A two-stage procedure for estimation of the common mean of several normal populations having unequal and unknown variances

Abstract

The problem of fixed-width confidence interval estimation of the common mean of several normal populations having unknown and unequal variances is considered. A two-stage procedure is developed to tackle the problem and its nice asymptotic properties are studied. Zacks (9) considered sequential procedures to construct fixed-width confidence interval for the common mean of two normal populations assuming one population's variance to be known. Mukhopadhyay and Narayan (6) generalized the results of Zacks (9) when both the variances were unknown. They also considered the problem of constructing fixed-width confidence interval for the common location parameter of two exponential populations having unequal and unknown scale parameters. Costanza, Hamdy and Son (4) dealt with the problem of constructing a confidence interval of pre-assigned width and coverage probability for the common location parameter of several exponential populations, assuming their scale parameters to be unknown and unequal. They established the non- existence of the fixed sample size procedure to deal with this estimation problem. In order to tackle with the problem, along the lines of Stein (8), they developed a 'routine-type' two-stage procedure. For the same estimation problem Chaturvedi, Tiwari and Pandey (2) developed three-stage, accelerated sequential and purely sequential procedures and obtained the associated second-order approximations. In a later communication, Chaturvedi, Tiwari and Pandey (3) considered the problem of bounded risk point estimation of the common location parameter of several exponential distributions and proposed some multistage procedures and derived the second-order approximations. In the present note, we consider the problem of fixed-width confidence interval estimation of the common mean of several normal populations having unknown and possibly unequal variances. In Section 2, we consider the set-up of the estimation problem and establish the failure of the fixed sample size procedure to deal with it. In Section 3, a two-stage procedure is developed for the fixed-width confidence interval estimation problem and its asymptotic properties are obtained.