The Statistics of Sample Minimums

Consider the distribution of sample minimums 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 minimum is just the sample value.

Analysis of the Sample Minimum

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.
 

This is the probability that an observation of the variable has a value of x or less. The probability that it is x or more is 1−P(x).

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

[1−P(x)]^n-1p(x)
 

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

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

(x₁^σ + … + x_n^σ)^1/σ
 

as the parameter σ increases goes to zero. For each positive 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 minimum decreases with the size of the sample. The following shows the relationship based upon, in each case, 2000 samples.

The mean value of the sample minimums thus decreases with sample size but not quite at the rate of the negative 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 minimum observed pressure. 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.)