Abstract Generated abstract
The paper studies the existence of regularly varying similar tests for a generalized Behrens-Fisher problem involving two independent normal samples with linear mean structures and unknown variances. It formulates invariance and sufficiency conditions extending Wald-type axioms, reduces admissible test functions to functions of transformed statistics, and derives the corresponding one-parameter family of densities under the null hypothesis. Using the polynomial structure of the denominator, properties of its negative real roots, and analytic continuation of the similarity condition in the variance ratio, the argument shows that the required integral identity cannot hold. The main conclusion is that no regularly varying similar test exists for this generalized Behrens-Fisher setting.
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MATHEMATICS
O. V. SHALAEVSKII
ON THE NONEXISTENCE OF REGULARLY VARYING TESTS FOR THE BEHRENS–FISHER PROBLEM
(Presented by Academician V. I. Smirnov on 7 II 1963)
The concept of a regularly varying test was introduced in \((^1)\). We shall consider the question of the existence of similar tests of this type for the Behrens–Fisher problem, treating the latter somewhat broadly.
Let the column vector \(L_1\) and the column vector \(L_2\) be drawn independently, respectively, from \(n_1\)- and \(n_2\)-dimensional normal populations with column vectors of means \(\Lambda_1,\Lambda_2\) and matrices of second moments \(\sigma_1^2 E_{n_1 n_1}, \sigma_2^2 E_{n_2 n_2}\). Let \(\Lambda_1=A_1\Xi_1\) and \(\Lambda_2=A_2\Xi_2\), where \(A_i=(A_i)_{n_i m_i}\), \(\operatorname{rank}(A_i)=m_i<n_i\), \(\Xi_i=(\Xi_i)_{m_i,1}\), \(i=1,2\). Introduce matrices \(G_i=(G_i)_{k m_i}\) of rank \(k\leq m_i\), \(i=1,2\), and construct the column vector \(H=G_1\Xi_1+G_2\Xi_2\). The vectors \(\Xi_1\) and \(\Xi_2\) will be regarded as unknown, as will \(\sigma_1,\sigma_2\) and the ratio \(\vartheta=\sigma_1^2/\sigma_2^2\). Under these conditions we are interested in tests suitable for testing the hypothesis \(H=0\) (\(0\) is the null vector) and satisfying certain restrictions.
We shall require that such a test for our problem be defined by an appropriately measurable function \(G(L_1,L_2)\), for which the critical regions \(\mathfrak R_C: G(L_1,L_2)\geq C\) would satisfy the following conditions:
- The regions \(\mathfrak R_C\) lie in the space of sufficient statistics. The likelihood function for \(L_1\) and \(L_2\) can be represented in the form
\[ \frac{1}{(2\pi)^{\frac{n_1+n_2}{2}}\sigma_1^{n_1}\sigma_2^{n_2}} \exp\left\{-\frac12\sum_{i=1}^2\frac{1}{\sigma_i^2} \left([vv]_i+X_i^T B_iX_i-2\Xi_i^T B_iX_i+\Xi_i^T B_i\Xi_i\right)\right\}, \]
where \(B_i=A_i^T A_i\), \(X_i=B_i^{-1}A_i^T L_i\), \([vv]_i=(L_i-A_iX_i)^T(L_i-A_iX_i)\). This gives the sufficient statistics \(X_1,X_2,[vv]_1,[vv]_2\). Condition 1 means that if \((L_1,L_2)\in\mathfrak R_C\), and for \((L_1',L_2')\) \(X_i'=X_i,\ [vv]_i'=[vv]_i,\ i=1,2\), then also \((L_1',L_2')\in\mathfrak R_C\).
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If \((L_1,L_2)\in\mathfrak R_C\), then \((L_1+A_1C_1,L_2+A_2C_2)\in\mathfrak R_C\) for any vectors \(C_i=(C_i)_{m_i,1}\) for which \(G_1C_1+G_2C_2=0\).
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If \((L_1,L_2)\in\mathfrak R_C\), then for any \(k\ne0\), \((kL_1,kL_2)\in\mathfrak R_C\).
Let us note that these conditions generalize in a natural way the well-known axioms of A. Wald \((^2)\). Moreover, by a method analogous to that given in \((^3)\), one can construct “approximate” similar regions of any size; these regions will satisfy the three conditions formulated.
