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The Distribution of Sample Maximums
as a Function of Sample Size for
an Unbounded Distribution

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) = ∫_{-∞}^{x}p(z)dz.

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

[P(x)]^{n-1}p(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:

(x_{1}^{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.

Likewise if the number of daily observations of temperature is changed there will be a corresponding
change in the daily maximum temperature even when no real change in climate has occurred.