Econ 140 Lecture 61 Inference about a Mean Lecture 6

Econ 140 Lecture 62 Today’s Plan Start to explore more concrete ideas of statistical inference Look at the process of generalizing from the sample mean to the population value Consider properties of the sample mean as a point estimator Properties of an estimator: BLUE (Best Linear Unbiased Estimator)

Econ 140 Lecture 63 Sample and Population Differences So far we’ve seen how weights connect a sample and a population But what if all we have is a sample without any weights? What can the estimation of the mean tell us? We need the sample to be composed of independently random and identically drawn observations

Econ 140 Lecture 64 Estimating the Expected value We’ve dealt with the expected value µ y =E(Y) and the variance V(Y) =  2 Previously, our estimation of the expected value of the mean was –But this is only only a good estimate of the true expected value if the sample is an unbiased representation of the population –What does the actual estimator tell us?

Econ 140 Lecture 65 BLUE We need to consider the properties of as a point estimator of µ Three properties of an estimator : BLUE –Best (efficiency) –Linearity –Unbiasedness –(also Consistency) We’ll look at linearity first, then unbiasedness and efficiency

Econ 140 Lecture 66 BLUE: Linearity is a linear function of sample observations The values of Y are added up in a linear fashion such that all Y values appear with weight equal

Econ 140 Lecture 67 BLUE: Unbiasedness Proving that is an unbiased estimator of the expected value of µ We can rewrite the equation for –This expression says that each Y has an equal weight of 1/n Since c i is a constant, the expectation of is

Econ 140 Lecture 68 Proving Unbiasedness Lets examine an estimator that is biased and inefficient We can define some other estimator m as We can then plug the equation for c’ into the equation for m and take its expectation The expected value of this new estimator m is biased if

Econ 140 Lecture 69 BLUE: Best (Efficiency) To look at efficiency, we want to consider the variance of We can redefine as Our variance can be written as –Where the last term is the covariance term –Covariance cancels out because we are assuming that the sample was constructed under independence. So there should be no covariance between the Y values –Note: we’ll see later in the semester that covariance will not always be zero

Econ 140 Lecture 610 BLUE: Best (Efficiency) (2) So how did we get the equation for the variance of ?

Econ 140 Lecture 611 Variance Our expression for variance shows that the variance of is dependent on the sample size n How is this different from the variance of Y?

Econ 140 Lecture 612 Variance (2) Before when we were considering the distribution around µ y we were considering the distribution of Y Now we are considering as a point estimator for µ y –The estimate for will have its own probability distribution much like Y had its own –The difference is that the distribution for has a variance of  2 /n whereas Y has a variance of  2

Econ 140 Lecture 613 Proving Efficiency The variance of m looks like this V(m) =  i c i ’ 2 V(Y) +  h  i c h ’c i ’C(Y h Y i ) Why is this not the most efficient estimate? We have an inefficient estimator if we use anything other than c i for weights

Econ 140 Lecture 614 Consistency This isn’t directly a part of BLUE The idea is that an optimal estimator is best, linear, and unbiased But, an estimator can be biased or unbiased and still be consistent Consistency means that with repeated sampling, the estimator tends to the same value for

Econ 140 Lecture 615 Consistency (2) We write our estimator of µ as We can write a second estimator of µ The expected value of Y* is

Econ 140 Lecture 616 Consistency (3) If n is small, say 10, –Y* will be a biased estimator of µ –But, Y* will be a consistent estimator –So as n approaches infinity Y* becomes an unbiased estimator of µ

Econ 140 Lecture 617 Law of Large Numbers Think of this picture: As you draw samples of larger and larger size, the law of large numbers says that your estimation of the sample mean will become a better approximation of µ y The law only hold if you are drawing random samples

Econ 140 Lecture 618 Central Limit Theorem Even if the underlying population is not normally distributed, the sampling distribution of the mean tends to normality as sample size increases –This is an important result if n < 30

Econ 140 Lecture 619 What have we done today? Examined the properties of an estimator. Estimator was for the estimation of a value for an unknown population mean. Desirable properties are BLUE: Best Linear Unbiased. Also should include consistency.