UDC 519.28

MATHEMATICS

I. S. DOBROKHOTOV

ON TESTS OF A GENERAL LINEAR HYPOTHESIS WITH UNKNOWN WEIGHTS OF OBSERVATIONS

(Presented by Academician Yu. V. Linnik on 30 III 1967)

In the book of Yu. V. Linnik \((^1)\), properties are indicated that characterize similar Scheffé-type tests as Neyman structures. Such tests have hitherto been known only in the Behrens—Fisher problem. The conditions found are satisfied, for example, by all tests based on the \(t\)-ratio, in particular by the well-known test of E. Barankin \((^3)\) for one problem on linear regression. In \((^1)\) the question is posed of extending the results obtained to the case of a general linear hypothesis with unknown weights of observations. The present note serves this purpose.

Let \(X_1,\ldots,X_N\), where \(X_i \in N(a_i,\sigma_i^2)\), be a repeated sample, and suppose that

\[ a_i=\sum_1^s \alpha_{ij}\beta_j \]

(\(\beta_j\) are new parameters, and the matrix \(\alpha=\|\alpha_{ij}\|\) is assumed given). Suppose that the \(\sigma_i^2\), generally speaking, are different. The general linear hypothesis \(H_0\) is the assertion

\[ \beta_1=B_1,\ldots,\beta_r=B_r, \]

where \(B_i\) \((i=1,\ldots,r)\) are given constants.

Denote by \(\mathfrak X\) the space generated by stochastically independent linear forms \(l_1,\ldots,l_\mu\) and \(L_1,\ldots,L_\nu\) such that

\[ E\{l_i\mid H_0\}=E\{L_j\}=0^* \quad (i=1,\ldots,\mu;\ j=1,\ldots,\nu), \]
\[ D\{l_i\}/D\{l_1\}=C_i \quad (i=1,\ldots,\mu), \]

and \(C_i>0\) does not depend on \(\sigma=(\sigma_1,\ldots,\sigma_N)\), whereas the forms \(L_j\) may have different variances.

Consider two continuous homogeneous functions \(T_1\) and \(T_2\) of degree \(n>0\) with the following properties:

\[ T_1(l_1,\ldots,l_\mu)>0,\quad \text{if }(l_1,\ldots,l_\mu)\ne(0,\ldots,0), \]
\[ T_2(L_1,\ldots,L_\nu)>0,\quad \text{if }(L_1,\ldots,L_\nu)=(0,\ldots,0), \]

and

\[ T_2>C>0, \tag{1} \]

if at least one of its arguments is equal to one. Then the following is valid.

Theorem. If a similar test \(\varphi\) (generally speaking, randomized) is defined on the space \(\mathfrak X\) and accepts the null hypothesis at least under the conditions

\[ T_1/T_2\leqslant \varepsilon \tag{2} \]

(\(\varepsilon>0\) sufficiently small) and

\[ \sqrt{Q_1}=\left[\sum_1^\mu \frac{l_i^2}{c_i}\right]^{1/2}<\delta \tag{3} \]

\[ {}^*\ \text{The mathematical expectations of the forms }L_j\text{ are equal to zero for any values }\beta=(\beta_1,\ldots,\beta_s). \]

(\(\delta>0\) is any fixed number), then the variances of the linear forms \(L_1,\ldots,L_\nu\) satisfy the relations

\[ D\{L_j\}/D\{l_1\}=d_j \qquad (j=1,\ldots,\nu), \tag{4} \]

and \(d_j>0\) do not depend on \(\sigma\).

It follows immediately from this that \(\varphi\) is a Neyman structure for the exponential family generated by the forms \(l_1,\ldots,l_\mu,L_1,\ldots,L_\nu\). For \(\mu=1\) and in the case of the Behrens—Fisher problem, we again obtain the result of Yu. V. Linnik \((^1)\).

