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STAT 450

Problems: Assignment 6

  1. Page 365. Q 21.

  2. Page 365 Q 24.

  3. Page 366. Q 28.

  4. Page 366. Q 29.

  5. Suppose are iid and are iid .

    1. Find complete and sufficient statistics.

    2. Find UMVUE's of and .

    3. Now suppose you know that . Find UMVUE's of and of . (You have already found the UMVUE for .)

    4. Now suppose and are unknown but that you know that . Prove there is no UMVUE for . (Hint: Find the UMVUE if you knew with a known. Use the fact that the solution depends on a to finish the proof.)

    5. Why doesn't the Lehmann-Scheffé theorem apply?

  6. Suppose iid Poisson( ). Find the UMVUE for and for .

  7. Let be the error sum of squares in the ith cell in the question 5 of Assignment 5.

    1. Find the joint density of the .

    2. Find the best estimate of of the form in the sense of mean squared error.

    3. Do the same under the condition that the estimator must be unbiased.

    4. If only are observed what is the MLE of ?

    5. Find the UMVUE of for the usual one-way layout model, that is, the model of the last two questions.

  8. Exponential families: Suppose are iid with density

    1. Find minimal sufficient statistics.

    2. If are the minimal sufficient statistics show that setting and solving gives the likelihood equations. (Note the connection to the method of moments.)

Richard Lockhart
Sun Oct 27 19:00:53 PST 1996