Referring to Problem 55, consider two other estimates of θ. (1) The expected number of c...

Referring to Problem 55, consider two other estimates of θ. (1) The expected number of counts in the first cell is n(2 + θ)/4; if this expected number is equated to the count X₁, the following estimate is obtained:

(2) The same procedure done for the last cell yields

Compute these estimates. Using that X₁ and X₄ are binomial random variables, show that these estimates are unbiased, and obtain expressions for their variances. Evaluate the estimated standard errors and compare them to the estimated standard error of the mle.

Reference

For two factors—starchy or sugary, and green base leaf or white base leaf—the following counts for the progeny of self-fertilized heterozygotes were observed (Fisher 1958):

a. Find the mle of θ and its asymptotic variance.

b. Form an approximate 95% confidence interval for θ based on part (a).

c. Use the bootstrap to find the approximate standard deviation of the mle and compare to the result of part (a).

d. Use the bootstrap to find an approximate 95% confidence interval and compare to part (b).