Page 334 - Fundamentals of Probability and Statistics for Engineers

Model Verification 317

alternative that the probability distribution of X is not of the stated type on the
basis of a sample of size n from population X. One of the most popular and most
versatile tests devised for this purpose is the chi-squared ( 2 )goodness-of-fit
test introduced by Pearson (1900).

10.2.1 THE CASE OF KNOWN PARAMETERS

Let us first assume that the hypothesized distribution is completely specified
with no unknown parameters. In order to test hypothesis H, some statistic
h(X 1 , X 2 ,.. . , X n ) of the sample is required that gives a measure of deviation of
the observed distribution as constructed from the sample from the hypothe-
sized distribution.
In the 2 test, the statistic used is related to, roughly speaking, the difference
between the frequency diagram constructed from the sample and a correspond-
ing diagram constructed from the hypothesized distribution. Let the range
space of X be divided into k mutually exclusive intervals A 1 , A 2 ,..., and A k ,
and let N i be the number of X j falling into A i , i ˆ 1, 2, . . . , k. Then, the observed
probabilities P(A i ) are given by

N i
observed P A i †ˆ ; i ˆ 1; 2; ... ; k: 10:1†
n
The theoretical probabilities P(A i ) can be obtained from the hypothesized
population distribution. Let us denote these by

theoretical P A i †ˆ p i ; i ˆ 1; 2; ... ; k: 10:2†

A logical choice of a statistic giving a measure of deviation is

k 2
N i
X
c i p i ; 10:3†
n
iˆ1
which is a natural least-square type deviation measure. Pearson (1900) showed
that, if we take coefficient c ˆ n/p i , the statistic defined by Expression (10.3)
i
has particularly simple properties. Hence, we choose as our deviation measure
k 2 k 2
X n N i X N i np i †
D ˆ p i ˆ
p i n np i
iˆ1 iˆ1
10:4†
k 2
X N i
ˆ n:
np i
iˆ1

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