x_l-R^2)+y_l^2\le R^2\) for some sufficiently large \(R\). The product of these disks forms a polycylinder \(Z\). We shall further require that the analytic set \(\{\Pi_j=0,\ j=1,2,\ldots,r\}\) split in \(Z\) into a finite number of connected analytic sets, each of real dimension \(2(s-r)\) and each containing within itself a real analytic set of real dimension \(s-r\).

Next, we require that the set of points of \(Z\) under the conditions

\[ \Pi_1(p_1,\ldots,p_s)=0,\ldots,\Pi_r(p_1,\ldots,p_s)=0;\qquad \operatorname{rang} \begin{Vmatrix} \dfrac{\partial\Pi_1}{\partial p_1}\cdots\dfrac{\partial\Pi_1}{\partial p_s}\\ \cdots\cdots\cdots\\ \dfrac{\partial\Pi_r}{\partial p_1}\cdots\dfrac{\partial\Pi_s}{\partial p_s} \end{Vmatrix}<r \]

consisted of a finite number of points. Under these conditions, from K. Oka’s theorem ((5), p. 26; (4)) one can deduce that the cotest ideal \(I_{H_0}\), defined by the relations (5) in the real domain, will have a basis \(\Pi_1, \Pi_2, \ldots, \Pi_r\); this basis will have easily computable prototypes—pretests \(F_1, F_2, \ldots, F_r\). We can now formulate the theorem.

Theorem. Let the conditions (5), describing the null hypothesis \(H_0\), be subject to the requirements explained above. Let the density \(h(T_1,\ldots,T_s)\) in formula (4) be differentiable and satisfy the conditions:

\[ h(T_1,\ldots,T_s)=O\bigl(\exp \varepsilon(T_1+\cdots+T_s)\bigr) \qquad (\varepsilon>0 \text{ arbitrarily small}), \]

\[ T_j>0\ (j=1,2,\ldots,s),\qquad \partial \ln h/\partial T_j = O\left(\frac{1}{T_1}+\cdots+\frac{1}{T_s}\right), \]

if some \(T_j\to\infty\). Then the \(\varepsilon\)-optimal cotests can be given in the form

\[ \psi=\frac{1}{h}\{F_1*G_1+\cdots+F_r*G_r\}, \tag{6} \]

where \(G_j=G_j(x)\) are smooth functions, and \(*\) is the convolution operation for the one-sided Laplace transform.

For example, in order to find an \(\varepsilon\)-optimal similar test in the Behrens–Fisher problem, depending only on \(|\bar{x}-\bar{y}|, s_1^2, s_2^2\), one must find a smooth function \(H(T_1,T_2,\ldots,T_s)\) such that
\[ -\alpha \le (F_1*H)T_1^{1/2}T_2^{-m_1}T_3^{-m_2} \le 1-\alpha, \]
and for which the cotest
\[ \psi=(F_1*H)T_1^{1/2}T_2^{-m_1}T_3^{-m_3} \]
makes the functional (3) as large as possible. Here \(F_1=n_2T_1+n_1T_2-T_3\); \(n_1,n_2\) are the sample sizes; \(m_i=\frac12(n_i-3)\) \((i=1,2)\). Here the ideal \(I_{H_0}\) is principal (generated by one element)*. The indicated restrictions can be substantially weakened and a description of \(\varepsilon\)-optimal similar tests can be given for significant classes of families. By the same method one can study questions of optimality and “\(\varepsilon\)-optimality” of unbiased estimators.

REFERENCES

1. E. Lehmann, H. Scheffé, Sankhya, 10, 305 (1950).
2. E. Lehmann, H. Scheffé, Sankhya, 15, 219 (1955).
3. H. Cartan, Variétés analytiques complexes et cohomologie, Liège—Paris, 1953.
4. H. Cartan, Bull. Soc. Math. France, 78, 28 (1950).
5. K. Oka, Bull. Soc. Math. France, 78, 1 (1950).
6. E. Lehmann, Testing Statistical Hypotheses, N. Y., 1959.

* In the terminology of algebraic number theory.