Transcript Lecture 17 - Statistics

Today
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Today: Chapter 9
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Assignment:
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Recommended Questions: 9.1, 9.8, 9.20, 9.23, 9.25
Chapter 9 - Estimation
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When given a model (e.g., N(μ, σ2)), would like to draw a sample
and estimate the parameter in the model
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Example
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The Neilson Corp. draws samples of TV viewers to estimate the
proportion of viewers watching various TV shows
If a sample of size n=1000 is taken on Thursday at 8:00 pm, what is the
distribution of the number of people watching Friends
How do we estimate the population proportion of people watching
Friends?
Errors in Estimation (9.1)
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Idea is to use samples of data to estimate parameters from models
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The estimators are sample statistics
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Not all estimators are necessarily good estimators
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This section looks at errors in estimators
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Will only consider the Mean Square Error (MSE)
Errors in Estimation (9.1)
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Suppose have a random sample (X1, X2,…,Xn ) from some
population and wish to estimate a parameter θ
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Let T be the statistic used to estimate θ
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Although T will vary from sample to sample, would like to be close
to (equal to) θ on average
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T will be viewed as a good estimator of θ if its sampling distribution
is centered at θ
Errors in Estimation (9.1)
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Unbiased estimator:
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Bias
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MSE:
Example
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Suppose X=(X1, X2,…,Xn) represents random sample from a
distribution with mean μ and finite variance σ2, then
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Show that the sample mean is unbiased
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What is the MSE of T
Example
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Suppose X=(X1, X2,…,Xn) represents random sample from a
distribution with mean μ and finite variance σ2
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Suppose the mean and variance are unknown
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How would you estimate the mean?
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Variance?
Example
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Unbiased estimate of the variance:
Standard Error
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The variance of the sample mean is:
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Often the population variance is unknown
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Can estimate the population variance
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The standard error for the sample mean is:
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The standard error for the population proportion is:
Example
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A newspaper poll reports that 60% percent of the American public
supports the President’s current policies
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What is the standard error of the sample proportion?
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What is the standard error estimating?
Estimation
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Can use the sample mean and sample variance to estimate the
population mean and variance respectively
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How do we estimate parameters in general?
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Will consider 2 procedures:
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Method of moments
Maximum likelihood
Method of Moments
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Suppose X=(X1, X2,…,Xn) represents random sample from a
population
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Suppose distribution of interest has k parameters
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The procedure for obtaining the k estimators has 3 steps:
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Conpute the first k population moments
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first moment is the mean, second is the variance, …
Set the sample estimates of these moments equal to the population
moment
Solve for the population parameters
Example
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Suppose X=(X1, X2,…,Xn) represents random sample from a
population
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Suppose the population is Poisson
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Find the method of moments estimator for the rate parameter
Example
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Suppose X=(X1, X2,…,Xn) represents random sample from a
population with pdf
f (x |  ) 
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x  x /
e
; for x  0

The mean and variance of X are:
E ( X )  2 ;
 2  2 2
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Find the method of moments estimator for the parameter
Maximum Likelihood
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Suppose X=(X1, X2,…,Xn) represents random sample from a Ber(p)
population
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What is the distribution of the count of the number of successes
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What is the likelihood for the data
Example
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Suppose X=(X1, X2,…,X10) represents random sample from a Ber(p)
population
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Suppose 6 successes are observed
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What is the likelihood for the experiment
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If p=0.2, what is the probability of observing these data?
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If p=0.5, what is the probability of observing these data?
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If p=0.6, what is the probability of observing these data?
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Maximum Likelihood Estimators
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Maximum likelihood estimators are those that result in the largest
likelihood for the observed data
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More specifically, a maximum likelihood estimator (MLE) is:
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Since the log transformation is monotonically increasing, any value
that maximizes the likelihood also maximizes the log likelihood
Example
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Suppose X=(X1, X2,…,Xn) represents random sample from a
population
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Suppose the population is Poisson
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Find the MLE for the rate parameter