1. t distribution Suppose Z ~ N(0,1) independent of X ~ χ2(n). Then,
that is, T has a t distribution with n degrees of freedom.
The probability density function of T is given by
Properties of the t distribution …
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2. Important Use of the t Distribution
Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ and variance σ2. Then,
Proof:
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3. Cauchy Distribution Let Z1, Z2 be independent N(0,1) variables. Then….
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4. F distribution Suppose X ~ χ2(n) independent of Y ~ χ2(m). Then,
Important use of this is ….
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5. Properties of the F distribution
The F-distribution is a right skewed distribution.
i.e.
Can use Table A.9 on page 794 to find percentile of the F- distribution.
Example…
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6. Claim
The square of random variable with t(n) distribution has an F distribution with (1, n) df. That is,
Proof:
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7. Parameters and Point Estimate
Distributions have parameters. Parameters are usually denoted using the letter θ.
A point estimate, is a single number, usually calculated from the sample data, that we use to estimate an unknown parameter θ.
A point estimator is a statistic (i.e. a function) that tells us how we can use the sample data to create a numeric point estimate.
A point estimator and a point estimate is usually denoted by
Examples:…
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8. Assessing an Estimator
For any parameter, there are many different point estimators of θ. How do we know which point estimators are “good”?
There are few criteria…
The bias of an estimator is
We would like our estimator to have zero (or very small) bias.
The variance of an estimator is
We would like our estimator to have small variance.
Small bias and small variance are usually competing goals, often we can’t minimize both properties.
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9. Mean Square Error of Point Estimators
The mean square error (MSE) of a point estimator is
The Mean Square Error of an estimator combines bias and variance, we want our estimator to have small MSE.
Claim:
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10. Example
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11. Minimum Variance Unbiased Estimator
MVUE is the unbiased estimator with the smallest possible variance. Look amongst all unbiased estimators for the one with the smallest variance.
Note, is called the Standard Error of a point estimator.
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