Question

Show that Tn=X+1/n+1 is a consistent estimator of the parameter theta of a binomial population

Answer 1

$T_n=\frac{x+1}{n+1}$

An estimator $T_n$  of parameter θ is consistent if it converges in probability:

$\displaystyle{\lim_{n\to \infin}}Pr(|T_n-\theta|>\varepsilon)=0$

for all $\varepsilon>0$

For a binomial population: $\theta=x/n$ - the proportion of successes.

Then:

$\displaystyle{\lim_{n\to \infin}}(T_n-\theta)=\displaystyle{\lim_{n\to \infin}}(\frac{x+1}{n+1}-\frac{x}{n})=0$

If

$(\frac{x+1}{n+1}-\frac{x}{n})=0$

then:

$Pr(|\frac{x+1}{n+1}-\frac{x}{n}|>\varepsilon)=0$

because $\varepsilon>0$ .

So:

$\displaystyle{\lim_{n\to \infin}}Pr(|\frac{x+1}{n+1}-\frac{x}{n}|>\varepsilon)=0$