Download Point Estimation

6.1 Point Estimation
What is an estimate?
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Want to study a population
Ideally the knowledge of the distribution,
Some parameters of the population may be the
first consideration:
θ
Use the information from a sample to estimate
Estimator θ̂ : defines the procedure (the formula)
Different estimators, each is a rv.
Examples of estimators
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want to estimate the mean µ of a normal dist
Possibilities of estimators:
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sample mean
X̄ =
sample median
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Xi
n
average of the smallest and the largest
10% trimmed mean
Which one is best (better)?
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Error in estimation
θ̂ = θ + error of estimation
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Error is random, depending on the sample
We would like to control the error
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mean 0 (unbiased)
smallest variance
Unbiased Estimator
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estimator: a particular statistic
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unbiased: E(θ̂) = θ
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If possible, we should always choose an unbiased
different estimators: some tend to over (under)
estimate
difference is called the bias
examples: sample mean (yes), sample proportion
(yes) if X is a binomial rv.
Sample Variance
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Need an estimator for the variance
Natural candidate: sample variance
Why divide by n-1, not n?
Make sure the estimator is unbiased
We can not say the same thing for sample standard
deviation
Unbiased Estimators for
Population Mean
• several choices (make things complicated)
• sample mean
• if distribution is continuous and symmetric,
sample median and any trimmed mean
Minimum Variance
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suppose we have two estimators, both unbiased,
which one do we prefer?
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pick the one with smaller variance
minimum variance unbiased estimator (MVUE)
for normal distribution, the sample mean X̄ is the
MVUE for the population mean.
Complications
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Which estimator is better? depending on the
distribution
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Cauchy dist: MVUE unknown, sample median is
better
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Uniform dist: X̄e is the best
Trimmed mean quite versatile, quite useful in
practice
Standard Error
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What quantity shall we use to report the error in
estimation?
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Standard error:
σθ̂ =
V (θ̂)
Not always available in theory
May involve some guess for other parameters