Let Y denote a Bernoulli() random variable with 0 < < 1. Suppose we are interested in estimating the odds ratio, = /

Let Y denote a Bernoulli(θ) random variable with 0 < θ < 1. Suppose we are interested in estimating the odds ratio, γ = θ/ (1 – θ) , which is the probability of success over the probability of failure. Given a random sample {γ₁, . . . .γ_n} we know that an unbiased and consistent estimator of θ is Y̅, the proportion of successes in n trials. A natural estimator of γ is G = Y̅/(1 – Y̅, the proportion of successes over the proportion of failures in the sample.

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

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