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

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