Today Today: Chapter 9 Assignment: Recommended Questions: 9.1, 9.8, 9.20, 9.23, 9.25

Chapter 9 - Estimation When given a model (e.g., N(μ, σ 2 )), would like to draw a sample and estimate the parameter in the model Example –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) Idea is to use samples of data to estimate parameters from models The estimators are sample statistics Not all estimators are necessarily good estimators This section looks at errors in estimators Will only consider the Mean Square Error (MSE)

Errors in Estimation (9.1) Suppose have a random sample (X 1, X 2,…,X n ) from some population and wish to estimate a parameter θ Let T be the statistic used to estimate θ Although T will vary from sample to sample, would like to be close to (equal to) θ on average T will be viewed as a good estimator of θ if its sampling distribution is centered at θ

Errors in Estimation (9.1) Unbiased estimator: Bias MSE:

Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a distribution with mean μ and finite variance σ 2, then Show that the sample mean is unbiased What is the MSE of T

Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a distribution with mean μ and finite variance σ 2 Suppose the mean and variance are unknown How would you estimate the mean? Variance?

Example Unbiased estimate of the variance:

Standard Error The variance of the sample mean is: Often the population variance is unknown Can estimate the population variance The standard error for the sample mean is: The standard error for the population proportion is:

Example A newspaper poll reports that 60% percent of the American public supports the President’s current policies What is the standard error of the sample proportion? What is the standard error estimating?

Estimation Can use the sample mean and sample variance to estimate the population mean and variance respectively How do we estimate parameters in general? Will consider 2 procedures: –Method of moments –Maximum likelihood

Method of Moments Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose distribution of interest has k parameters The procedure for obtaining the k estimators has 3 steps: –Conpute the first k population moments 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 Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose the population is Poisson Find the method of moments estimator for the rate parameter

Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population with pdf The mean and variance of X are: Find the method of moments estimator for the parameter

Maximum Likelihood Suppose X=(X 1, X 2,…,X n ) represents random sample from a Ber(p) population What is the distribution of the count of the number of successes What is the likelihood for the data

Example Suppose X=(X 1, X 2,…,X 10 ) represents random sample from a Ber(p) population Suppose 6 successes are observed What is the likelihood for the experiment If p=0.2, what is the probability of observing these data? If p=0.5, what is the probability of observing these data? If p=0.6, what is the probability of observing these data?

Maximum Likelihood Estimators Maximum likelihood estimators are those that result in the largest likelihood for the observed data More specifically, a maximum likelihood estimator (MLE) is: Since the log transformation is monotonically increasing, any value that maximizes the likelihood also maximizes the log likelihood

Example Suppose X=(X 1, X 2,…,X n ) represents random sample from a population Suppose the population is Poisson Find the MLE for the rate parameter