Estimation of population parameters using sample extremes from nonconstant sample sizes

We examine the accuracy and precision of parameter estimates for both the exponential and normal distributions when using only a collection of sample extremes. That is, we consider a collection of random variables, where each of the random variables is either the minimum or maximum of a sample of nj independent, identically distributed random variables drawn from a normal or exponential distribution with unknown parameters. Previous work derived estimators for the population parameters assuming the nj sample sizes are constant. Since sample sizes are often not constant in applications, we derive new unbiased estimators that take into account the varying sample sizes. We also perform simulations to assess how the previously derived estimators perform when the constant sample size is simply replaced with the average sample size. We explore how varying the mean, standard deviation, and probability distribution of the sample sizes affects the estimation error. Overall, our results demonstrate that using the average sample size in place of the constant sample size still results in reliable estimates for the population parameters, especially when the average sample size is large. Our estimation framework is applied to a biological example involving plant pollination.