Chapter 9a: One-Sample Estimation Problems
97
CHAPTER 9a — ONE-SAMPLE ESTIMATION PROBLEMS
In our development of probability theory leading to the Central Limit Theorem, we have
assumed complete knowledge of the population of interest.
Based on this knowledge, we
considered the
sampling distribution
of the sample mean.
Now we turn to a more realistic situation:
the population (or at least some aspect of the
population) is unknown to us.
We now explore the ways in which knowledge of the behavior of
common statistics (i.e., their sampling distributions) provides insight when we wish to use
sample information to draw conclusions about a population.
In short, we now begin the study of
statistical inference
.
Statistical inference
refers to the process of making generalizations (drawing conclusions) about
an unknown population.
There are two pervasive types of statistical inference:
•
Classical
– population parameters are estimated using only sample data; no outside
information is considered
•
Bayesian
– population parameters are estimated using a combination of sample data and
prior subjective knowledge
Our focus shall be on
classical
inference.
ESTIMATION
Suppose we are seeking the value of an unknown population parameter,
θ
.
To estimate the value of this parameter, we propose gathering a sample from the population:
n
X
X
X
,
,
,
2
1
K
, and calculating a
statistic
(or
estimator
)
,
( 29
n
X
X
X
g
,
,
,
ˆ
2
1
K
=
Θ
.
Notation:
Recall that random variables are denoted using uppercase letters, and particular
observations are lowercase.
Therefore, one particular sample that we observe would be
represented by
n
x
x
x
,
,
,
2
1
K
, and its related
estimate
is
( 29
n
x
x
x
g
,
,
,
ˆ
2
1
K
=
.
What are some desirable properties of
Θ
ˆ
?