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Exam Question

  1. a

    The random variables X1,X2,,Xn are independent and identically distributed with the geometric distribution

    f(x|θ)=θx(1-θ),x=0,1,2,

    where θ is a parameter in the range of 0θ1 to be estimated. The mean of the above geometric distribution is θ/(1-θ).

    1. i

      Write down formulae for the maximum likelihood estimator for θ and for Fisher’s information;

    2. ii

      Write down what you know about the distribution of the maximum likelihood estimator for this example when n is large.

  2. b

    In a particular experiment, n=10, i=1nxi=10.

    1. i

      Compute an approximate 95% confidence interval for θ based on the asymptotic distribution of the maximum likelihood estimator;

    2. ii

      Compute the deviance D(θ) and sketch it over the range 0.1θ0.9. Use your sketch to describe how to use the deviance to obtain an approximate 95% confidence interval for θ;

    3. iii

      If you were asked to produce an approximate 95% confidence interval for the mean of the distribution θ/(1-θ), what would be your recommended approach? Justify your answer.