MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR Yu. V. LINNIK, I. V. ROMANOVSKII,
V. N. SUDAKOV

A NONRANDOMIZED HOMOGENEOUS TEST IN THE BEHRENS–FISHER PROBLEM

For two repeated normal samples:

\[ x_1,\ldots,x_{n_1}\in N(a_1,\sigma_1^2);\qquad y_1,\ldots,y_{n_2}=N(a_2,\sigma_2^2) \]

the Behrens–Fisher problem of testing the hypothesis \(H_0:\ a_1=a_2\) is considered by means of tests \(\phi(x_1,\ldots,x_{n_1};y_1,\ldots,y_{n_2})\) similar with respect to the nuisance parameters. For the general problem of describing all such randomized tests, at present one can construct a system of integro-differential equations with peculiar boundary conditions. One can also construct many types of nonrandomized similar tests (among which the best known is the Bartlett–Scheffé test \((^1)\)). In the present note we deal with the narrower, unsolved question of the existence of nonrandomized homogeneous similar tests:

\[ \phi=\phi\left(\frac{\bar{x}-\bar{y}}{s_2},\,\frac{s_1}{s_2}\right), \]

where \(\bar{x},\bar{y},s_1,s_2\) are sufficient statistics of the problem in the usual notation. Attempts at a formal construction of such tests have not given convincing results (see the literature up to 1955 in the survey of G. Breni \((^2)\)). A. Wald \((^3)\) constructed an approximately similar test of this type. If the boundary separating the region of values \(\phi=\phi\left(\frac{\bar{x}-\bar{y}}{s_2},\,\frac{s_1}{s_2}\right)=1\) from \(\phi=0\) satisfies certain conditions of piecewise smoothness, then nonrandomized tests with a critical region of the form

\[ \frac{\bar{x}-\bar{y}}{s_2}=\Phi\left(\frac{s_1}{s_2}\right) \]

do not exist (Yu. V. Linnik \((^{4,5})\)); concerning families of homogeneous tests see O. V. Shalaevskii \((^6)\). However, if only measurability is required of a nonrandomized homogeneous test \(\phi\), then, as it turns out, it exists for any level.

Theorem 1. For any level \(\alpha\in(0,1)\) and pairs of samples of sizes \(n_1,n_2\) of different parity, there exists a measurable nonrandomized similar test for the Behrens–Fisher problem with critical region determined by the values

\[ \frac{|\bar{x}-\bar{y}|}{s_2}\quad\text{and}\quad \frac{s_1}{s_2}. \]

Theorem 1 can be somewhat strengthened:

Theorem 2. Suppose a finite number \(K\) of pairs of samples of sizes \(n_{1i}, n_{2i}\) \((i=1,2,\ldots,K)\) of different parity is given. Then there exists a measurable nonrandomized homogeneous test

\[ \phi=\phi\left(\frac{|\bar{x}-\bar{y}|}{s_2},\,\frac{s_1}{s_2}\right), \]

which is similar simultaneously for all these pairs of samples and has the prescribed level.

We shall briefly outline the main steps in the construction of the test indicated in Theorem 1. Let a level \(\alpha\in(0,1)\) be given. To construct the desired test \(\phi\), consider the “test complement” \(\psi=\phi-\alpha\), a measurable function taking only the values \(1-\alpha\) and \(-\alpha\). Introduce the “critics” of the family of measures induced by our pair of samples:

\[ A=A\left(\frac{|\bar{x}-\bar{y}|}{s_2},\,\frac{s_1}{s_2}\right);\quad B=B\left(\frac{|\bar{x}-\bar{y}|}{s_2},\,\frac{s_1}{s_2}\right) \]

(see (5, 7)). Then the question reduces to the construction of a measurable function \(\psi(A,B)\), taking only the values \(1-\alpha\) and \(-\alpha\), such that:

\[ J(\theta)=\iint_{\Pi}\frac{\psi(A,B)(AB)^{(n_1-3)/2}(B-A)\,dA\,dB}{\sqrt{1-A}\sqrt{B-1}(\theta+A)^N(\theta+B)^N}=0 \tag{1} \]

for all \(\theta>0\). Here \(\Pi\) is the half-strip \(0\leq A\leq 1;\ 1\leq B<\infty;\ N=(n_1+n_2-1)/2\); in what follows we shall assume \(n_2>n_1\geq 3\) (the latter condition is introduced here to simplify the reasoning).

We divide the half-strip \(\Pi\) into disjoint half-strips \(\Pi_k\): \(1-2^{-k}\leq A<1-2^{-k-1};\ 1\leq B<\infty;\ \Pi=\bigcup_{k=0}^{\infty}\Pi_k\). We shall construct the function \(\psi(A,B)\) so that for every \(k=0,1,2,\ldots\) one has

\[ J_k(\theta)=\iint_{\Pi_k}\frac{\psi(A,B)(AB)^{(n_1-3)/2}(B-A)\,dA\,dB}{\sqrt{1-A}\sqrt{B-1}(\theta+A)^N(\theta+B)^N}=0 \tag{2} \]

for all values \(\theta>0\). Since the function \((B-A)/\sqrt{1-A}\sqrt{B-1}\) has only an integrable singularity at \((1,1)\) of our half-strip, and from the condition \(n_2>n_1\geq 3\), the relations (2) for \(k=0,1,2,\ldots\) will imply (1) for all \(\theta>0\). We shall consider the left-hand side of (2) for some value \(k\geq 0\). Since \(n_1\) and \(n_2\) have different parity, \(N=(n_1+n_2-1)/2\) is an integer, and the expression

