The Statistics of Sample Maximums

Consider the distribution of sample maximums for samples of a random variable normally distributed with a mean of 0.0 and a standard deviation of 0.1. For n=1 the sample maximum is just the sample value.

Analysis of Sample Maximum

If p(x) is the probability density function for a random variable x, let P(x) be the cumulative probability function; i.e.,

P(x) = ∫_-∞^xp(z)dz.
 

The probability that the maximum of a sample of size n is x is given by

[P(x)]^n-1p(x)
 

This is the probability density function q(x) for the sample maximums. For the case of a random variable uniformly distributed between -0.5 and +0.5 see Sample Maximums.

The sample maximum can be considered the limit of the following function:

(x₁^σ + … + x_n^σ)^1/σ
 

as the parameter σ increases without bound. For each finite value of σ the distribution of the σ-powers would approach a normal distribution as the sample size n increases without bound. The distribution of the σ-root would then be a transformation of a normal distribution.

For an unbounded distribution, such as the normal distribution used above, the expected value of the sample maximum increases with the size of the sample. The following shows the relationship based upon, in each case, 2000 samples.

The mean value of the sample maximums thus increases with sample size but not quite at the rate of the logarithm of sample size. The effect is important for observations such as the meteorological where such things as hurricanes are classified on the basis of maximum observed wind speed. Over time the number of observations of individual hurricanes has increased dramatically. This would lead to a systematic increase in the rating of hurricanes.

(To be continued.)