Large deviations in a linear model with nuisance parameters

The well-known approcimative approach to the problem of studentization in testing linear hypothesis due to B. L. Welch (Biometrika, 1947, v. 34, p. 28) permits a number of generalizations. One of them is connected with examination of linear models when the weights of observations are nuisance parameters, one else with constructing multiplicative analogues to the Welch's approach in the sense attached to this term by O. V. Šalaevskii (Zap. nauchn. sem. LOMI, 1978, v. 79, p. 88). In the present paper such an analogue is constructed for a general linear model when population variances are unknown.