Definition of Estimator and Choosing
among Estimators: Lecture XVII
I.
What is An Estimator?
A.
In the next several lectures we will be discussing statistical estimators and
estimation.
The book divides this discussion into the estimation of a single
number such as a mean or standard deviation or the estimation of a range
such as a confidence interval.
B.
At the most basic level, the definition of an estimator involves the
distinction between a sample and a population.
1.
In general we assume that we have a random variable,
X
, with some
distribution function.
2.
Next, we assume that we want to estimate something about that
population, for example we may be interested in estimating the mean of
the population or probability that the outcome will lie between two
numbers.
a)
For example, in a farmplanning model we may be interested in
estimating the expected return for a particular crop.
b)
b.
In a regression context, we may be interested in estimating the
average effect of price or income on the quantity of goods consumed.
3.
This estimation is typically based on a sample of outcomes drawn from the
population instead of the population itself.
C.
Common point estimators are the sample moments:
1.
Sample Mean
n
i
i
X
n
X
1
1
2.
Sample Variance
2
1
2
1
2
2
1
1
X
X
n
X
X
n
S
n
i
i
n
i
i
X
3.
Sample
th
k
moment around zero
n
i
k
i
X
n
1
1
4.
Sample
th
k
moment around the mean
n
i
k
i
X
X
n
1
1
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AEB 6933
–
Mathematical Statistics for Food and Resource Economics
Lecture XVII
Professor Charles Moss
Fall 2005
2
5.
Sample Covariance
n
i
i
i
xy
Y
Y
X
X
n
S
Y
X
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 Fall '09
 CARRIKER
 Variance, Probability theory, Cumulative distribution function, Professor Charles Moss

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