Question

Suppose that X_1,…….,X_n form a random sample from the Bernoulli distribution with unknown parameter p. Show that the variance of every unbiased estimator of (1-p)² must be at least 4p(1-p)³/n.

Short answer

The variance of every unbiased estimator must be at least

\begin{aligned}\frac{{4p{{\left({1-p}\right)}^3}}}{n}\end{aligned}

Step-by-step solution

Given the information

Suppose that X_1,…….,X_n from a random sample from the Bernoulli distribution with an unknown parameter p.

Showing the variance

Here, (1-p)² is an unbiased estimator.

So,

m(p) = (1-p)²

Then,

m’(p) = -2(1-p)

[m’(p)]² = (-2(1-p))²

= 4(1-p)²

So, the fisher information for Bernoulli distribution is,

L (p) = 1/[p(1-p)]

Therefore, if the T is an unbiased estimator of m(p), it follows the relation,

Var_e (T) ≥ [m’(p)]²/ nl(p)

So,

Var (T) ≥ 4(1-p)²/ n1/p(1- p)

= 4p(1-p)³/n

That is,

Var (T) ≥ 4p(1-p)³/n

Hence, proved.