Let Y denote a Bernoulli(θ) random variable with 0<θ<1. Suppose we are interested in...

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 the probability of failure. Given a random sample {Y1, …, Yn}, we know that an unbiased and consistent estimator of θ is , the proportion of successes in n trials. A natural estimator of γ is G = θ /(1 -), the proportion of successes over the proportion of failures in the sample.

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

(ii) Use PLIM.2(iii)

PLIM.2 If plim (Tn) = αand plim (Un) =β, then

(i) plim(Tn _+Un) = α+ β;

(ii) plim(TnUn) = α β;

(iii) plim(Tn /Un) = α/β, provided β≠ 0.

to show that G is a consistent estimator of γ.