Page 340 - Fundamentals of Probability and Statistics for Engineers

Model Verification 323

doing so, however, a complication arises in that theoretical probabilities p i
defined by Equation (10.2) are, being functions of the distribution parameters,
functions of the sample. The statistic D now takes the form

k 2 k 2
^
X n N i X N i
D ˆ ^ P i ˆ ^ n; 10:10†
iˆ1 P i n iˆ1 nP i
^
where P i is an estimator for p i and is thus a statistic. We see that D is now
a much more complicated function of X 1 , X 2 ,..., X n . The important question
to be answered is: what is the new distribution of D?
The problem of determining the limiting distribution of D in this situation
was first considered by Fisher (1922, 1924), who showed that, as n !1 , the
distribution of D needs to be modified, and the modification obviously depends
on the method of parameter estimation used. Fortunately, for a class of
important methods of estimation, such as the maximum likelihood method,
the modification required is a simple one, namely, statistic D still approaches a
r
chi-squared distribution as n !1 but now with (k 1) degrees of free-
dom, where r is the number of parameters in the hypothesized distribution to be
estimated. In other words, it is only necessary to reduce the number of degrees
of freedom in the limiting distribution defined by Equation (10.5) by one for
each parameter estimated from the sample.
We can now state a step-by-step procedure for the case in which r parameters
in the distribution are to be estimated from the data.
. Step 1: divide range space X into k mutually exclusive and numerically con-
venient intervals A i , i ˆ 1, ..., k. Let n i be the number of sample values fall-
ing into A i . As a rule, if the number of sample values in any A i is less than 5,
combine interval A i with either A i 1 or A i 1 .
‡

. Step 2: estimate the r parameters by the method of maximum likelihood from
the data.
. Step 3: compute theoretical probabilities P(A i ) ˆ p i , i ˆ 1, ..., k, by means of
the hypothesized distribution with estimated parameter values.
. Step 4: construct d as given by Equation (10.7).
. Step 5: choose a value of and determine from Table A.5 for the 2
r
distribution of (k 1) degrees of freedom the value of 2 k r 1, . It is

assumed, of course, that k r 1 > 0.
. Step 6: reject hypothesis H if d > 2 . Otherwise, accept H.

k r 1,

Example 10.3. Problem: vehicle arrivals at a toll gate on the New York State
Thruway were recorded. The vehicle counts at one-minute intervals were taken
for 106 minutes and are given in Table 10.4. On the basis of these observations,
determine whether a Poisson distribution is appropriate for X, the number of
arrivals per minute, at the 5% significance level.

TLFeBOOK