08InferenceUnder

08InferenceUnder - Inference Under the Normal Theory...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Inference Under the Normal Theory Gauss-Markov Linear Model I Remember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) I Weve answered all questions except the last one: 3) When does t = [( y 1- y 2 )- ( 1- 2 )] / p s 2 * 2 / n follow a T distribution? I This underlies all statistical inference about y 1- y 2 I Math Stat tells us the random variable T defined as T = Z p V / has the distribution known as the T distribution when: I Z has a standard normal distribution, Z N ( , 1 ) I V has a Chi-square distribution with d.f., V 2 I and Z and V are independent Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 42 Inference (cont.) I To relate our t-statistic in Q3 to the Math Stat requires answering four additional questions I 1) What is the distribution of y 1- y 2 or C ? I 2) What is the distribution of s 2 ? I 3) If [( y 1- y 2 )- ( 1- 2 )] / p s 2 * 2 / n follows a t distribution, which t distribution? I.e., how many d.f.? I 4) Are y 1- y 2 and s 2 independent? Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 42 Inference (cont.) The next series of slides will show that: Under the normal GM model, y = X + , where N ( , 2 I ) and C is estimable: I the distribution of C , the OLS estimator of C is C N ( C , 2 C ( X X )- C ) . I s 2 has a 2 distribution I k = tr ( I- P x ) is the d.f. associated with the distribution of s 2 I C and s 2 are independent I Hence, ( y 1- y 2 ) / p s 2 * 2 / n follows a t distribution with k d.f. I Deriving these answers requires an extended discussion on the multivariate normal distribution Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 42 Basic Facts about Multivariate Normal Distributions I Suppose Z 1 ,..., Z n i . i . d . N ( , 1 ) and Z = [ Z 1 ,..., Z n ] . I The vector Z has the standard multivariate dn: Z N ( , I ) . I What if we shift and scale Z , i.e. multiply Z i by a i and add i , i.e. W i = i + a i Z i I As a matrix operation, W = + A Z . is a k 1 vector of constants; A is a k n matrix of constants. I W = + A Z has a multivariate normal distribution with mean and variance = A A . I Our notation for the multivariate normal dn will be W N ( , ) . I When is nonsingular, the density of W is 1 ( 2 ) K / 2 | | 1 / 2 exp- 1 / 2 ( W- ) - 1 ( W- ) Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4 / 42 What if we apply a linear transformation, a i + b i W i , i.e. a + B W ? I If a is an 1 vector of constants and B is a k matrix of constants, then a + B W N ( a + B , B B ) ....
View Full Document

Page1 / 42

08InferenceUnder - Inference Under the Normal Theory...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online