Question

Suppose that X₁,….., X_n forms a random sample from the Bernoulli distribution with unknown parameter p. Show thatX̄_n is an efficient estimator of p.

Short answer

X̄_nis an efficient estimator of p.

Step-by-step solution

Given the information

Given that X_1,….,X_n are iid random variables from Bernoulli distribution with parameter p. therefore, X_i ~ Ber (p)

Define an efficient estimator

The most efficient estimator among a group of unbiased estimators is the one with the most minor variance.

An efficient estimator also fetches a small variance or mean square error. Therefore, there is a small deviation between the estimated and parameter values.

Find the unbiased estimator for p

In our case, the population mean is: E(X) = p

Also, an estimatorᵟ(X) of g(p) is unbiased if E[ᵟ(X) ] = g (p) for all possible values of p.

Now, we know that the unbiased estimator of a population mean is the sample mean.

Therefore,

E[ᵟ(X) ] = g (p)

E[X̄_n] = p

It satisfies the equation andX̄_n is the unbiased estimator.

Find the variance of the unbiased estimator

To check the efficiency, we have to check if the deviation between the actual and estimated value decreases as n tends to zero.

We first know that

E [X̄]→p, n→∞

Therefore, E ( p - X̄_n)→0, n→∞

Also, let us check for the variance,

Var [X̄_n] = p( 1- p)/ n

As

n→∞

p(1-p)/n→0

Therefore, the variance is minimized here.

So X̄ is an efficient estimator of p