Ch.7:
Point Estimation
1
Introduction
The objective of data collection is to learn about the distribution, or aspects of the dis
tribution, of some characteristic of the units of a population of interest. In Chapter 6,
we saw how to estimate certain key descriptive parameters of a population distribution,
such as the mean, the variance, percentiles, and probabilities (or proportions), from the
corresponding sample quantities. In this chapter we will learn another approach to esti
mation of such population parameters. This approach is based on the assumption that
the population distribution belongs in a certain family of distribution models, and on
methods for
fitting
a particular family of distribution models to data.
Different fitting
methods will be presented and discussed.
The estimators obtained from this other approach will, occasionally differ from the es
timators we saw in Chapter 6. For example, under the assumption that the population
distribution is normal, estimators of population percentiles and proportions depend only
on the sample mean and sample variance, and thus differ from the sample percentiles and
proportions; the assumption of a uniform distribution yields an estimator of the popu
lation mean value which is different from the sample mean; the assumption of Poisson
distribution yields an estimator of the population variance which is different from the
sample variance.
Another learning objective of this chapter is to develop criteria for selecting the best
among different estimators of the same quantity, or parameter. For example, should the
alternative estimators, mentioned in the preceding paragraph, be preferred over those
which were discussed in Chapter 6? The same criteria can also help us decide whether
a stratified sample is preferable to simple random sample for estimating the population
mean or a population proportion. Finally, in this chapter we will learn how to report the
uncertainty of estimators through their
standard error
and how that leads to
confidence
intervals
for estimators which have (or have approximately) a normal distribution.
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Overview, Notation and Terminology
Many families of distribution models, including all we have discussed, depend on a small
number of parameters; for example, a Poisson distribution model is identified by the single
parameter
λ
(the model mean number of counts), and normal models are identified by
two parameters,
μ
and
σ
2
. Such families of distribution models are called
parametric
.
An approach to extrapolating sample information to the population is to
assume
that
the population distribution is a member of (or belongs in) a specific parametric family of
distribution models, and then
fit
the assumed family to the data, i.e. identify the member
of the parametric family that best fits the data.
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 Fall '00
 Akritas
 Normal Distribution, Standard Deviation, Probability theory, θ

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