# Thread: Is this an unbiased estimator for Xi?

1. ## Is this an unbiased estimator for Xi?

So here's the question:

Suppose we observe Xi (as i=1,..., 10) which are identically distributed with mean E(Xi) = mu and variance V(Xi)=sigma^2. Then is x(bar) = X1 an unbiased estimator of mu?

Logically speaking, it doesn't make sense that it would be. If i=1,..., 10 is the entire population, then the mean is 55/10 (sum of Xi as i goes to 10, divided by 5) = 5.5. To say that an estimator for the mean, x(bar), is equal to X1 (which if i=1), in turn, equals 1. Is this unbiased? It doesn't seem so - population mean is 5.5 and that is not equal to 1. Am I understanding this correctly?

2. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by elay422
So here's the question:

Suppose we observe Xi (as i=1,..., 10) which are identically distributed with mean E(Xi) = mu and variance V(Xi)=sigma^2. Then is x(bar) = X1 an unbiased estimator of mu?

Logically speaking, it doesn't make sense that it would be. If i=1,..., 10 is the entire population, then the mean is 55/10 (sum of Xi as i goes to 10, divided by 5) = 5.5. To say that an estimator for the mean, x(bar), is equal to X1 (which if i=1), in turn, equals 1. Is this unbiased? It doesn't seem so - population mean is 5.5 and that is not equal to 1. Am I understanding this correctly?
x(bar) = X1 is not an unbiased estimator of mu. Apply the definition of unbiased estimator that you have been given.

3. ## Re: Is this an unbiased estimator for Xi?

Is something wrong with my logic? And are you sure that it is not unbiased?

4. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by elay422
So here's the question:

Suppose we observe Xi (as i=1,..., 10) which are identically distributed with mean E(Xi) = mu and variance V(Xi)=sigma^2. Then is x(bar) = X1 an unbiased estimator of mu?

Logically speaking, it doesn't make sense that it would be. If i=1,..., 10 is the entire population, then the mean is 55/10 (sum of Xi as i goes to 10, divided by 5) = 5.5. To say that an estimator for the mean, x(bar), is equal to X1 (which if i=1), in turn, equals 1. Is this unbiased? It doesn't seem so - population mean is 5.5 and that is not equal to 1. Am I understanding this correctly?
For a function of a sample to be an unbiased estimator of some parameter requires that the expectation be equal to the parameter value.

$\displaystyle f(X_1,X_2,...,X_n)=X_1$

$\displaystyle E[f(X_1,X_2,...,X_n)]=E(X_1)=\mu$

(you are told this) so this is an unbiased estimator of $\displaystyle \mu$

CB

5. ## Re: Is this an unbiased estimator for Xi?

For some reason I don't get why that first part is correct - that f(X1, X2,...,X10) = X1. Shouldn't that be Xi? Could you explain this?

6. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by CaptainBlack
For a function of a sample to be an unbiased estimator of some parameter requires that the expectation be equal to the parameter value.

$\displaystyle f(X_1,X_2,...,X_n)=X_1$

$\displaystyle E[f(X_1,X_2,...,X_n)]=E(X_1)=\mu$

(you are told this) so this is an unbiased estimator of $\displaystyle \mu$

CB
Whoops. I didn't see that bit.

7. ## Re: Is this an unbiased estimator for Xi?

^ Could you possible explain that? Thanks so much, guys.

8. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by elay422
^ Could you possible explain that? Thanks so much, guys.
You are told E(Xi) = mu. Therefore E(X1) = mu. Now - again - go back and review the defintion of an unbiased estimator.

9. ## Re: Is this an unbiased estimator for Xi?

I understand the definition but where did that E(X1) come from? Doesn't that X1 mean it's the first value of the 10 that are in the population? How is that close to the mean of the population?

10. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by elay422
I understand the definition but where did that E(X1) come from? Doesn't that X1 mean it's the first value of the 10 that are in the population? How is that close to the mean of the population?
Look, it comes straight from the problem statement!:

Suppose we observe Xi (as i=1,..., 10) which are identically distributed with mean E(Xi) = mu
[snip]
So let i = 1: E(X1) = 1.

There is nothing more can be said!

11. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by elay422
For some reason I don't get why that first part is correct - that f(X1, X2,...,X10) = X1. Shouldn't that be Xi? Could you explain this?
You are extimating $\displaystyle \mu$ by the value of $\displaystyle X_1$, But the expectation of each of the $\displaystyle X_i$ 's $\displaystyle (i=1,...,10)$ is $\displaystyle \mu$, so the expectation of your estimator is $\displaystyle \mu$, hence it is unbiased.

CB

12. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by CaptainBlack
You are extimating $\displaystyle \mu$ by the value of $\displaystyle X_1$, But the expectation of each of the $\displaystyle X_i$ 's $\displaystyle (i=1,...,10)$ is $\displaystyle \mu$, so the expectation of your estimator is $\displaystyle \mu$, hence it is unbiased.

CB
I guess there was something more could be said

13. ## Re: Is this an unbiased estimator for Xi?

Originally Posted by mr fantastic
I guess there was something more could be said
For some people hearing a thing with many words seems more comprehensible than hearing it with few words. And as we also know; the conviction of the many is more easily believed by some than that of the few (no matter how preposterous that conviction, but in this case we have the real TRUTH(tm))

CB