Asymptotica behaviour of an estimator

[_joey]

I have to determine if the estimator for Poisson distrubtion expectation $$\lambda_1$$ is asymptotically biased or unbiased.

The estimator is $$( \sum_{i=1}^n \frac{\sqrt{y}}{n})^2$$

It's easy to do the algebra and show that the mean is asymptotically unbiased. I am not sure how to start with the estimator presented above.

Any suggestions and hints will be much appreciated.

Thanks!

 

[Stephen Tashi]

You haven't defined what $y$ is. Should it have a subscript?

 

[_joey]

You haven't defined what $y$ is. Should it have a subscript?

Hi!

y should have a subscript. Y is a iid random variable

 

[Stephen Tashi]

I'll assume each $y_i$ is a non-negative sample. One thought is:

$$( \sum_{i=1}^n \frac{\sqrt{y_i}}{n})^2= (\sum_{i=1}^n \sqrt{ \frac{y_i} {n^2}})^2 \geq \sum_{i=1}^n (\sqrt{ \frac{ y_i }{n^2} })^2 = \sum_{i=1}^n \frac{y_i}{n^2}$$

This is a "thought", not a "hint" because I haven't worked the problem.

 

[_joey]

I'll assume each $y_i$ is a non-negative sample. One thought is:

$$( \sum_{i=1}^n \frac{\sqrt{y_i}}{n})^2= (\sum_{i=1}^n \sqrt{ \frac{y_i} {n^2}})^2 \geq \sum_{i=1}^n (\sqrt{ \frac{ y_i }{n^2} })^2 = \sum_{i=1}^n \frac{y_i}{n^2}$$

This is a "thought", not a "hint" because I haven't worked the problem.

Thanks for your thoughts on this problem. I can't see how this inequality could help me to solve the problem. If I take the expectation of the lower bound I will get $$\frac{\lambda}{n}$$. But that's a lower bound and the value of the estimator could be greater than $$\frac{\lambda}{n}$$. It could be $$\lambda$$

 

[_joey]

Perhaps we could set up inequality for upper bound and show that the estimator could not be greater of the upper bound. What could the upper bound be? :)

 

[Stephen Tashi]

It might work to multiply out the expression [tex]( \sum_{i=1}^n \frac{\sqrt{y_i}}{n})^2 [/itex] since the expectation of each term involving the product of independent random samples is just the product of the expectations. All terms have the same expectation. How many of them are there?

I suppose the critical question will be "What is the expectation of $\sqrt{y_i}$ ?".

 

[_joey]

It might work to multiply out the expression [tex]( \sum_{i=1}^n \frac{\sqrt{y_i}}{n})^2 [/itex] since the expectation of each term involving the product of independent random samples is just the product of the expectations. All terms have the same expectation. How many of them are there?

I suppose the critical question will be "What is the expectation of $\sqrt{y_i}$ ?".

Thanks for your thoughts. Here is my solution. Correct me if it's wrong, please.

http://img51.imageshack.us/img51/4955/exerc.jpg

 

[bpet]

Thanks for your thoughts. Here is my solution. Correct me if it's wrong, please.

http://img51.imageshack.us/img51/4955/exerc.jpg

No there's a factor of n missing in the last line. However the estimator can be reduced to
$$E[\hat{\lambda}_2] = \frac{n-1}{n}E[\sqrt{Y}]^2 + \frac{1}{n}E[Y]$$
so we just need to determine if $E[\sqrt{Y}]=\sqrt{\lambda}$.