Exponential fit to grouped data with progressive censoring. For motivation, first see the distribution transformer data in Table 3.1 of Chapter 10. Each group there is assumed here to have the same exponential life distribution with common mean θ,. Suppose group k with nk units was observed through year k, and the yearly numbers of failures are.
and the number of survivors is 
(a) Calculate an actuarial estimate (Chapter 4) of the common distri-bution and plot it on Weibull probability paper.
(b) Write the sample log likelihood ek(θ) for group k in terms of θ
(c) Write the sample log likelihood f(θ) for the K independent groups.
(d) Derive af?/a1θ, and solve the likelihood equation for θ
(e) Calculate θ for the transformer data.
(f) (f) Derive θ' C/M2 and its expectation when the true mean is θ,.
(g) Calculate the local estimate of the asymptotic variance of θ.
(h) Calculate the ML estimate of the asymptotic variance ofθ.
(i) Calculate two-sided positive approximate 95% confidence limits
(j) (j) Do (i), using (h)
(k) Plot the fitted exponential distribution and 95% confidence limits for percentiles (parallel straight lines) on the Weibull plot from (a).
Table
