Inference about a Mean

1
Inference about a Mean

• Lecture 6

2
Todays Plan

• Start to explore more concrete ideas of
  statistical inference
• Look at the process of generalizing from the
  sample mean to the population value
• Consider properties of the sample mean as a
  point estimator
• Properties of an estimator BLUE (Best Linear
  Unbiased Estimator)

3
Sample and Population Differences

• So far weve seen how weights connect a sample
  and a population
• But what if all we have is a sample without any
  weights?
• What can the estimation of the mean tell us?
• We need the sample to be composed of
  independently random and identically drawn
  observations

4
Estimating the Expected value

• Weve dealt with the expected value µyE(Y) and
  the variance V(Y) ?2
• Previously, our estimation of the expected value
  of the mean was

• But this is only only a good estimate of the true
  expected value if the sample is an unbiased
  representation of the population
• What does the actual estimator tell us?

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BLUE

• We need to consider the properties of as a
  point estimator of µ

• Three properties of an estimator BLUE
• Best (efficiency)
• Linearity
• Unbiasedness
• (also Consistency)
• Well look at linearity first, then unbiasedness
  and efficiency

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BLUE Linearity

• is a linear function of sample observations

• The values of Y are added up in a linear fashion
  such that all Y values appear with weight equal

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BLUE Unbiasedness

• Proving that is an unbiased estimator of the
  expected value of µ
• We can rewrite the equation for

• This expression says that each Y has an equal
  weight of 1/n
• Since ci is a constant, the expectation of is

8
Proving Unbiasedness

• Lets examine an estimator that is biased and
  inefficient
• We can define some other estimator m as

• We can then plug the equation for c into the
  equation for m and take its expectation
• The expected value of this new estimator m is
  biased if

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BLUE Best (Efficiency)

• To look at efficiency, we want to consider the
  variance of

• We can redefine as

• Our variance can be written as

• Where the last term is the covariance term
• Covariance cancels out because we are assuming
  that the sample was constructed under
  independence. So there should be no covariance
  between the Y values

• Note well see later in the semester that
  covariance will not always be zero

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BLUE Best (Efficiency) (2)

• So how did we get the equation for the variance
  of ?

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Variance

• Our expression for variance shows that the
  variance of is dependent on the sample size
  n
• How is this different from the variance of Y?

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Variance (2)

• Before when we were considering the distribution
  around µy we were considering the distribution of
  Y
• Now we are considering as a point estimator
  for µy
• The estimate for will have its own
  probability distribution much like Y had its own
• The difference is that the distribution for
  has a variance of ?2/n whereas Y has a variance
  of ?2

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Proving Efficiency

• The variance of m looks like this
• V(m) Sici2V(Y) ShSichciC(YhYi)

• Why is this not the most efficient estimate?

• We have an inefficient estimator if we use
  anything other than ci for weights

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Consistency

• This isnt directly a part of BLUE
• The idea is that an optimal estimator is best,
  linear, and unbiased
• But, an estimator can be biased or unbiased and
  still be consistent
• Consistency means that with repeated sampling,
  the estimator tends to the same value for

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Consistency (2)

• We write our estimator of µ as

• We can write a second estimator of µ

• The expected value of Y is

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Consistency (3)

• If n is small, say 10,

• Y will be a biased estimator of µ
• But, Y will be a consistent estimator
• So as n approaches infinity Y becomes an
  unbiased estimator of µ

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Law of Large Numbers

• Think of this picture

As you draw samples of larger and larger size,
the law of large numbers says that your
estimation of the sample mean will become a
better approximation of µy
The law only hold if you are drawing random
samples
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Central Limit Theorem

• Even if the underlying population is not normally
  distributed, the sampling distribution of the
  mean tends to normality as sample size increases
• This is an important result if n lt 30

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What have we done today?

• Examined the properties of an estimator.
• Estimator was for the estimation of a value for
  an unknown population mean.
• Desirable properties are BLUE Best Linear
  Unbiased.
• Also should include consistency.