Using the sample maximum to estimate the parameters of the underlying distribution

We propose novel estimators for the parameters of an exponential distribution and a normal distribution when the only known information is a sample of sample maxima; i.e., the known information consists of a sample of m values, each of which is the maximum of a sample of n independent random variables drawn from the underlying exponential or normal distribution. We analyze the accuracy and precision of the estimators using extreme value theory, as well as through simulations of the sampling distributions. For the exponential distribution, the estimator of the mean is unbiased and its variance decreases as either m or n increases. Likewise, for the normal distribution, we show that the estimator of the mean has negligible bias and the estimator of the variance is unbiased. While the variance of the estimators for the normal distribution decreases as m, the number of sample maxima, increases, the variance increases as n, the sample size over which the maximum is computed, increases. We apply our method to estimate the mean length of pollen tubes in the flowering plant Arabidopsis thaliana, where the known biological information fits our context of a sample of sample maxima.