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STA 35: Multivariate Analaysis
创建日期：2024-04-12 17:04:19
所属分类：统计学statistics
标签： STA 35
Multivariate Analaysis

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STA 35: Multivariate Analaysis 

Practice Problems Instructor: Dr. Maxime Pouokam Disclaimer: 

These are just Practice Problems. This is NOT meant to look just like the test, 

and it is NOT the only thing that you should study. Make sure you know all the material from the notes, 

suggested homework, and the corresponding chapters in the book. You may use the following results 

without any proof. Given X ∼ Np(µ,Σ) and (p× q) matrix A with q < p, ATX ∼ Nq(ATµ,ATΣA). 

The density of X is f(X) = 1 (2pi)p/2|Σ| 12 e− 1 2 (X−µ)TΣ−1(X−µ) 1. Let X1 and X2 be

 measurements of X1= tail length and X2= wing length (both in mm) from the Hook-billed kites. Suppose 

that it is known X = ( X1 X2 ) ∼ N [( 195 280 ) , ( 120 125 125 210 )] What is the distribution of wing lengths

 for those birds that have tail lengths of 190 mm ? 2. Suppose that X ∼ Np(µX ,ΣX) and Y ∼ Np(µY ,ΣY ) 

and X and Y are independent. What is the distribution of X − CY where C is a p× p matrix of constants. 

3. Suppose X ∼ Np(µ,Σ) is partitioned as X = ( X1 X2 ) where X1 is (q × 1) with q < p and the corre- sponding partitions of µ and Σ are µ = ( µ1 µ2 ) and Σ = ( Σ11 Σ12 Σ21 Σ22 ) where Σ11 and Σ22 are positive semi-definite symmetric matrices. Show that the marginal distribution of X1 is Nq(µ1,Σ11). 4. Suppose we have data X1, X2, . . . , Xn ∼ Np(µx,Σ) with Σ unknown. (a) State the form of the test statistic for testing H0 : µx = µ0 vs. H1 : µx 6= µ0. (b) What is the null distribution for this test statistic ? 5. Suppose Y = AX + b where A is a (p× p) non-singular matrix and b is a (p× 1)-vector. (a) Show that your test statistic in the previous problem for testing H0 : µx = µ0 is identical to that used in testing H0 : µy = Aµ0 + b. (b) What does this tell you about the properties of your test statistic with regards to non-singular linear transformations of the variables? 6. Given iid observations Z1, Z2, . . . , Zn ∼ N(0, 1) state how these can be used to obtain a single obser- vation X ∼ N(µ,Σ) for a given (p×1) vector µ and for positive definite non-singular (p×p) matrix Σ. Hence, or ortherwise, derive the density function of the p-dimensional multivariate normal distribution with mean µ and covariance matrix Σ 2-1 2-2 Lecture 2: Practice Problems 7. Two samples were taken from an experiment where two characteristics X1 and X2 were measured. Assume the samples 1 and 2 are taken from N2(µ1,Σ) and N2(µ2,Σ), respectively. The summary statistics of the two samples for the observations are x¯2 = 13 2 , n1 = 16 and x¯2 = 24 2 , n2 = 16 respectively. Let the pooled sample covariance matrix be Spooled = 13 −4 2−4 13 −2 2 −2 10 . (a) Test the hypothesis H0 : µ1 = µ2 V S. H1 : µ1 6= µ2 at 5% significant level. (b) Find the 95% Bonferroni simultaneous confidence interval for the individual mean difference. 8. Consider the regression model: Yi = β0 + β1Zi1 + β2Zi2 + β3Zi3 + β4Zi4 + i where i are i.i.d with mean 0 and variance σ. Let recall the least square estimate of ~β =  β0 β1 β2 β3 β4  is given as ~ˆβ = (ZTZ)−1ZT ~Y where Z is defined as in class. (a) Derive the distribution of ~ˆβ. Make sure you clearly elaborate all the steps. (b) Assume we want to test the hypothesis: H0 : β1 = β3 = β4 = 0. Please derive the test statistic and precisely identify its distribution under H0. Test at α = 0.05. 9. Suppose that Y is random with mean µ and variance Σ. Show that E(Y TAY ) = µTAµ+ tr(AΣ). Hint: Y TAY = (µ+ (Y − µ))TA(µ+ (Y − µ) 10. Suppose X comes from one the two populations: pi1: Normal with mean µ1 and covariance matrix Σ1 pi2: Normal with mean µ2 and covariance matrix Σ2 If the respective density functions are denoted by f1(X) and f2(X), find the expression for the quadratic discriminator Q = ln( f1(X) f2(X) If Σ1 = Σ2 = Σ, for instance, verify that Q becomes (µ1 − µ2)TΣ−1X − 1 2 (µ1 − µ2)TΣ−1(µ1 + µ2) 11. Suppose the population covariance matrix of X = ( X1 X2 ) is Σ = ( 3 1 1 2 ) and the mean is µ = (−1 2 ) Assume a bivariate normal distribution, please derive the distribution of X1 −X2.

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