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Chapter 7 Estimation
Maximum likelihood estimates
Example:
If X1, X2, …, X16 are observations of a random sample of size
16 from a normal distribution N(50,100), Find
Blah blah blah
How do we know that the mean is 50 and the variance is 100?
In this chapter, we consider random variables for which the function form of
pdf is known, but of the parameter of the pdf, say
θ
,
is unknown.
Parameter space
Ω
:
all possible values of
θ
.
Example:
f
(
x;
θ
)
=
(
1/
)
e
-x/
0 < x <
∞
θ
∈
Ω
= {
θ
:0 <
θ
<
∞
}
Objective:
Choose one member of
Ω
as the most likely value of
θ
How?
Take a random sample from the distribution to elicit some information
about the unknown parameter of
θ
X1, X2,…, Xn is a random sample of size n from the distribution
x
1
, x
2
,…, x
n
are observed values
Estimator:
The function of
X1, X2,…, Xn
used to estimate
θ
, say the statistics
u(
X1, X2,…, Xn), is called an estimator of
θ
Fall 2005
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