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

What is it about?

In this article, we propose new formulas for how to estimate the mean of an exponential distribution when the only data available consists of 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 distribution. We also propose new formulas for how to estimate the mean and variance of the normal distribution for an analogous setting.

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Photo by Carlos Muza on Unsplash

Photo by Carlos Muza on Unsplash

Why is it important?

Our article is the first to our knowledge to present explicit formulas for computing parameters of an exponential or normal distribution when the only known information is a sample of sample maxima. Moreover, we demonstrate that our estimators are unbiased in the exponential setting and have negligible bias in the normal distribution setting, with the estimation precision increasing as the number of sample maxima increases.