Page 244 - Elements of Distribution Theory

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230 Distribution Theory for Functions of Random Variables

the statistic Y ≡ Y(Z 1 ,..., Z n ) defined by the following equation:
n
Y

Z − 1 log Z j = n.
j
j=1
This statistic arises in connection with estimation of the parameters of the Weibull distri-
bution.
Since an explicit expression for Y in terms of Z 1 ,..., Z n is not available, an exact
expression for the distribution of Y is difficult, if not impossible, to determine. However, the
Monte Carlo method is still easily applied; all that is needed is an algorithm for determining
Y from a given value of X = (Z 1 ,..., Z n ).

There are a number of disadvantages to the Monte Carlo method. One is that, if the
ˆ
method is repeated, that is, if a new set of observations is generated, a new result for P N is
ˆ
obtained. Although the variation in different values of P N may be decreased by choosing a
very large value of N, clearly it would be preferable if two sets of “identical” calculations
would lead to identical results. This is particularly a problem in complex settings in which
generation of each Y j is time-consuming and, hence, N must be chosen to be relatively small.
In view of this variation, it is standard practice to supplement each estimate of Pr(Y ≤ y)
with its standard error.

Example 7.33 (Ratio of exponential random variables to their sum). Consider the proba-
bility considered in Example 7.29. When the Monte Carlo analysis was repeated, the results
ˆ
ˆ
P N = 0.246 for N = 1,000 and P N = 0.2428 for N = 10,000 were obtained. These may
be compared to the results obtained previously.
A second drawback of the Monte Carlo method is that, because no formula for Pr(Y ≤ y)
is available, it may be difficult to see how the probability varies as different parameters in
the problem vary.

Example 7.34 (Ratio of exponentials random variables to their sum). Recall
Example 7.29. The probability of interest was Pr(Y ≤ 1/4), calculated under the assumption
that λ = 1. Suppose that we now want the same probability calculated under the assumption
that λ = 5. Note that the distribution of the statistic
X 1
X 1 + X 2
does not depend on the value of λ.To see this, note that we may write X j = λZ j , j = 1, 2,
where Z 1 , Z 2 areindependentstandardexponentialrandomvariables;thisresultalsofollows
from the general results on invariance presented in Section 5.6.
When the Monte Carlo approach was used with λ = 5 and N = 10,000, the result was
ˆ
P N = 0.2500. Although this is close to the result calculated under λ = 1(0.2464), it is not
clear from these values that the two probabilities are exactly equal.

Despite these drawbacks, the Monte Carlo method is a very useful and powerful method.
It is invaluable in cases in which an exact analysis is not available. Furthermore, the gen-
erality of the Monte Carlo method gives the statistical analyst more flexibility in choosing
a statistical model since models do not have to be chosen on the basis of their analytical
tractability. Also, even in cases in which an exact analysis is possible, results from a Monte