# Good Estimates of the Square of Bernoulli Probability of Success? [closed]

I am trying to understand the metrics of a good estimator. For example, the Bernoulli probability of success takes the parameter p. But for $$X_1,\ldots, X_n \overset{\textrm{iid}}{\sim}\textsf{Ber}(p^2),$$ how would you estimate the $$p^2?$$

How would you find $$\hat {p^2}= \frac1n\sum_{i=1}^n X_i =\bar X?$$ Or similarly $$Y_1,\ldots, Y_n \overset{\textrm{iid}}{\sim}\textsf{Ber}(p),$$ how would you find, working forwards not backwards, that$$\hat p = \frac1n \sum_{i=1}^n Y_i^2 = y^2$$ is a GOOD estimator? I am not following these examples.

• Please edit your question to use Mathjax (mathjax.org) for the math; it makes it SO much easier for us to read! Commented Apr 16, 2019 at 20:26
• Are you assuming a Bernoulli $p$ or Bernoulli $p^2$ distribution, as written? In the latter case $p^2$ is merely a strange way of specifying the usual Bernoulli parameter and therefore is estimated as always--usually as the mean of the data. Maybe you're trying to estimate $p=\sqrt{p^2}$? Or maybe you're in the former case and trying to estimate $p^2$ for a Bernoulli $p$ distribution??
– whuber
Commented Sep 5 at 13:56

The bias of an estimator $$\hat p$$, estimating a parameter $$p$$, is defined as
$$Bias(\hat p) = \mathbb{E}(\hat p) - p .$$
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.