Question

, , Yn is a random sample from a distribution with pdf f,0% θ)-22, 3. (20 points) If Y., Y2, 0 Syse, a. find cÝ, where c is a constant, that is an unbiased estimator of θ; and b. show that the variance of is less than the Cramér-Rao lower bound for fr (y; 0) c. Why isn't this a violation of the Cramér-Rao inequality?

Answer 1

Given the PDF .

2y

a) The mean of the given random variable is

ду 20

For the estimator $\widehat{\theta }=c\overline{Y}$ to be unbiased,

32 5

b) The variance of the given random variable is

0 E (Y2 E (Y) =乏 Var ( Y) Var (Y) = 18 0 2 2 9 θ2

The variance of the estimator is

Var (a) = Var (cy) Var (θ) = C2 Var (Y) /n Var@) = 4 18n Var (θ) 8n

Cramer-Rao Lower bound is $Var\left (\widehat{\theta } \right )\geqslant \frac{1}{nE\left [ \left ( \frac{\partial \log f(x;\theta )}{\partial \theta } \right )^2 \right ]}$

Now,

$E\left [ \left ( \frac{\partial \log f(x;\theta )}{\partial \theta } \right )^2 \right ]=E\left [ \left ( \frac{\partial (2y-2\ln \theta )}{\partial \theta } \right )^2 \right ]\\ E\left [ \left ( \frac{\partial \log f(x;\theta )}{\partial \theta } \right )^2 \right ]=E\left [\frac{4}{\theta ^2}\right ]\\ E\left [ \left ( \frac{\partial \log f(x;\theta )}{\partial \theta } \right )^2 \right ]=\frac{4}{\theta ^2}$

$Var\left (\widehat{\theta } \right )\geqslant \frac{\theta ^2}{4n}$

We can see that $\frac{\theta ^2}{4n}> \frac{\theta ^2}{18n}$ . The variance is less than Cramer's-Rao lower bound.

c) The condition

$\frac{\partial }{\partial \theta }\left [ \int \int ...\int \widehat{\theta} \prod_{i=1}^{n}f_Y\left ( y_i;\theta \right ) dy_1dy_2...dy_n \right ]=\int \int ...\int \widehat{\theta} \frac{\partial \prod_{i=1}^{n}f_Y\left ( y_i;\theta \right ) }{\partial \theta }dy_1dy_2...dy_n$

Is not met. Hence this is not a violation of the Cramer's-Rao in equality.