- This is part of probstat.
Previously we tried to estimate the population means and variances using the sample means and variances. In this section, we shall see the justification why what we did makes sense.
There are many ways to estimate parameters.
Method of moments estimators
- See also wikipedia article.
This is probably the simplest estimators. However, they are often biased (as we shall show in the example).
Definition: For a random variable
,
is called the k-th moment of
. Note that the first moment is the mean
. The variance of a random variable depends on the first and the second moments.
If we want to estimate a parameter
, using the method of moments, we start by writing the parameter as a function of the moments, i.e.,
We then estimate the sample moments
for
. Our estimate
is thus
EX1: We show how to estimate the variance with the method of moments. Recall that the variance
We first estimate the first moment
and the second moment
. The estimator is
Note that the estimate
is biased, because
.
As this example shows, other estimation techniques are usually preferred over the method of moments.
Maximum likelihood estimators
Suppose that we want to estimate parameter
based on observations (or sample)
. If we can find a joint-distribution function
,
that gives the probability that we observe
given a particular value
. The maximum likelihood estimator is
such that
is maximum, i.e.,
.
Examples
EX1: A box contains 5 balls; they are blue balls and white balls. We randomly choose a ball from the box 3 times with replacement and get 2 blue balls and 1 white balls. Estimate the number of blue balls in the box using the maximum likelihood method.
Solution: Let
denote the number of blue balls. The possible values are 0,1,2,3,4, and 5. For each value of
the probability that we observe the outcome of getting 2 blue balls and 1 while balls is
.
We put the probabilities in the following table.
|
0
|
1
|
2
|
3
|
4
|
5
|
Probability
|
0
|
0.096
|
0.288
|
0.432
|
0.384
|
0
|
Since the probability is maximum at
, the maximum likelihood estimator for the number of blue balls is 3.
EX2: Maximum likelihood estimator for a Bernoulli parameter
A coin has probability
of turning up heads. We perform
independent trials and obtain random variables
such that
if in the i-th trial, we get a Head, and
otherwise. What is the maximum likelihood estimator for
?
Solution: Note that
. We rewrite the probability that the random variable
takes value
for
as
.
(Remarks: The random variable is
. We denote the actual value of the variable with
(i.e., with a small x).)
Therefore, the probability of obtaining the data
given parameter
is
Since
maximizes also when
is maximum, we can take a logarithm of the above term and get
To find its maximum, we differentiate the term and get
Setting
, we solve the equation and get that
implying that
which is the maximum likelihood estimator for
.
EX3: maximum likelihood estimator for the Poisson mean
- To be added...
Bayes estimators
Let the observed data be
. We fist compute the conditional probability
With that distribution function, the Bayes estimator is