Ein allgemeines Konstruktionsverfahren für Konfidenzbereiche

Summary A theorem ofTakács concerning interchangeable random variables is used to derive a simple method for the construction of confidence regions. Applying this method to a location parametera we get a.s. convergence of the confidence interval toa if the sample sizen increases while its probability is (n–1)/(n+1). Under certain conditions the interval contains always the maximum-likelihood estimate and another estimate which results from a least squares postulate. Lower bounds are given for the probability that our intervals become shorter than the intervals we would get relying on the central limit theorem. In order to avoid an assumption of finite support needed first to derive the a.s. convergence, we modify our method omitting extreme values.The modified intervals converge forn with probability 1 to the true parameter value under weaker conditions. A lower bound for the probability and-using a result due toRényi-theasymptotic probability of the modified interval is given. For the two kinds of intervals a formula concerning the velocity of their convergence to the length 0 is derived.Finally, the results are extended to a shift parameter in the two-sample case. Here we derive for equal sample sizes the exact probability of the modified interval and give upper and lower bounds fir its asymptotic probability. The method is practicable also if one sample size is an integer multiple of the other.