Question

B2. (a) Suppose θ is an unknown parameter which is to be estimated from a single measurement X, distributed according to some probability density function f(r0). The Fisher information I(0) is defined by de Show that, under some suitable regularity conditions, the variance of any unbi- ased estimator θ of θ is then bounded by the reciprocal of the Fisher information Var | θ 1(8) Note that the suitable regularity conditions, which are not specified here, allow the interchange of the order of integration and differentiation. (b) Let Xi,X2, .. X be a random sample from a Bernoulli() distribution with probability function Note that, for a random variable X with a Bernoulli(9) distribution. EX) = θ and Var [X (1 ) i) Obtain the log-likelihood function, ), and hence show that the maximum likelihood estimator of θ is -1 ii Show that ii Calculate the expected information I(e)-E[(8)] iv) Show that θ is the minimum variance unbiased estimator of θ, ie, that it v) For a particular set of observed outcomes 1,0.0. 1, 1, calculate an estimated vi) In a trial on the effectiveness of nicotine patches for stopping smoking, out is unbiased and has variance that attains the Cramer-Rao lower bound standard error for θ of a random sample of 120 coronary heart disease patients 30 were still not smoking after 6 months. Obtain a point estimate and approximate 95% confidence interval for the probability of nicotine patches being effective at stopping smoking in the population of coronary heart disease patients.

Answer 1

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