Download Mathematical Statistics Lecture Notes

Mathematical Statistics
Lecture Notes
Chapter 8 – Sections 8.1-8.4
General Info
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I’m going to try to use the slides to help save my voice.
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First homework is now posted – covers 8.1-8.5 and is
due next Wednesday, Feb. 2.
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We should be finished that material by Friday.
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Structure of these notes: series of definitions, etc. then
examples (by hand on board with more comments)
Chapter 8 – Estimation
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What is an estimator?
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A rule, often expressed as a formula, that tells how to calculate
the value of an estimate based on measurements in a sample.
What estimators are you already familiar with? Two
should come to mind.
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If you want to estimate the proportion of Amherst College
students who participated in community service over Winter
Break, what estimator would you use?
If you want to estimate the average amount of money spent in
traveling expenses by Amherst College students over Winter
Break, what estimator would you use?
Questions about Estimators to Think About
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What estimator should I use?
If I have multiple estimators, how do I pick one? How do
we know which is best?
How do we place bounds on the estimate or the error of
the estimate?
How well does the estimator perform on average?
We’ll look at all these questions in Chapter 8 and 9.
8.2 helps with question 2.
8.4 helps with question 3.
8.2 – Bias and Mean Square Error (MSE) of
Point Estimators
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What is a point estimator?
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It is an estimator that is a single value (or vector). It is NOT an
interval of possible values (like a confidence interval). The
estimators you are familiar with are most likely point
estimators.
For notation, let  be the parameter to estimate and let
ˆ be the estimator for a general estimation setting.
Definition of Bias
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ˆ is an unbiased estimator for  if E (ˆ)   .
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The bias of a point estimator ˆ is given by:
 (ˆ)  E (ˆ)  
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How are we going to compute bias? We need some basic
information about the distribution of ˆ . This may mean
using methods of transformations (from Probability) to
obtain a pdf, etc. (For reference in current text: Chapter
6)
Why not look at the variance of the
estimator?
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Well, you could.
You would want the variance of the estimator to be small.
It turns out there is a another quantity, called the mean
square error that can be examined to gain information
about the bias and variance of your estimator.
Definition of Mean Square Error (MSE)
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The MSE of a point estimator ˆ is given by:
MSE (ˆ)  E[(ˆ   ) 2 ]
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We note (as a useful result) that:
MSE (ˆ)  Var (ˆ)  [  (ˆ)]2
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This means that if our estimator is unbiased, the MSE is
equal to the variance of the estimator.
Proof of Useful Result
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See Board
Please bear with me as I try to make notes about the
computation so you can follow.
8.3 – Some Common Unbiased Estimators
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See chart on board
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Standard deviation vs. Standard error
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Standard deviations involve the unknown parameters
Standard errors mean you have plugged in some logical
estimators for those parameters.
Your book will use them interchangeably (more than I would
like). Basically if you need an actual value for a calculation, go
ahead and use the standard error.
Fortunately for us, due to large sample asymptotics, the math
still works out.
8.4 – Definition of Error of Estimation
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The error of estimation is the distance between an
estimator and its target parameter:
 | ˆ   |
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The error of estimation is a random variable because it
depends on the estimator!
The norm is usual Euclidean distance.
More on Error of Estimation
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We can make probabilistic statements about  ; well
really, they are statements about the estimator and
parameter. This concept leads to confidence intervals
(8.5).
For example:
P(| ˆ   | b)  P(b  ˆ    b)  P(ˆ  b    ˆ  b)  
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Set  , and then find b so that (if you have the pdf of the
ˆ b
estimator):

ˆ b
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f (ˆ)dˆ  
Could also use Tchebysheff’s.
Examples
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See board.
Bear with me again as I try to make notes so the
computation is easy to follow.