LR test for K binomial samples Suppose that Yk has a binomialdistribution with sample size n, and proportion p k , k = I , . . . , K . All Y,are statistically independent of each other.

(a) Derive the (log) likelihood ratio test statistic for equality p ...=  P K

(b) How many degrees of freedom does the approximate chi-square distribution for this statistic have?

(c) Apply the test to the capacitor data of Chapter 10.

(d) Do you think that the asymptotic theory is adequate for the Explain why. capacitor data?

(e) Apply the test to the appliance component data of Problem 10.5, doing parts (b) through (e) of the problem.

( f ) Do you think that the asymptotic theory is adequate for the appliance component data? Explain why.

Problem 10.5

Appliance component (renewal data). The data in the accompanying table show the number of appliances that entered each month of service and the number of failures of a particular component. On failing, acomponent was replaced (renewed) with a new one, and an appliance can have any number of such failures over the months it is in service. Regard each month of service of an appliance as a binomial trial in which the

component either fails or survives with the probability for that month.

(a) For each month, calculate the estimate of the binomial proportion failing (expressed as a percentage). Month 1 has a high percentage. typical of many appliance components.

(b) Test for equality of the binomial proportions of all months. Calculate the pooled estimate of the proportion.

(c) Repeat (b) for months 2 through 12.

(d) Repeat (b) for months 13 through 24.

(e) Repeat (b) for months 25 through 29.

(f) Cumulate the percentage failing for each month to estimate the cumulative percentage failing through each month. Such cumulative percentages could exceed loo%, as an appliance can have more than one failure. Plot the sample cumulative percentage failing (called the "renewal function") on log-log paper.

(g) Estimate the cumulative percentage failing on the 12-month warranty and during the 15-year life of the appliance.

(h) Criticize the preceding analyses.

(i) Comment on the accuracy of the 15-year estimate in view of the fact that data from months 1 through 12 come from 65% of the population, data from months 13 through 24 come from about 8% of the population that is on service contract (unlimited repair service, owner choses to pay a single premium for), and data from months 25 through 29 come from 2% of the population, whose owners elect a second year of service contract.

(j) Devise a better 15-year estimate of the cumulative percentage failing.