Reference no: EM132296284
Assignment - PRINCIPLES OF STATISTICAL INFERENCE (PSI)
Practical Exercises - These practical exercises make use of some of our previous examples from module notes and extended examples, as well as modifications of these examples, and require the justification and computation of various test statistics.
Exercise - This question deals with multiple parameter tests for the fertility study example. Review Exercise 5 from Module 1, and Exercise 3 from Module 2, for the fertility study where mothers having premature births have different fertility to mothers having full term births. In those exercises we studied a likelihood function that was a function of two parameters, the fertility parameter for mothers with premature births (p2), and the relative fertility rate (γ). Start by writing down the log-likelihood function from part b of Exercise 5 in Module 1. Our goal is to test the hypothesis that the fertility rates are identical, that is:
H0 : γ = 1 versus H1 : γ ≠ 1
a. For the data set described in Exercise 4 of Module 1 evaluate the MLEs of p2 and γ. Hence evaluate the maximum value achieved by the log-likelihood under the full model, l(p^2, γ^).
b. Write down the log-likelihood function under the assumption that the null hypothesis is true. Hence derive and evaluate the MLE of p2 under the assumption that the null hypothesis is true and evaluate the maximum value achieved by the log-likelihood under the null model.
c. Based on your results in parts a and b, write down the likelihood ratio test statistic for testing the above hypotheses, and carry out a likelihood ratio test. Evaluate the p-value for this test.
d. Using your results from part d of Exercise 3 in Module 2, evaluate the approximate Wald test statistic for testing the above hypotheses. Carry out the test and calculate the p-value.
e. Recall your results from parts d and e in Exercise 3 of Module 2. Explain why these are consistent with the tests conducted above.
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