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STAT 426 – Categorical Data Analysis

Homework 1

2020

Below exercises give you a practice in R Markdown while reviewing maximum likelihood esti-mation, hypothesis testing and the Poisson distribution.

1. Characteristics of Poisson distribution

Let X1, X2, ..., Xn be a random sample from a Poisson distribution with mean λ.

(i) Use mathematical expressions in R Markdown to show what the maximum likelihood estimator λb is.

(ii) What is the exact variance of λb?

(iii) Using Fisher Information find the asymptotic variance of λb. Is it same as in (ii)?

2. Suppose you obtain a random sample from a Poisson distribution that is (9, 16, 17, 14, 17, 11, 12, 15, 13, 9). Find the 95 percent Wald confidence interval for λb.

3. Using the Wald test statistic and the data from 2., find the p-value for testing H0 : µ = 10 vs. Ha : µ > 10.

4. Find the p-value of an exact test for the same hypotheses. Hint: If X1, X2, ..., Xn is a random sample from a Poisson distribution with mean λb, then P n i=1 Xi is Poisson with mean nλ.

5. Plot the density functions of a Poisson variable with mean 6 and a Binomial variable with n = 200 and p = 3/100 side-by-side for x = 0, 1, 2, ..., 10. What do you observe?