2_Probability and Statistics

9112012 p kolm 34 an example estimating the population

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Unformatted text preview: la to calculate q from sample data is called an estimator VER. 9/11/2012. © P. KOLM 34 An Example Estimating the population mean by the sample average n Let us start with the formula for E (X ) = å x i f (x i ) i =1 Since each point has an equal chance of being included in the sample, we 1 for f (x i ) (the probability weight). Then, the sample average for n our sample takes the form substitute 1n ˆ mX º X º å Xi n i =1 We will see many more examples throughout the lecture sequence VER. 9/11/2012. © P. KOLM 35 What Makes a Good Estimator? Some estimators are better than others. The search for good estimators is a big part of econometrics Estimators are compared based on a number of attributes: Unbiasedness Efficiency Mean Square Error (MSE) Asymptotic properties (for large samples) Consistency We discuss each one of these attributes next VER. 9/11/2012. © P. KOLM 36 Unbiasedness Intuition: We want our estimator “to be right on average” ˆ Formally, we say that an estimator, q = h(Y ) , of the parameter q is unbiased if the mean of its sampling distribution is q . Mathematically, this means that ˆ E (q) = q VER. 9/11/2012. © P. KOLM 37 Example: The Sample Mean and Variance Let Y = (Y1,Y2,...,Yn ) be a random sample from a population with mean mY and 2 variance sY 1n The sample mean Y = åYi is unbiased: n i =1 We take the expectation of the sample mean æ1 n ö 1 n 11 ÷= ç E (Y ) = E ç åYi ÷ E (Yi ) = å n mY =mY ÷ ÷ ç n i =1 ø n å n i =1 è i =1 Note: The variance of Y (the sample variance) is given by 2 n æ1 n ö sY 2 ÷= 1 ç ÷ Var (Y ) = Var ç åYi ÷ s= ç n i =1 ÷ n 2 å Y n è ø i =1 Therefore, as n increases Y gets more precise VER. 9/11/2012. © P. KOLM 38 1n The estimator for the variance s = å (Yi -Y )2 is unbiased: n - 1 i =1 First, observe that 2 Y n n å (Yi -Y ) = å ((Yi - mY ) - (Y - mY )) 2 i =1 2 i =1 n n = å (Yi - mY ) - 2(Y - mY )å (Yi - mY ) + n(Y - mY )2 2 i =1 n i =1 = å (Yi - mY )2 - 2n(Y - mY )2 + n(Y - mY )2 i =1 n = å (Yi - mY )2 - n(Y - mY )2 i =1 Therefore, æ1 n ö 2÷ ç E (s ) = E ç (Y -Y ) ÷ ÷ ÷ çn - 1 å i è ø i =1 æ1 n ö n 2÷ ç ÷ E (Y - mY )2 =Eç (Yi - mY ) ÷ å ÷ n -1 ç n - 1 i =1 è ø 2 Y ( ) n n E (Yi - mY )2 E (Y - mY )2 n -1 n -1 2 ææ n öö çç 1 n n ÷÷ çç ÷÷ ÷ = Var (Yi ) E ç å (Yi - mY )÷ ÷ çç ÷÷ ÷ n -1 n - 1 çè n i =1 øø è = VER. 9/11/2012. © P. KOLM ( ) ( ) 39 Since Y = (Y1,Y2,...,Yn ) are iid, we obtain 2 ææ n öö çç 1 n n 2 ÷÷ çç ÷÷ E (sY ) = Var (Yi ) E ç å (Yi - mY )÷ ÷ çç n i =1 ÷÷ n -1 n - 1 çè ø÷ ÷ è ø æ1 n ö n n 2÷ ç = Var (Yi ) E ç 2 å (Yi - mY ) ÷ ÷ ÷ n -1 n - 1 ç n i =1 è ø n n n Var (Yi ) = ⋅ 2 E (Yi - mY )2 n -1 n -1 n n n n Var (Yi ) = ⋅ 2 E (Yi - mY )2 n -1 n -1 n = Var (Yi ) ( ) ( ) which shows the unbiasedness of the estimator. 1n ˆ Note: The estimator s = å (Yi -Y )2 is biased downward. In particular, n i =1 n -1 2 n -1 1 2 ˆ2 ˆ2 s . However, Var (sY ) = 2 2 s 4 < Var (sY ) = 2 E (sY ) = s 4 (For you: n n -1 n Show this.) This means that the biased estimator has a smaller variance. The 2 Y difference is negligible in large samples, but, for example is about 10% in a sample of 16. VER. 9/11/2012...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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