S Sequential confidence intervals of given maximal length, Charles M. Stein 1945. Let X1;X2; : : : be independent, Nm;v-distributed random variables with unknown parameters m and v. Furthermore, let n 2 and 0 1I1˛=2 the ˛=2-fractile of the tn1-distribution. Consider the unbiased estimators

of m and v after n experiments. Finally, let " > 0 and consider the random sample size N D max¹n; d.t="/2V
n eº (where dxe denotes the smallest integer x) as well as the associated estimator MN D PN iD1 Xi=N of m after N experiments. Show the following.

(a) .MN m/
p N=v is independent of V
n and has standard normal distribution N0;1.

(b) .MN m/
p N=V
n is tn1-distributed, and MN t p V
n =N; MN Ct p V
n =NOE is a confidence interval for m with error level ˛ and total length at most 2".

Mn X and V (Xi-Mn) n