As in \((^1)\), the proof is by contradiction and is based on consideration of the test-similarity condition

\[ E_\sigma\{\varphi\mid H_0\}=\alpha \quad \text{uniformly in } \sigma . \tag{5} \]

Let us briefly explain the proof. Suppose that (4) holds for \(j=1,\ldots,p<\nu\), and transform (5) as described in \((^1)\). As a result, \(\sigma\) are replaced by new parameters \(\theta\), which we regard as complex. Then we consider a parametric point that is singular for both sides of the transformed relation (5). In a neighborhood of this point the stated relation has the form

\[ \int_{\mathfrak x}\cdots\int \varphi\, \frac{dl_1\cdots dl_\mu\,dL_1\cdots dL_\nu}{Q^{\tau+(\mu+\nu)/2}} \]

\[ = A_1G_1(\tau)\xi^{-\tau}(i\xi)^{-\tau-(\nu-p)/2} \prod_{p+1}^{\nu}\left[d_{jN-1}i\xi(d_{jN}-d_{jN-1})\right]^{1/2}, \tag{6} \]

where

\[ Q=Q_1+\sum_{1}^{p}\frac{L_j^2}{d_j} +\sum_{p+1}^{\nu}L_j^2 \frac{i\xi}{d_{jN-1}i\xi+(d_{jN}-d_{jN-1})} +i\xi^2; \]

\(A_1\) is a positive constant; \(G_1(\tau)\) is a function regular in \(\operatorname{Re}(\tau)>0\), and \(\xi>0\) is a number which we henceforth make arbitrarily small. We are now interested in the behavior of the moduli of both sides of (6) as \(\xi\downarrow 0\). A lower estimate for the modulus of the right-hand side has the form

\[ B\xi^{-2\tau-(\nu-p)/2}, \]

where the symbol \(B\) denotes a bounded quantity. To obtain an upper estimate for the modulus of the left-hand side of (6), we divide the space \(\mathfrak x\) into “layers”:

I. \(\;2^{m-1}\xi\le \sqrt{Q_1}<2^m\xi,\)

II. \(\;\dfrac{1}{2^m}\xi\le \sqrt{Q_1}<\dfrac{1}{2^{m-1}}\xi\) inside (3),

III. \(\;2^{m-1}\delta\le \sqrt{Q_1}<2^m\delta\) in the remaining part of the space \(\mathfrak x\).

The estimate is carried out separately in each “layer.” In doing so, the properties of the functions \(T_1\) and \(T_2\) and the inequalities

\[ |Q|\ge |\operatorname{Re}Q|\ge Q_1,\qquad |Q|\ge |\operatorname{Im}Q|\ge B\xi^2 \]

are used for \(|L_j|<\xi\). In the end we obtain the desired estimate

\[ B\xi^{-2\tau}. \]

To avoid a contradiction in the behavior of both sides of (6), we must have \(\nu-p=0\).

Since the exponential family generated by the forms \(l_1,\ldots,l_\mu,L_1,\ldots,L_\nu\) is one-parametric under \(H_0\), it follows from the Lehmann—Scheffé theory that \(\varphi\) is a Neyman structure with respect to the statistic

\[ \sum_{1}^{\mu}\frac{l_i^2}{\sigma_i}+\sum_{1}^{\nu}\frac{L_j^2}{d_j}. \]

The forms \(l_i\) are naturally to be chosen in such a way that the indicated statistic is not sufficient under the specified alternative \(H_1\); then the test \(\varphi\) is nontrivial.

In many practical cases the sample consists of \(m\) independent subsamples of sizes \(n_k\) \((k=1,\ldots,m)\), drawn respectively from

\[ N(a_k,\sigma_k^2). \]

Then \(N=\sum_1^m n_k\), and \(\mu,\nu\) must satisfy the condition

\[ \mu+\nu \leq \min n_k . \]

The optimal choice of the forms \(l_1,\ldots,l_\mu,L_1,\ldots,L_\nu\) depends on the particular form of the functions \(T_1\) and \(T_2\) and can be carried out, for example, on the basis of the same considerations as in (2) or (3), if tests based on the \(F\)-distribution are constructed.

The result obtained admits a multivariate generalization. In this case \(T_1\) and \(T_2\) are replaced respectively by the matrices \(Q_1\) and \(Q_2\). Instead of inequality (2), acceptance of the null hypothesis is postulated under the condition

\[ \operatorname{sp} Q_1 Q_2^{-1} \leq \varepsilon . \]

Condition (1) takes the form

\[ |Q_2| > c > 0, \]

if all entries of the matrix \(Q_2\) are bounded.

Received
28 III 1967

REFERENCES

\(^{1}\) Yu. V. Linnik, Statistical Problems with Nuisance Parameters, “Nauka,” 1966.
\(^{2}\) H. Scheffe, Ann. Math. Statistics, 14, No. 1, 35 (1943).
\(^{3}\) E. W. Barankin, Proc. Berkeley Symp. on Math. Stat. and Prob., 1949.