\[ \frac{1}{(\theta+A)^N(\theta+B)^N} \]

can be decomposed into simple rational fractions. As a result, for any bounded and measurable \(\psi(A,B)\) we find:

\[ \begin{aligned} J_k(\theta)= &\sum_{m=1}^{N}D_m \int_{1-2^{-k}}^{1-2^{-k-1}} dA \int_{1}^{\infty} dB\, \frac{\psi(A,B)(AB)^{(n_1-3)/2}(B-A)^m} {\sqrt{1-A}\sqrt{B-1}(B-A)^{2N-1}(\theta+A)^m} \\ &+\sum_{m=1}^{N}E_m \int_{1}^{\infty} dB \int_{1-2^{-k}}^{1-2^{-k-1}} dA\, \frac{\psi(A,B)(AB)^{(n_1-3)/2}(B-A)^m} {\sqrt{1-A}\sqrt{B-1}(B-A)^{2N-1}(\theta+B)^m}. \end{aligned} \tag{3} \]

Here \(D_m,E_m\) are constants. Such a decomposition is possible, since \(A\ne B\) in the half-strip \(\Pi_k\). We next use the following lemma:

Lemma (I. V. Romanovskii, V. N. Sudakov). Let a finite number of measurable probability densities \(p_m(x,y)\), \(m=1,2,\ldots,M\), be given on the rectangle \(Q:\ a\leq x\leq b;\ c\leq y\leq d\). Then for any prescribed \(\alpha\in(0,1)\) there exists a measurable function \(I(x,y)\), taking only the values \(0\) and \(1\), such that for almost all \(x\) (respectively \(y\))

\[ E^{(m)}(I(x,y)\mid x)=\alpha;\qquad E^{(m)}(I(x,y)\mid y)=\alpha \]

for \(m=1,2,\ldots,M\); \(E^{(m)}(\cdot\mid\cdot)\) denotes conditional expectation under the probability density \(p_m(x,y)\).

Since \(\alpha\) is different from 0 and 1, \(I(x,y)\) is obviously nontrivial (not almost everywhere constant). Setting \(\xi(x,y)=I(x,y)-\alpha\), we obtain a nontrivial measurable function with values \(1-\alpha\) and \(-\alpha\), such that

\[ \int_{a}^{b} p_m(x,y)\xi(x,y)\,dy=0;\qquad \int_{c}^{d} p_m(x,y)\xi(x,y)\,dx=0 \tag{4} \]

for almost all \(x\) (respectively \(y\)). It is clear that the \(p_m(x,y)\) may also be non-normalized, but only such nonnegative functions that

\[ \iint_Q p_m(x,y)\,dx\,dy<\infty \]

for \(m=1,2,\ldots,M\).

Turning to the half-strip \(\Pi_k\), we divide it into rectangles \(Q_s\): \(1-2^{-k}\leq A<1-2^{-k-1}\); \(s\leq B<s+1\); \(s=1,2,\ldots\). In each rectangle \(Q_s\), according to the lemma, construct a function \(\psi(A,B)\) with properties of type (4), where

\[ p_m(A,B)=\frac{(AB)^{(n_1-3)/2}(B-A)^m}{\sqrt{1-A}\sqrt{B-1}(B-A)^{2N-1}} \]

is the unnormalized probability density in \(Q_s\). Thus:

\[ \int_{1-2^{-k}}^{1-2^{-k-1}} dA\,\psi(A,B)p_m(A,B) = \int_s^{s+1} dB\,\psi(A,B)p_m(A,B)=0 \tag{5} \]

for almost all \(B\) (respectively, \(A\)); \(m=1,2,\ldots,N\).

From expression (3) we see that, for the function \(\psi(A,B)\) defined in this way, the integral \(J_k(\theta)\) will vanish for all \(\theta>0\), so that (2) will be true and, on the basis of what was said above, (1) is satisfied.

Thus the test \(\phi(A,B)=\psi(A,B)+\alpha\) will be a nonrandomized measurable similar test of level \(\alpha\), which proves Theorem 1.

To prove Theorem 2, instead of \(N\) functions \(p_m(A,B)\) we consider a finite number of functions \(p_{mi}(A,B)\), where \(i=1,2,\ldots,K\); for each \(i\),

\[ m\leq N_i=\frac{n_{1i}+n_{2i}-1}{2}; \]

for given \(i\) and \(m\), \(p_{mi}(A,B)\) are constructed with the aid of \(n_{1i}, n_{2i}\), and \(m\) in the same way as \(p_m(A,B)\) with the aid of \(n_1, n_2\), and \(m\). Of course, the sufficient statistics \(\bar{x},\bar{y},s_1,s_2\) will be taken for the given pair of samples, so that the test will depend on the number \(i\) of the pair of samples through them. We also note that, for the level \(\alpha=1/2\), a measurable nonrandomized similar test can be constructed by means of the theory of primes of a Gaussian field. Such a construction requires the use of the distribution law of the primes of a Gaussian field and is rather cumbersome.

References

1. H. Scheffe, Ann. Math. Stat., 14, No. 1, 35 (1943).
2. H. Brény, Travaux de Statistica, 6, Cuaderno, 2, 111 (1955).
3. A. Wald, Selected Papers in Probability and Statistics, N. Y., 1955, p. 669.
4. Yu. V. Linnik, DAN, 152, No. 3, 548 (1963).
5. Yu. V. Linnik, DAN, 150, No. 2, 254 (1963).
6. O. V. Shalaevskii, DAN, 151, No. 3, 509 (1963).
7. Yu. V. Linnik, Theory of Probability and Its Applications, vol. 1, 16 (1964).