This research explores many analytical features of Fisher information (FI) in censored samples from univariate and bivariate populations and discusses their applications. We primarily focus on the FI contained in Type-II censored samples. The FI plays a significant role in determining an optimal sample size in a life-testing experiment while taking the expected duration of the experiment into account.
In Chapter 2 we investigate the linkage between unfolded and folded distributions in terms of FI in a single order statistic and in Type-II censored samples for symmetric distributions. We exploit this connection to simplify the efforts in finding the FI in order statistics and in Type-II censored samples from an unfolded distribution that is symmetric about zero. We have shown that 4n-3 independent computations of the expectations of special functions of order statistics from the folded distribution are needed to obtain FI in all single order statistics and all Type-II (right or left or doubly) censored samples for all random samples of size m up to n. We use this efficient approach to find the FI in order statistics and Type-II censored samples from the Laplace distribution using the expectations of functions of exponential order statistics that can be easily obtained.
We present in Chapter 3 the FI matrix (FIM) in censored samples from a mixture of two exponentials when the mixing proportion is unknown and when it is known. It is proved that every entry of the FIM is finite. As closed form expressions do not exist we pursue the simulation approach to generate reliable, close approximations to the elements of the FIM. However we found that the determinants of FIM are almost zero for any assumed values of the parameters when n is small. This supports the general knowledge that a very large sample is needed for a precise estimation of parameters in a finite mixture of exponential distributions.
Let Xi:n be the ith order statistic and Yi:n be its concomitant obtained from a random sample from the absolutely continuous Block-Basu (1974) bivariate exponential random variable (X,Y). For this model, Chapter 4 provides expressions for the elements of the FIM in censored samples (Xi:n, Yi:n,~1 Less than or equal to; i Less than or equal to; r} and studies the growth pattern of the FI relative to the total FI in the sample on the three parameters as r/n changes in (0,1). This is done for small and large sample sizes. The results show that the FI on the parameter associated with X, is always greater than the FI on the parameter associated with the concomitant Y. We calculate the FI per unit of experimental duration to suggest optimal sample sizes for life-testing experiments. We describe its implications on the design of censored trials. In all of our investigations we also consider left and doubly censored samples.