STAT511 — Summer 2009
Lecture Notes
1
Chapter 6
July 16, 2009
Point Estimation
6.1
Point Estimation
Chapter Overview
•
Point estimate and point estimator
–
Define point estimate and estimator
–
Unbiased estimators
–
Minimum variance estimators
•
Finding the standard error of an estimator
–
Deriving directly
–
Bootstrapping
•
Methods of point estimation
–
The method of moments
–
The method of maximum likelihood
–
Estimating functions of parameters
Point Estimate and Point Estimator
Example 6.1.1
Not in textbook. Suppose we want to find the ratio
p
of FIV infected
cats in a specific area. Impossible to check all feral cats in the area. Maybe we can do
the following:
1. let
X
be the rv that:
X
= 1 if cat has FIV and
X
= 0 if not.
2. Distribution of
X
? Bernoulli,
p
unknown.
3. Now the question becomes: how to estimate the value of the parameter
p
of a
Bernoulli distribution?
RV =
⇒
Distribution =
⇒
Parameter of interest
Point Estimate and Point Estimator
Example 6.1.1 continued...
To estimate the value of
p
:
1. We go catch 25 feral cats randomly and check them. This is a random sample of
size
n
= 25 of the Bernoulli distribution
X
1
, X
2
, ..., X
25
.
2. Use statistic ˆ
p
=
X
1
+
...
+
X
25
25
. This statistic is a random variable, its value could be
used to estimate
p
, and this statistic is called a
point estimator
of
p
.
Purdue University
Chapter6˙print.tex; Last Modified: July 16, 2009 (W. Sharabati)
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STAT511 — Summer 2009
Lecture Notes
2
3. Suppose we found that cat number 1,5,10,15,23 are infected with FIV, then ˆ
p
=
5
25
= 0
.
2 Now the actual value of ˆ
p
is 0
.
2 and this is called the
point estimate
of
p
.
Random sample =
⇒
Estimator =
⇒
Estimate
Definitions and Differences
Definition 1.
A
point estimate
of a parameter
θ
is a single number that can be
regarded as a sensible value for
θ
. A point estimate is obtained by selecing a suitable
statistic and computing its value from given sample data. The selected statistic is called
the
point estimator
.
•
Difference between point estimate and point estimator: point estimate is a
value
,
point estimator is a
statistic
.
•
Usually we use
ˆ
θ
to denote the point estimator of a parameter
θ
.
•
Different statistic can be used to estimate the same parameter, i.e., a parameter
may have multiple point estimators.
Example 6.1.2
Example 6.1.2
Example 6.2 in textbook. Assume the dielectric breakdown voltage
for pieces of epoxy resin to be normally distributed. Now we want to estimate the mean
μ
of the breakdown voltage.
We randomly check 20 breakdown voltages, and denote
them
X
1
, X
2
, ..., X
20
. This is a random sample of size 20 from this normal distribution
of interest. Suppose the observed voltage values are:
{
24.46, 25.61, 26.25, 26.42, 26.66,
27.15, 27.31, 27.54, 27.74, 27.94, 27.98, 28.04, 28.28, 28.49, 28.50, 28.87, 29.11, 29.13, 29.50,
30.88
}
Which point estimators could be used to estimate
μ
?
1. Sample mean: ˆ
μ
=
X
2. Sample median: ˆ
μ
=
e
X
3. Average of the extremes: ˆ
μ
=
min
(
X
i
)+
max
(
X
i
)
2
4. Trimmed mean, say, 10% trimmed mean (discard the smallest and largest 10% of
the sample data and then take an average): ˆ
μ
=
X
tr
(10)
5. etc..
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 Spring '08
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 Normal Distribution, Variance, Maximum likelihood, Estimation theory, Purdue University, W. Sharabati

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