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Unformatted text preview: Inference Under the Normal Theory GaussMarkov Linear Model I Remember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) I We’ve answered all questions except the last one: 3) When does t = [(¯ y 1 ¯ y 2 ) ( μ 1 μ 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 /ν has the distribution known as the T distribution when: I Z has a standard normal distribution, Z ∼ N ( , 1 ) I V has a Chisquare distribution with ν d.f., V ∼ χ 2 ν 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 tstatistic in Q3 to the Math Stat requires answering four additional questions I 1) What is the distribution of ¯ y 1 ¯ y 2 or C ˆ β ? I 2) What is the distribution of s 2 ? I 3) If [(¯ y 1 ¯ y 2 ) ( μ 1 μ 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 β + , where ∼ N ( ,σ 2 I ) and C β is estimable: I the distribution of C ˆ β , the OLS estimator of C β is C ˆ β ∼ N ( C β , σ 2 C ( X X ) C ) . I s 2 has a χ 2 distribution I k = tr ( I P x ) is the d.f. associated with the distribution of s 2 I C ˆ β 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 μ i , i.e. W i = μ i + a i Z i I As a matrix operation, W = μ + A Z . μ is a k × 1 vector of constants; A is a k × n matrix of constants. I W = μ + A Z has a multivariate normal distribution with mean μ and variance Σ = A A . I Our notation for the multivariate normal d’n will be W ∼ N ( μ , Σ) . I When Σ is nonsingular, the density of W is 1 ( 2 π ) K / 2  Σ  1 / 2 exp 1 / 2 ( W μ ) Σ 1 ( W μ ) 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 μ , B Σ B ) ....
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 Spring '08
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 Normal Distribution, Dept. of Statistics

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