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as a Function of Sample Size |
Let x be a stochastic variable with a probability density function p(x) and a cumulative probability distribution function of P(x). Thus the probability of an observation being less than or equal to x is P(x). This function P(x) necessarily has an inverse function P^{-1}(z). Note that P(½)=x_{median}. Let x_{max} be defined to be the lowest value of x such that P(x)=1. Likewise x_{min} is the largest x such that P(x)=0. Note that x_{max} and x_{min} may or may not be finite.
The probability density function for the sample maximum of a sample of size n, q_{n}(x) is given by the probability of getting (n-1) observations which are less than or equal to x and one that is exactly x. The one observation that is exactly x can occur at any one of n places in the sample. Thus the probability density is
Note that
The expected value of the sample maximum is
Let z=[P(x)]^{n} so x=P^{-1}(z^{1/n}). Then changing the variable of integration in the above expression to z results in
Now consider the limit of M_{n} as n increases without bound and note that the limit of a function of a variable is equal to the function of the limit of the the variable; i.e.,
But for all z in the interval 0<z≤1, lim_{n→∞}z^{1/n} = 1. Therefore
Note that M_{1} = x_{mean}.
The characteristic function for sample maximums can be defined as
where i is the square root of −1.
From the previous work this reduces to
Integration by parts can be applied to this expression to obtain
Since lim_{x→∞}P(x)=1` and lim_{x→−∞}P(x)=0 the above reduces to
The integral on the right is the characteristic function of the n-th power of P(x) which can be expressed as the the n-th convolution product of the characteristic function of P(x). The characteristic function of P(x) is the characteristic function of p(x) divided by iω.
(To be continued.)
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