Showing papers on "Coverage probability published in 1969"

A General Method of Determining Fixed-Width Confidence Intervals

Abstract: 0. Summary. A general method for determining stopping rules to obtain a fixed-width confidence interval of prescribed coverage probability for an unknown parameter of a distribution is obtained. Asymptotic theory in the sense of Chow and Robbins [4] is discussed. The sequential procedure obtained is asymptotically consistent and efficient in the sense of Chow and Robbins [4]. 1. Introduction. Fixed-width confidence interval estimation for the mean of a normal distribution has been considered by Ray [6] and Starr [7] etc. The analogous problem for the variance of a normal population has been considered by Graybill and Connell [5] by using two stage sampling. Chow and Robbins [4] have considered the problem of determining a confidence interval of prescribed width and prescribed coverage probability for the unknown mean of a population with unknown finite variance. They constructed a stopping rule and thereby developed an asymptotic theory in a certain sense. When there are some nuisance parameters present, presumbaly unknown, fixed sample size procedure will usually not work to obtain a fixed-width interval with a given coverage probability. But there are examples where there are no nuisance parameters and still the fixed sample size procedure does not work, e.g., for the variance of a normal population with zero mean. In all such cases a stopping rule can be adopted which will provide a bounded length confidence interval of given coverage probability. However, bounded length confidence intervals with prescribed coverage probability have been treated in few special cases. The object of this note is to give a general method of constructing sequential procedure for obtaining fixed-width confidence intervals of prescribed coverage probability for an unknown parameter of a distribution involving possibly some unknown nuisance parameters. The distribution involved will be assumed to be known except for the parameters. For the sake of simplicity, the discussion is restricted to the case of a single nuisance parameter since the case of several nuisance parameters is immediate. Let p (x, 01., 02) be the probability density function of a random variable X (for convenience with respect to Lebesque measure) with real valued parameters 01 and 02 where 02 is regarded as nuisance parameter. We want to determine a confidence interval of fixed-width 2d (d > 0) for 01 when both 01 and 02 are unknown, with preassigned coverage probability 1 - a (O < a < 1). ASSUMPTION. We assume that all the regularity assumptions of maximum likelihood estimation are satisfied.