Consider a censored sample from the exponential distribution with rate parameter λ and fixed censoringtime c, common to all individuals that were in the study. That is, individual i is censored iffailure time, Xi > c. Otherwise Xiis observed.
(a) Suppose that only the number of failures, D, is observed. Show that D has a binomial distributionwith parameters n and p = 1 - exp[-λc]. Using the relationship between the variance ofmaximum likelihood estimators and expected information obtain an expression for the varianceof the maximum likelihood estimate of λ. This should be a true variance not a standard error;i.e. an expression that depends upon the parameters of the model, not data.
(b) Suppose now the usual case: (Piti, D) are observed, where the ti are the failure and censoringtimes. You showed on Assignment 4 that l00(λ) = -D/λ2 and λˆ = D/Piti. Using the relationshipbetween the variance of maximum likelihood estimators and expected information obtainan expression for the variance of the maximum likelihood estimate of λ. Again, this shoulddepend on the parameters of the model, not data.
(c) The efficiency of an estimator λˆ1 to another estimator λˆ2 is the ratio of the large sample variancesof the two estimators: Var(λˆ2)/Var(λˆ1). Obtain an expression for the efficiency of the estimatorin (a) to the estimator in (b). Note that the efficiency depends upon λ and c. In R, produce aplot of the efficiency as a function of λ for c = 1. For what values of λ does the method in (a)give almost as good results as method (b), if any?