Find a one-dimensional sufficient statistics. Is the MLE estimator you derived unbiased? What is its MSE?

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ANSWER ALL QUESTIONS

FINAL EXAM

All problem parts have equal weight. In budgeting, your time expect that some problems will take longer than others.

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1. Assume that X1, . . . , Xn are i.i.d. Gamma(3, λ), where λ > 0.

(a) Find a one-dimensional sufficient statistics.

(b) Find the MM estimator.

(c) Find the MLE.

(d) Is the MLE estimator you derived unbiased? What is its MSE?

(e) Assuming that the prior is λ Γ(a, b). Find the posterior and the posterior Bayes estimator.

(f) Find the CRLB for estimating λ. 

2. The following are results of independent repetitions of an experiment. (Success is denoted by 1 and failure by 0.)

1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0

Test H0 : p = 0.7 against H1 : p > 0.7 at α = 0.05. 

3. Let X1, . . . , Xn be i.i.d. N (µx, σ2) and Y1, . . . , Ym be i.i.d.  N (µy, σ2). x yThe Xs are independent of Ys.

(a) Find the GLRT for testing H0 : µx = µy, σ2 = σ2 at α = 0.01 Assume that n, m are large.

(b)  We have observed n = 121, X¯  = 0.065, S2 = 0.806 and m = 101,

Y¯   =  −0.176,  S2  =  9.689.   Does  the  GLRT  test  you  derived  in part 3a reject the null hypothesis?

(c) Using the same data set as in part 3b test H0 : σ2= σ2vs.Ha : σ2 /= σ2 at α = 0.05.

(d) Using the same data set as in part 3b find the 99% confidence interval for µy.

4. An experiment was conducted to compare the mean number of tapeworms in the stomachs of sheep that had been treated for worms against the mean number in those that were untreated. A sample of 14 worm– infected lambs was randomly divided into 2 groups. Seven were injected with the drug and the remainder were left untreated. After a six–month period, the lambs were slaughtered and the following worm counts were recorded:

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