Ch.7:
Estimation
1
Introduction
The objective of data collection is to learn about the distribution, or aspects of the
distribution, 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 the
estimation 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 esti
mators 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, which were mentioned in the preceding paragraph, be preferred
over those which were discussed in Chapter 6?
The same criteria can also help us de
cide 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. The above estimation concepts will be developed here in the context of a
single sample, but will be applied in later chapters to samples from several populations.
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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
, 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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