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I am trying to understand the metrics of a good estimator. For example, the Bernoulli probability of success takes the parameter p. But for X1...Xn iid Ber(p^2) how would you estimate the p^2. How would you find p^2(hat)= (1/n) Summation from 1 to n (Xi) = X(bar). Or similarly Y1...Yn iid Ber(p), how would you find, working forwards not backwards, that p(hat) = (1/n)summation from 1 to n Yi^2 = y^2 is a GOOD estimator. I am not following these examples. Thank you.

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  • $\begingroup$ Please edit your question to use Mathjax (mathjax.org) for the math; it makes it SO much easier for us to read! $\endgroup$ – jbowman Apr 16 at 20:26
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There are a number of ways to evaluate an estimator, in the cases you have mentioned unbiasedness is a good one to start with.

The bias of an estimator $\hat p$, estimating a parameter $p$, is defined as

$$Bias(\hat p) = \mathbb{E}(\hat p) - p .$$

An unbiased estimator is an estimator with a bias of zero. For a lot of cases (and before one learns the other ways of evaluating an estimator), unbiased means good.

In your first example you are estimating the parameter $p^2$ with the estimator $\hat{p^2} = \frac{1}{n}\sum_{i=1}^{n} X_i$, where $X_i \overset{i.i.d}{\sim} Ber(p^2)$. Using the definition above, and noting $\mathbb{E}(X_i) = p^2$, you should find the bias of $\hat{p^2}$ to be zero.

In the second example you are estimating the parameter (which I have intentionally renamed to avoid confusion with the first example) $q$ with the estimator $\hat{q} = \frac{1}{n} \sum_{i=1}^{n} Y_i^2$, where $Y_i \overset{i.i.d}{\sim} Ber(q)$. Again using the definition of bias, and noting $\mathbb{E}(Y_i^2) = q$, you should find the bias of $\hat{q}$ to be zero.

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