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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).