Conditions 1–3 determine the form of the function \(G\). Applying condition 2 first with
\(C_1=-X_1+G_1^T(G_1G_1^T)^{-1}G_1X_1\) and \(C_2=0\), and then with
\(C_1=G_1^T(G_1G_1^T)^{-1}G_2X_2\) and \(C_2=-X_2\); applying condition 3 with \(k=1/\sqrt{[vv]_2}\), we find \(G=g(H,h)\), where
\(H=(G_1X_1+G_2X_2)/\sqrt{[vv]_2}\), \(h=\sqrt{[vv]_1/[vv]_2}\), and
\((H,h)\in\Omega=(-\infty<H<\infty,\ 0\leq h<\infty)\). In the space \(\Omega\) we have a one-parameter family of densities
\[ C\vartheta^{\frac{n_2-m_2+k}{2}} |M|^{\frac{n_1-m_1+n_2-m_2+k-1}{2}} \frac{h^{n_1-m_1-1}} {\left[\,|M|(\vartheta H^T M^{-1}H+h^2+\vartheta)\,\right]^{\frac{n_1-m_1+n_2-m_2+k}{2}}}, \]
\[ M=\vartheta G_1B_1^{-1}G_1^T+G_2B_2^{-1}G_2^T. \]
The denominator
\[ |M|\bigl(\vartheta H^T M^{-1}H+h^2+\vartheta\bigr) \]
is a polynomial of degree \(k+1\); all its roots are real and negative. For the root whose absolute value is greater than the maximal eigenvalue of the regular pencil of forms
\[ G_2B_2^{-1}G_2^T-\lambda G_1B_1^{-1}G_1^T, \]
we always have
\[ |\vartheta|>H^T(G_1B_1^{-1}G_1^T)^{-1}H+h^2 . \]
By the boundary measure, at least one such root exists. Let \(\vartheta_i=\vartheta_i(H,h)\), \(i=1,\ldots,k+1\), be the roots, with \(|\vartheta_{k+1}|\) greater than the indicated maximal eigenvalue.
Consider a regularly varying similar test \(g(H,h)\). For definiteness, let
\[ \operatorname*{vrai\,max}_{K} g(H,h)<\operatorname*{vrai\,max}_{\Omega} g(H,h) \]
whatever half-ball \(K\subset\Omega\) with center at the origin is taken. By choosing the radius of the half-ball large, one can push the root \(\vartheta_{k+1}\) arbitrarily far away. Define the function
\[ \Psi(z)= \begin{cases} 1, & z>\operatorname*{vrai\,max}_{K} g(H,h),\\ 0, & z<\operatorname*{vrai\,max}_{K} g(H,h). \end{cases} \]
From the condition of similarity it follows that
\[ \int_{\Omega}\Psi[g(H,h)]\, \frac{h^{\,n_1-m_1-1}\,dH\,dh} {\bigl[(\vartheta-\vartheta_1)\cdots(\vartheta-\vartheta_{k+1})\bigr]^{ \frac{n_1-m_1+n_2-m_2+k}{2}}} = \]
\[ = C_{\Psi}\,\vartheta^{-\frac{n_2-m_2+k}{2}} \left[\prod_{i=1}^{k}(\vartheta+\lambda_i)\right]^{ -\frac{n_1-m_1+n_2-m_2+k-1}{2}}, \tag{1} \]
where \(\lambda_1,\ldots,\lambda_k\) are the eigenvalues of the pencil
\[ G_2B_2^{-1}G_2^T-\lambda G_1B_1^{-1}G_1^T . \]
The integral relation (1) admits analytic continuation in \(\vartheta\), in any case into a small angle containing the positive half-axis. Meanwhile, after moving slightly from a sufficiently distant point of the half-axis, one can find that the right- and left-hand sides of (1) cannot coincide. Let \(\vartheta=Re^{i\varphi}\), with \(R\) sufficiently large and \(\varphi>0\) sufficiently small. Then the values of the integrand in (1) (and, consequently, the integral over \(\Omega\) itself) will lie inside a certain angle, whereas the right-hand side of (1) will lie outside this angle. This follows from the inequality
\[ (n_1-m_1+n_2-m_2+k)\sum_{i=1}^{k+1}\arg(\vartheta-\vartheta_i)\le \]
\[ \le \varepsilon+(n_1-m_1+n_2-m_2+k)\,k\,\arg\vartheta < (n_2-m_2+k)\arg\vartheta+ \]
\[ +(n_1-m_1+n_2+m_2+k-1)\,k\,\arg\!\left(\vartheta+\max_i \lambda_i\right), \]
which holds for a large radius of the half-ball \(K\), large \(R\), and small \(\varphi>0\).
Thus, the following assertion is true:
Theorem. A regularly varying similar test for the Behrens–Fisher problem cannot exist.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
2 I 1963
CITED LITERATURE
- Yu. V. Linnik, O. V. Shalaevskii, DAN, 150, No. 1 (1963).
- A. Wald, Selected Papers in Statistics and Probability, N. Y., 1955, p. 669.
- O. V. Shalaevskii, DAN, 130, No. 1 (1960).