Marshall University Math Colloquium
March 9, 2012
University of New Hampshire
In this presentation, I am going to talk about the problems of computing confidence interval, tolerance interval and prediction interval based on the samples with non-detectable values, i.e. Type I censored samples from Normal and related distri- butions. Firstly two types of imputation approach has been investigated: one based on the maximum likelihood estimates (MLEs) of the parameters, and the second uses some ad hoc estimates, that are particularly suitable for sample sizes that are small or moderately large. Monte Carlo simulation is used to investigate the performance of these procedures. For a given percentage of values below the DL, this proposed imputation approach exhibits excellent performance when the sample size is small to moderately large. However, as the sample size gets large, the ad hoc procedure performs poorly; but the MLE based procedure continues to perform reasonably well unless the sample size gets very large. However, the conï¬dence levels can be cali- brated so that the MLE based imputation approach continues to provide coverage probabilities close to the nominal level.
I will also talk about the inferential problems concerning the mean and the quan- tiles of a lognormal distribution based on a Type I censored sample. Here procedures based on generalized conï¬dence intervals and modiï¬ed signed log-likelihood ratio test (MSLRT) statistics are used. Performance of these two procedures along with that of the signed log-likelihood ratio test (SLRT) statistic is compared using Monte Carlo simulation. Based on numerical results, it is found that the generalized conï¬dence interval and the MSLRT based conï¬dence interval are both satisfactory for inference concerning a lognormal quantile, when the percentage of non-detects is fairly large, as large as 70%. However, the conclusion is not so clear cut for inference on the lognormal mean. In fact, this work shows that the routine application of the MSLRT must be approached with caution, the procedure may even give results that are less satisfactory compared to the SLRT based solution. A ï¬nal point to note is that the generalized conï¬dence interval idea is easier to understand and implement, especially for a practitioner, and it provides accuracy very similar to that of the MSLRT for estimating the lognormal quantiles. For each of the problems considered, the results the illustrated using practical examples.