- Given a population described by a
random variable Y. Suppose
that the distribution of Y is known except for one or more
- Suppose Y is the number of cigarettes and we know (or assume)
that Y is normally distributed, but we don't know Y's mean
. In this
case, the unknown parameter
- We observe N observations from the population, called a sample:
- Sample =
- Each observation in the sample can be thought of as a different
random variable. The actual value in the data set for observation 10
is a realization of the corresponding random variable Y_10.
- We can think of the sample space
here as all possible outcomes in a data set:
- Estimation from a Sample
- An Estimator is in general terms simply
- a function of a sample space
- Basic Properties that all Estimator share
- Because an estimator is a function of the sample space, then
it is a random variable.
- Because an estimator is a random variable,
it has a distribution, a mean, a variance, etc.
- Because estimators have distributions, they have statistical properties
- Because estimators are functions of sample information, they
are usually represesnted by formulas or procedures
- Desirable Properties than Estimators May or May not Have.
be an estimate of some population parameter
- In words: Across samples, average value of
the value of the population parameter
- Asymptotically Unbiased (with sample size n)":
- In words: With ``enough" data, the estimator is unbiased.
- Asymptotically Normal
- In words: With "enough" data, the distribution of the
estimator becomes the normal distribution.
- Consistent - requires two things of
- 1. Asymptotically Unbiased
- 2. Asymptotically Variance of estimator goes to 0:
- In words: With enough data the distribution of the
estimator collapses onto the true value.
- Example: Estimation of the population mean
- The usual estimator for the population mean is
sample mean (let's call it
is a function (formula).
is a function of
random variables, it is a random variable itself. Its value
is not determined until the sample is drawn from the population.
The usual estimator for the population variance
- For a given
sample (realization of all the Yi's),
takes on a particular
value. But the sample mean, as a concept, is a function and
a random variable. It is not conceptually a number.
- Statistical Properties of
- In words: The mean of the sample mean is the population
mean. It is an unbiased estimator.
- In words: The variance of the sample mean is the variance
of the underlying random variable divided by the
sample size. It follows directly that
is a consistent estimate of
follows directly from the first two properties.
- In words: the sample variance is an unbiased
- The Central Limit Theorem:
- In words: The distribution of the sample mean for any
population converges to a normal distribution as the
sample size approaches infinity.
distributed. This imples that
- Note that the variance of
upon the variance of the random variable itself
- The Estimated Variance of the sample mean
- The Estimated Standard Error of the sample mean
- You can see the Central Limit Theorm in action
by running the
tutorial for week 2.
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Document Last revised: 1997/1/5