Introduction to Probability and Statistics for Engineers and Scientists

11.2Goodness of Fit Tests When All Parameters are Specified 491

REMARKS

(a)To utilize the foregoing simulation approach to determine whether or not to accept
H 0 whenTis observed, we need to specify how one can simulate, or generate, a random
variableYsuch thatP{Y=i}=pi,i=1,...,k. One way is as follows:

Step 1:Generate a random numberU.
Step2: If

p 1 +···+pi− 1 ≤U<p 1 +···+pi

setY=i(wherep 1 +···+pi− 1 ≡0 wheni=1). That is,

U<p 1 ⇒Y= 1

p 1 ≤U<p 1 +p 2 ⇒Y= 2

..

.

p 1 +···+pi− 1 ≤U<p 1 +···+pi⇒Y=i

..

.

p 1 +···+pn− 1 <U⇒Y=n

Since a random number is equivalent to a uniform (0, 1) random variable, we have that

P{a<U<b}=b−a,0<a<b< 1

and so

P{Y=i}=P{p 1 +···+pi− 1 <U<p 1 +···+pi}=pi

(b) A significant question that remains is how many simulation runs are necessary. It has
been shown that the valuer=100 is usually sufficient at the conventional 5 percent level
of significance.*

EXAMPLE 11.2c Let us reconsider the problem presented in Example 11.2b. A simulation
study yielded the result

PH 0 {T≤9.52381}=.95

and so the critical value should be 9.52381, which is remarkably close toχ.05,4^2 =9.488
given as the critical value by the chi-square approximation. This is most interesting since
the rule of thumb for when the chi-square approximation can be applied — namely, that

• See Hope, A., “A Simplified Monte Carlo Significance Test Procedure,”J. of Royal Statist. Soc., B. 30, 582–598,