4.4.1 Stein's Procedure for Estimating the Mean of a Normal Distribution with Unknown Variance

It is known that there does not exist a fixed-sample size procedure for estimating the mean of a normal population (when the variance is unknown) with a confidence interval of fixed-width and specified confidence coefficient. Stein (1945) has presented a two-sample procedure, in which the size of the second sample depends upon the result of the first sample, for the problem of determining confidence intervals of preassigned length and confidence coefficient for the mean of a normal population with unknown variance. In order to make the length of the confidence interval free of the variance, it seems necessary to "waste" a small portion of the information contained in the sample.

Thus, in practical applications one would, if possible, modify this procedure still preserving this property, and use an interval of the same length, whose confidence coefficient (although a function of ?) is always greater than the desired value and at the same time, reducing the expected number of observations by a small amount. The two-sample procedure will be a special case of sequential estimation. It is further shown by Stein (1945) that if the variance and initial sample size are sufficiently large, the expected number of observations differs only slightly from the number of observations required by a single sample interval estimation procedure when the variance is known.

Let X _i ( i=1, 2,...) be independent normal variables having mean