
Here's what you'll find in this section:
If I have a known distribution, but I don't know what the parameter values (e.g., , , etc.) are, how can I estimate these population parameters using a sample of size n (sample statistics)? Given a population parameter of interest, e.g., the population mean or the population proportion , we want to use a sample to compute a number that represents a ``good'' guess for the true value of the parameter. This number is called a point estimate. We will use the Greek letter (``theta'') to represent any of the parameters we will study such as , , , etc.
To emphasize:
As we have seen before, we have several natural point estimates for the various situations we will consider:
to estimate .
Consider three marksmen,
Here we have three different situations.
It is not always obvious what the best way to estimate a parameter is. For example, the sample mean and sample median are both natural estimators of the mean of a normal population. We want our estimator to be like target 3: unbiased with the smallest possible variance. Thus, we only look at unbiased estimators and choose the one with the smallest variance. This we call the minimum variance unbiased estimator/ (MVUE) or the best unbiased estimator/ (BUE), because it is the ``best'' estimator we can get using only unbiased estimators.
NOTE:\ For the normal distribution, the mean, median, and mode all occur at the same location. Why do we usually use the mean, , instead of the median, ? It turns out that both and are unbiased, but has a smaller variance than . In fact, is the MVUE. See the ``Minimum variance estimation'' concept lab for some examples of this.
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