Nonparametric comparison with grouped data and progressive censoring. For motivation, first see Problem 9.6, where interval data on K groups of transformers were pooled to fit a nonparametric multinomial model to the data. The following (log) likelihood ratio test compares the K groups for equality of corresponding proportions failing in each year. Suppose that the groups differ. Then, for group 

denotes the expected proportion failing in year m  denotes the expected proportion surviving the latest year ( k ) in service. Similarly ykm denotes the observed number failing in year m, and yL. k + I denotes the observed number surviving the latest year ( k ) in service.

(a) For each transformer group, plot the sample cumulative distribution function on the same plotting paper. (Connect plotted points so that the plot is easier to see.) Do the distributions differ convincingly? How? Confidence limits for the cumulative distributions may help.

(b) Write the general expression for the log likelihood for the multinomial model for group k , k = 1 , . . :, K.

(c) Write the general total log likelihood for all groups (assuming that they differ).

(d) Derive the formulas for the ML estimates for the proportions

(e) Evaluate the estimates (c) for the transformer data of Problem 9.6.

( f ) Give the general expression for the maximum log likelihood for differing groups, and evaluate it for the transformer data.

(8) Use the results of Problem 9.6 to calculate the maximum log likelihood for the'transformer data under the model there, with a common proportion failing in each year, all 

(h) Calculate the log likelihood ratio statistic for testing for equality of group proportions within each year, 

(i) Give a general formula for the number of degrees of freedom of the test statistic, and evaluate it for the transformer data.

(j) Test whether the groups differ significantly. How do they differ?

(k) Do you t h n k that the asymptotic theory is adequate for the transformer data? Explain why.

Problem 9.6

Nonparametric fit to grouped data with progressive censoring. For motivation, first see the distribution transformer data in Table 3.1 of Chapter 10. Suppose each of the K groups there is assumed here to have the same proportion T] failing in year 1, the same proportion πr2 failing in year 2, etc. The following method yields ML estimates of the proportions π~, π',. . . from the data on all groups.

(a) Write the separate multinomial log likelihoods for groups K through 1, denoting the number from group k failing in year m by yknl and the number surviving the last year by yI,L+I.

(b) From the total log likelihood calculate the likelihood equations.

(c) Solve the equations to obtain the ML estimates of T,, ..., nK, ,.

(d) Derive formulas for all second partial derivatives of the total log

Table 3.1