Question

For each n, let Xn be a binomial random variable with n trials and probability of success p Yn Use the Weak Law of Large Numbers to show that is a consistent estimator of p. (b) Explain why it follows from (a) that (1 isaconsistent estimator of p(1 -p) and that 1) is a consistent estimator of p(-P) 7l V n

Answer 1

a) A statistic T is said to be a consistent estimator of parameter$\theta$ if T converges in probability

$i.e\, P\left \{ \begin{vmatrix} T-\theta \end{vmatrix}}<\epsilon\right \}\rightarrow 1$

Bernoulli Law of Large Numbers:

In n trails let X denotes the number of successes with constant p of success for each trial, then for any $\epsilon>0$,however small

$P\left \{ \frac{X_n}{n}-p<\epsilon \right \}\rightarrow 1\, as\, n\rightarrow \infty$

let kTR--e

b) Also $P\left \{ (1-\frac{Y_n}{n})-(1-p)<\epsilon \right \}\rightarrow 1\, as\, n\rightarrow \infty$

Property:

If t is consistent for $\theta\, then\, T^2 \, is \, consistent\, for \, \theta^2$

Then by the property of consistency $\frac{Y_n}{n}(1-\frac{Y_n}{n})$ is a consistent estimator for p(1-p)

Also $\sqrt{\frac{Y_n}{n}(1-\frac{Y_n}{n})}$ is consistent for $\sqrt{p(1-p)}$