Cramer-von Mises Type Statistics for the Goodness-of-Fit of Censored Data -- Simple Hypothesis.

摘要

This paper considers modifications of Cramer-von Mises type goodness-of-fit statistics based on the sample distribution function when observations are censored. The asymptotic distributions of the modified statistics, the Cramer-von Mises statistic W sub n squared, the Anderson-Darling statistic A sub n squared and the Watson statistic U sub n squared, are found to be sums of weighted chi-squared random variables, where the weights are eigenvalues of integral equations. The small sample distribution of the modified W sub n squared statistics are approximated by three-moment fits. Also the asymptotic power of the statistics is considered for shifts of location for the normal population, and shifts of scale for the normal and exponential populations. The modified statistics can also be partitioned into components as Durbin and Knott showed the standard statistics could be. These components are considered and their asymptotic relative efficiencies given for the same cases as the asymptotic power for for complete statistics above. Finally the use of the statistics is illustrated with some hypothetical data.