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STAT 801

Problems: Assignment 5

  1. Suppose tex2html_wrap_inline49 are iid tex2html_wrap_inline51 and tex2html_wrap_inline53 are iid tex2html_wrap_inline55 .

    1. Find complete and sufficient statistics.
    2. Find UMVUE's of tex2html_wrap_inline57 and tex2html_wrap_inline59 .
    3. Now suppose you know that tex2html_wrap_inline61 . Find UMVUE's of tex2html_wrap_inline63 and of tex2html_wrap_inline65 . (You have already found the UMVUE for tex2html_wrap_inline67 .)
    4. Now suppose tex2html_wrap_inline69 and tex2html_wrap_inline71 are unknown but that you know that tex2html_wrap_inline73 . Prove there is no UMVUE for tex2html_wrap_inline75 . (Hint: Find the UMVUE if you knew tex2html_wrap_inline77 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?

  2. Suppose tex2html_wrap_inline83 iid Poisson( tex2html_wrap_inline85 ). Find the UMVUE for and for tex2html_wrap_inline89 .
  3. Suppose iid with

    displaymath93

    for tex2html_wrap_inline95 . For n=1 and 2 find the UMVUE of tex2html_wrap_inline99 . (Hint: The expected value of any function of X is a power series in divided by . Set this equal to and deduce that two power series are equal. Since this implies their coefficients are the same you can see what the estimate must be. )

  4. Suppose tex2html_wrap_inline109 are independent tex2html_wrap_inline111 random variables. (This is the usual set-up for the one-way layout.)

    1. Find the MLE's for tex2html_wrap_inline113 and .
    2. Find the expectations and variances of these estimators.

  5. Let tex2html_wrap_inline117 be the error sum of squares in the ith cell in the previous question.

    1. Find the joint density of the .
    2. Find the best estimate of of the form tex2html_wrap_inline125 in the sense of mean squared error.
    3. Do the same under the condition that the estimator must be unbiased.
    4. If only tex2html_wrap_inline127 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.

  6. Exponential families: Suppose are iid with density

    displaymath135

    1. Find minimal sufficient statistics.
    2. If tex2html_wrap_inline137 are the minimal sufficient statistics show that setting tex2html_wrap_inline139 and solving gives the likelihood equations. (Note the connection to the method of moments.)

  7. In question ? take tex2html_wrap_inline141 for all i and let tex2html_wrap_inline145 . What happens to the MLE of ?
  8. Suppose that are independent random variables and that tex2html_wrap_inline151 are the corresponding values of some covariate. Suppose that the density of tex2html_wrap_inline153 is

    displaymath155

    where tex2html_wrap_inline157 , and tex2html_wrap_inline159 are unknown parameters.

    1. Find the log-likelihood, the score function and the Fisher information.
    2. For the data set in /home/math/lockhart/teaching/801/data1 fit the model and produce a contour plot of the log-likelihood surface, the profile likelihood for and an approximate 95% confidence interval for .

  9. Consider the random effects one way layout. You have data tex2html_wrap_inline165 and a model tex2html_wrap_inline167 where the tex2html_wrap_inline169 's are iid tex2html_wrap_inline171 and the tex2html_wrap_inline173 's are iid tex2html_wrap_inline175 .

    1. Write down the likelihood.
    2. Find minimal sufficient statistics.
    3. Are they complete?
    4. Find method of moments estimates of the three parameters.
    5. Can you find the MLE's?

  10. For each of the doses tex2html_wrap_inline177 a number of animals tex2html_wrap_inline179 are treated with the corresponding dose of some drug. The number dying at dose d is Binomial with parameter h(d). A common model for h(d) is tex2html_wrap_inline187

    1. Find the likelihood equations for estimating tex2html_wrap_inline189 and .
    2. Find the Fisher information matrix.
    3. Define the parameter LD50 as the value of d for which h(d)= 1/2; express LD50 as a function of tex2html_wrap_inline197 and .
    4. Use a Taylor expansion to find large sample confidence limits for LD50.
    5. At each of the doses -3.204, -2.903, 2.602, -2.301 and -2.000 a sample of 40 mice were exposed to antipneumonococcus serum. The numbers surviving were 7, 18, 32, 35, and 38 respectively. Get numerical values for the theory above. You can use glm or get preliminary estimates based on linear regression of the MLE of tex2html_wrap_inline201 against dose.

  11. Suppose are a sample of size n from the density

    displaymath207

    In the following question the digamma function tex2html_wrap_inline209 is defined by tex2html_wrap_inline211 and the trigamma function tex2html_wrap_inline213 is the derivative of the digamma function. From the identity tex2html_wrap_inline215 you can deduce recurrence relations for the digamma and trigamma functions.

    1. For tex2html_wrap_inline217 known find the mle for .
    2. When both tex2html_wrap_inline221 and are unknown what equation must be solved to find tex2html_wrap_inline225 , the mle of tex2html_wrap_inline227 ?
    3. Evaluate the Fisher information matrix.
    4. A sample of size 20 is in the file
      /home/math/lockhart/teaching/801/gamma.
      Use this data in the following questions. First take tex2html_wrap_inline229 and find the mle of subject to this restriction.
    5. Now use tex2html_wrap_inline233 and tex2html_wrap_inline235 to get method of moments estimates tex2html_wrap_inline237 and tex2html_wrap_inline239 for the parameters.
    6. Do two steps of Newton Raphson to get MLEs.
    7. Use Fisher's scoring idea to redo the previous question.
    8. Compute standard errors for the MLEs and compare the difference between the estimates in the previous 2 questions to the SEs.
    9. Do a likelihood ratio test of tex2html_wrap_inline241 .

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Richard Lockhart
Tue Apr 15 22:54:18 PDT 1997