Estimation of the parameters of a population from a multi-censored sample

The specific problems in this thesis were the estimation of parameters of univariate populations from samples where multiple-points of censorship occur. The case of a sample subjected to multiple points of censorship (on the right) may be described as follows: A sample of n items is tested; when the first one fails at time x1, a random sample of k1 is withdrawn from the n-1 items still in test; the remaining items are observed until the second item fails at time x2, when k2 items are withdrawn; and the process of withdrawing a prescribed number k i at the time xi when the i-th failure occurs continues until the r-th failure occurs at time xr, at which time the remainder of the items are withdrawn;The method of maximum likelihood is employed to estimate the parameters for the exponential, the normal, and the gamma distributions. These estimates are, in certain cases, difficult to obtain. They require iteration; therefore, certain practical limitations exist for their use. A new method of solving the likelihood equations for the normal distribution is introduced, and a Delta function is tabulated to facilitate the solution. An extension of the censorship procedure to another general type is considered for estimation by the method of maximum likelihood;The non-parametric estimate of the probability of surviving (quantiles) is obtained, and a general method of estimation based on the quantiles is presented, which will yield reasonable results when the method of maximum likelihood cannot be used, and which will be reasonably efficient in comparison to the maximum likelihood estimates when these are available for comparison. It is shown that the method of estimation from the quantiles yields the maximum-likelihood estimate for the exponential distribution for all rules of censorship and the uniform distribution for a random sample. The quantile method is asymptotically equivalent to the methods of maximum likelihood for the parameters of the normal distribution. The method yields a simple result (best linear unbiased estimate) for the uniform distribution with single or multi-censorship. This is an advantage over the maximum-likelihood method, which does not furnish a simple result;The above results are illustrated by a number of examples taken from industrial experiments. It is possible, through the techniques presented, to utilize small samples such as exist in industry, and also, although curtailment exists, to have assurance of a certain number of complete life times from which to make estimates even where no prior knowledge---other than the distributional form---exists on the life times of the items tested