Efficient Confidence Interval Methodologies for the Noncentrality Parameters of Noncentral T-Distributions

Abstract

The problem of constructing a confidence interval for the noncentrality parameter of a noncentral t-distribution based upon one observation from the distribution is an interesting problem with important applications. A general theoretical approach to the problem is provided by the specification and inversion of acceptance sets for each possible value of the noncentrality parameter. The standard method is based upon the arbitrary assignment of equal tail probabilities to the acceptance set, while the choices of the shortest possible acceptance sets and UMP unbiased acceptance sets provide even worse confidence intervals, which means that since the standard confidence intervals are uniformly shorter than those of UMPU method, the standard method are "biased". However, with the correct choice of acceptance sets it is possible to provide an improvement in terms of confidence interval length over the confidence intervals provided by the standard method for all values of observation.

The problem of testing the equality of the noncentrality parameters of two noncentral t-distributions is considered, which naturally arises from the comparison of two signal-to-noise ratios for simple linear regression models. A test procedure is derived that is guaranteed to maintain type I error while having only minimal amounts of conservativeness, and comparisons are made with several other approaches to this problem based on variance stabilizing transformations. In summary, these simulations confirm that the new procedure has type I error probabilities that are guaranteed not to exceed the nominal level, and they demonstrate that the new procedure has size and power levels that compare well with the procedures based on variance stabilizing transformations.