Computational Probability Applications by Andrew G. Glen & Lawrence M. Leemis

Author:Andrew G. Glen & Lawrence M. Leemis
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(9.8)

In a similar manner, and using Eq. (9.6), it can be shown that a general expression for c 5 is

Burr [25] also presents c 5 in terms of c 4 via the relationship

Therefore, if we are successful in finding c 4 we can easily evaluate c 5, narrowing the focus of evaluation to c 4. Substituting n = 2 into 9.8, we conclude that the numerator, E[S], is

The bias correction factor is then calculated via

Given that the parameter σ X appears in the denominator of the expression, we require it to also appear in the numerator forcing a cancellation and a numerical c 4 value that is independent of σ X . Unfortunately, this only occurs for distributions in which a single parameter involving the standard deviation appears. The next example highlights such an occurrence.

Example 9.4.

Given that X 1, and X 2 are iid exponential(λ) random variables, find the bias correction factor c 4 for the sample standard deviation. The APPL statements

> X := ExponentialRV(lambda):

> Y := Difference(X, X):

> g := [[x -> -x, x -> x], [-infinity, 0, infinity]]:

> Z := Transform(Y, g):

> Mean(Z) / sqrt(2 * Variance(X)):

yield

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