San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

The Expected Value of of Sample Maximums
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)=xmedian. Let xmax be defined to be the lowest value of x such that P(x)=1. Likewise xmin is the largest x such that P(x)=0. Note that xmax and xmin may or may not be finite.

The probability density function for the sample maximum of a sample of size n, qn(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

qn(x) = n[P(x)]n-1p(x)

Note that

qn(x)dx = n[P(x)]n-1p(x)dx = d[P(x)n]

The expected value of the sample maximum is

Mn = ∫-∞ xqn(x)dx

Let z=[P(x)]n so x=P-1(z1/n). Then changing the variable of integration in the above expression to z results in

Mn = ∫01 P-1(z1/n)dz

Now consider the limit of Mn 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.,

limn→∞ Mn = ∫01 P-1(limn→∞z1/n)dz

But for all z in the interval 0<z≤1, limn→∞z1/n = 1. Therefore

limn→∞ Mn = ∫01 P-1(1)dz = P-1(1)
which can be expressed as
limn→∞ Mn = xmax

Note that M1 = xmean.

The Characteristic Function for the Distribution of Sample Maximums

The characteristic function for sample maximums can be defined as

Φn(ω) = ∫-∞ exp(−iωx)qn(x)dx

where i is the square root of −1.

From the previous work this reduces to

Φn(ω) = ∫-∞ exp(−iωx)d([P(x)]n)

Integration by parts can be applied to this expression to obtain

Φn(ω) = [exp(−iωx)[P(x)]n]-∞ − (−iω) ∫-∞ exp(−iωx)[P(x)]ndx

Since limx→∞P(x)=1` and limx→−∞P(x)=0 the above reduces to

` Φn(ω) = iω ∫-∞ exp(−iωx)[P(x)]ndx

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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