Suppose that we have a random sample of size n from a Bernoulli(7) distribution. We will...

 

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Suppose that we have a random sample of size n from a Bernoulli(7) distribution. We will use a complementary log-log model: cloglog() = log(-log(1)). The mle (maximum likelihood estimate) for is = log(-log(1)), where = (number of successes)/(number of trials) is the sample proportion. (a) Use the delta method, to show that the estimated variance for is V[0] = == n (1 ) [log(1 )]* (b) Using (a), compute the estimated (asymptotic) variance for cloglog(), if the sample proportion is = 0.25 with n = 100. (c) Suppose that == 0.25, with n = 100. Give the 95% cloglog confidence interval for T, which is obtained by first computing the 95% Wald interval for cloglog() and evaluating the limits of the interval in the inverse function of the cloglog. Suppose that we have a random sample of size n from a Bernoulli(7) distribution. We will use a complementary log-log model: cloglog() = log(-log(1)). The mle (maximum likelihood estimate) for is = log(-log(1)), where = (number of successes)/(number of trials) is the sample proportion. (a) Use the delta method, to show that the estimated variance for is V[0] = == n (1 ) [log(1 )]* (b) Using (a), compute the estimated (asymptotic) variance for cloglog(), if the sample proportion is = 0.25 with n = 100. (c) Suppose that == 0.25, with n = 100. Give the 95% cloglog confidence interval for T, which is obtained by first computing the 95% Wald interval for cloglog() and evaluating the limits of the interval in the inverse function of the cloglog.