Fundamentals of Probability and Statistics for Engineers

Model Verification 323doing so, however, a complication arises in that theoretical probabilities p idefined by Equation (10.2) are, being functions of the distribution parameters,functions of the sample. The statistic D now takes the formD ˆ Xkiˆ1n^P iN in^P i2ˆ Xkiˆ1N 2 in ^P in; …10:10†where ^P i is an estimator for p i and is thus a statistic. We see that D is nowa much more complicated function of X 1 ,X 2 ,...,X n . The important questionto be answered is: what is the new distribution of D?The problem of determining the limiting distribution of D in this situationwas first considered by Fisher (1922, 1924), who showed that, as n !1, thedistribution of D needs to be modified, and the modification obviously dependson the method of parameter estimation used. Fortunately, for a class ofimportant methods of estimation, such as the maximum likelihood method,the modification required is a simple one, namely, statistic D still approaches achi-squared distribution as n !1but now with (k r 1) degrees of freedom,where r is the number of parameters in the hypothesized distribution to beestimated. In other words, it is only necessary to reduce the number of degreesof freedom in the limiting distribution defined by Equation (10.5) by one foreach parameter estimated from the sample.We can now state a step-by-step procedure for the case in which r parametersin the distribution are to be estimated from the data.. Step 1: divide range space X into k mutually exclusive and numerically convenientintervals A i ,i ˆ 1,...,k. Let n i be the number of sample values fallinginto 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 fromthe data.Step 3: compute theoretical probabilities P(A i ) ˆ p i ,i ˆ 1,...,k, bymeansofthe hypothesized distribution with estimated parameter values.Step 4: construct d as given by Equation (10.7).. 2Step 5: choose a value of and determine from Table A.5 for the 2distribution of (k r 1) degrees of freedom the value of k r 1, .It isassumed, of course, that k r 1 > 0.. 2Step 6: reject hypothesis H if d > k r 1, . Otherwise, accept H.Example 10.3. Problem: vehicle arrivals at a toll gate on the New York StateThruway were recorded. The vehicle counts at one-minute intervals were takenfor 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 ofarrivals per minute, at the 5% significance level.TLFeBOOK