Let Y denote a Bernoulli(u) random variable with 0 , u , 1. Suppose we are interested in estimating the odds ratio, g 5 u/11 2 u 2, which is the probability of success over the probability of failure.

Given a random sample 5Y1,

c, Yn6, we know that an unbiased and consistent estimator of u is Y, the proportion of successes in n trials. A natural estimator of g is G 5 Y/11 2 Y2, the proportion of successes over the proportion of failures in the sample.

(i) Why is G not an unbiased estimator of g?

(ii) Use PLIM.2 (iii) to show that G is a consistent estimator of g.