Not All Sample Statistics
Approximate a Normal Distribution
as Sample Size Increases

Consider the distribution of sample minimums for samples of a random variable uniformly distributed between -0.5 and +0.5. For n=1 the sample minimum is just the sample value.

The above distributions indicate the necessity that for an extension of the central limit theorem to apply, the sample statistic must be representable as a sum.

Analysis of 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.
 

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

n[1-P(x)]^n-1p(x)
 

The quantity (1-P(x)) represents the probability that the random variable has a value greater than or equal to x. The (n-1) power of (1-P(x)) is the probability that all but one value of the sample has a value greater than or equal to x. The factor of n represents the fact that the minimum could occur for anyone of the n sample values.

This is the probability density function q(x) for the sample minimum. When p(x) is the uniform density function

p(x) = 1 for -0.5≤x≤+0.5
p(x) = 0 for all other values of x
 

then P(x) = (x-(-0.5) = x+0.5 for -0.5≤x≤+0.5, 1-P(x)=0.5-x. Thus the probability density function for the sample minimum is given by:

q(x) = 0 for x≤-0.5
q(x) = n(0.5-x)^n-1 for -0.5≤x≤+0.5
q(x) = 0 for +0.5≤x
 

As n increases without bound this distribution goes to a spike function at x=-0.5. There is thus there appears to be no tendency for the distribution of sample minimums to approach a normal distribution. The character of the sample distribution is dictated by the bounded nature of the the distribution of the sample variables; i.e, the minimum cannot be less than -0.5. For analysis of the case in which the sample variable has an unbounded distribution see Unbounded Sample Minimum.