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08InferenceUnder(1)

08InferenceUnder(1) - Inference Under the Normal Theory...

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Inference Under the Normal Theory Gauss-Markov Linear Model Remember our dairy cow study (2 treatments, 3 reps per trt) and our questions (Introduction, slide 3) We’ve answered all questions except the last one: 3) When does t = [(¯ y 1 ¯ y 2 ) ( μ 1 μ 2 )] / s 2 2 / n follow a T distribution? This underlies all statistical inference about ¯ y 1 ¯ y 2 Math Stat tells us the random variable T defined as T = Z V has the distribution known as the T distribution when: Z has a standard normal distribution, Z N ( 0 , 1 ) V has a Chi-square distribution with ν d.f., V χ 2 ν and Z and V are independent Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 42 Inference (cont.) To relate our t-statistic in Q3 to the Math Stat requires answering four additional questions 1) What is the distribution of ¯ y 1 ¯ y 2 or C ˆ β ? 2) What is the distribution of s 2 ? 3) If [(¯ y 1 ¯ y 2 ) ( μ 1 μ 2 )] / s 2 2 / n follows a t distribution, which t distribution? I.e., how many d.f.? 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 ( 0 , σ 2 I ) and C β is estimable: 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 ) . s 2 has a χ 2 distribution k = tr ( I P x ) is the d.f. associated with the distribution of s 2 C ˆ β and s 2 are independent Hence, y 1 ¯ y 2 ) / s 2 2 / n follows a t distribution with k d.f. 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 Suppose Z 1 , . . . , Z n i . i . d . N ( 0 , 1 ) and Z = [ Z 1 , . . . , Z n ] . The vector Z has the standard multivariate dn: Z N ( 0 , 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 As a matrix operation, W = μ + A Z . μ is a k × 1 vector of constants; A is a k × n matrix of constants. W = μ + A Z has a multivariate normal distribution with mean μ and variance Σ = A A . Our notation for the multivariate normal d’n will be W N ( μ , Σ) . 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
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What if we apply a linear transformation, a i + b i W i , i.e. a + B W ? 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 ) . Why? a + B W = a + B ( μ + A Z ) = a + B μ + B A Z a + B W is a vector of constants plus a matrix of constants times Z . So a + B W is multivariate normal. E ( a + B W ) = E ( a + B μ + B A Z ) = a + B μ + B A E ( Z ) = a + B μ .
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