 | Point
Estimation
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.
 | 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. |
|
To emphasize:
|
 | Point Estimator = Statistic (formula)
| Point Estimate = Number (calculated value) |
|
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| 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
to estimate
and 
to estimate
.
- For independent samples from continuous populations, we use
to estimate the difference 
and 
to estimate 
. If we know that the variances of the two populations are the same,
that is, 
, say, then we use the pooled estimate of variance,
to estimate
.
- 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
differences.
- For one sample from a 0-1 population we use the sample proportion p
to estimate the population proportion
, while for two independent samples we use
to estimate 
.
- In the correlation setting, we use the sample correlation
coefficient r to estimate the population correlation
coefficient
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.
| 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. |
| |
In statistical terminology, we say that
| Target 1 is biased/ with a small variance.
| Target 2 is unbiased/ with a large variance.
| Target 3 is unbiased/ with a small variance. |
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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.''
| 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
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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