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

Consider the distribution of sample maximums for samples of a random variable uniformly distributed between -0.5 and +0.5. For n=1 the sample maximum 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 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 maximum. 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. Thus the probability density function for the sample maximum is given by:

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

where the factor of n is to take into account the n different ways the n-1 factors below x and the one value of x can be arranged.

To check that q(x) is a proper probability density function consider its integration over the interval [-0.5,x];i.e.,

Q(x) = ∫_-0.5^xn(z+0.5)^n-1dz = [n(x+0.5)ⁿ]/n = (x+0.5)ⁿ.

The value of Q(x) at x=0.5 should be 1.0 and indeed Q(0.5) = 1ⁿ = 1.0.

As n increases without bound this distribution goes to a spike function at x=0.5. There is thus no tendency for the distribution of sample maximums to approach a normal distribution.

The purpose of the above analysis was to establish that there are sample statistics whose distribution does not approach a normal distribution as the sample size increases without bound. It was convenient for achieving this purpose to use a uniform, bounded distribution for the random variable. For the case of an unbounded distribution see Unbounded distribution. For an analysis of the general case see Sample Maximum.