Here's what you'll find in this section:
If I have a known distribution, but I don't know what the parameter
, 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
- Estimate A point estimate/ of a parameter
is a single number based on the sample data that we can consider to be
the most plausible value of
- Estimator The statistic (formula) used to obtain the point estimate.
- Point Estimator = Statistic (formula)
- Point Estimate = Number (calculated value)
Point Estimators for Different Situations
As we have seen before, we have several natural point estimates for
the various situations we will consider:
- For one sample from a continuous population, we use
- For independent samples from continuous populations, we use
to estimate the difference
. If we know that the variances of the two populations are the same,
, say, then we use the pooled estimate of variance,
- For paired data, we use the mean,
, and variance,
of the n differences
in the pairs to estimate the mean and variance of the population of
- For one sample from a 0-1 population we use the sample proportion
to estimate the population proportion
, while for two independent samples we use
- In the correlation setting, we use the sample correlation
coefficient r to estimate the population correlation
and the slope and the intercept of the least squares line to
estimate the slope and intercept of the true population line.
Properties of Point Estimators
Consider three marksmen,
Here we have three different situations.
In statistical terminology, we say that
- Target 1 has all its shots clustered tightly together, but none
of them hit the bullseye.
- Target 2 has a large spread, but on average the bullseye is hit.
- Target 3 has a tight cluster around the bullseye.
If you were hiring for the police department, which shooter would you
want? In general in statistics, we want both unbiased and small
variance--an estimator that almost always is ``on target.''
- Target 1 is biased/ with a small variance.
- Target 2 is unbiased/ with a large variance.
- Target 3 is unbiased/ with a small variance.
Minimum Variance Unbiased Estimation
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
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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