APPLIED MULTIVARIATE ANALYSIS

Transcription

1 ALIED MULTIVARIATE ANALYSIS FREQUENTLY ASKED QUESTIONS AMIT MITRA & SHARMISHTHA MITRA DEARTMENT OF MATHEMATICS & STATISTICS INDIAN INSTITUTE OF TECHNOLOGY KANUR

2 X = X X X [] The variace covariace atrix of a 3-diesioal rado vector ( ) is give by 5 4 Σ= (a) Fid the correlatio atrix X 3 (b) Fid the correlatio betwee X ad X + [] Suppose X = ( X ) X is a p-diesioal rado vector with ( ) Cov( X ) = Σ = where c j R 3 E X = µ ad Z cx ck X ; Fid the covariace atrix of the rado vector ( ) are vectors of costats [3] Show that S = s spp R where S is the saple variace covariace atrix ad R is the saple correlatio atrix [4] Suppose the rado vector X is such that E( X ) = µ ad Cov( X ) = Σ E( XX ) Let Y be aother rado vector with E( Y) = δ ad ( ) Derive E( YX ) Fid Cov X Y = Σ XY [5] Suppose the observed data atrix for a 3-diesioal rado vector is give by - 5 X= (a) For the observatios o variable X fid the projectio o (b) Fid the deviatio vectors ad lik the with the saple stadard deviatios (c) Calculate the agle betwee the deviatio vectors d ad d (d) Usig the deviatio vectors d d ad d3 fid X x ad verify whether it is of full rak (e) Fid the geeralized saple variace ad the total saple variace

3 X = X X X3 X4 is give by µ = 0 0 Σ= Let X( ) ( ) = X X3 ad X ( ) ( ) = X X4 be subvectors; A = ( ) ad B = (a) Fid Cov ( A X ( ) ) Cov ( B X ) ( ) ad Cov( A X ) ( ) B X ( ) (b) Fid the joit distributio of AX ( ) ad BX ( ) if X ~ N 4 ( µ Σ) (c) With X ~ N 4 ( µ Σ) fid the argial distributios of X ( ) ad X ( ) ad the coditioal distributio of X ( ) give X ( ) [6] Suppose the ea vector ad covariace atrix of ( ) [7] Suppose the covariace atrix of a 3-diesioal rado vector X is give by σ ρσ 0 Σ= ρσ σ ρσ ; ρ < 0 ρσ σ Suppose the uderlyig rado vector is N 3 ( 0 Σ ) fid the joit distributio ad the argial distributios of the pricipal copoets [8] Deterie the populatio pricipal copoets Y ad Y for the covariace atrix 5 Σ= Further fid ρ Y X ad ρ Y X 5 [9] Let X X N 0 Σ Σ> 0 Defie the p data atrix X as X = ( X X ) (a) Fid the distributio of U I U where ( ) U = U U with Ui = axi p i = () a R a 0 (b) Fid the distributio of Xb b R such that bb= (c) Fid the distributio of b X Σ Xb be a rado saple fro ( )

4 [0] Let Y0 Y Y p be idepedet ad idetically distributed rado variables with ea 0 ad variace σ Defie X = Y0 + Y; i = () p (a) Show that there is a pricipal copoet of ( ) i X = X X p that is proportioal to X = X p (b) Show that the above pricipal copoet is i fact the first pricipal copoet [] Let X X be a rado saple fro a p-diesioal ultivariate populatio with ea vector µ ad covariace atrix Σ Let X = ( X X ) be the p data atrix rove or disprove S = XI X where S is the saple variace covariace atrix with divisor [] Let X ~ N ( 0 Σ) i where Σ is a sigular atrix of rak r < p ad a o sigular p p p atrix H I 0 HΣ H = r 0 0 If B is a g-iverse of Σ fid the distributio of [3] Let X X X T BB = I k (a) Fid the distributio of (i) be iid N( µσ I) T BXj (ii) ( X j µ ) ( X j µ ) (b) Let Y= BX idepedet? Are Z ad Y idepedet? X BX ad B is k atrix of costats with ad (iii) ( X j µ )( X j µ ) T T fid the distributio of Z = ( X X Y Y) T Are Z ad X [4] Let Xi i = be idepedetly distributed as N ( µ Σ) Fid the distributio of ai Xi ; where a a are real i=

5 X = X X X [5] Let ( ) 3 be distributed as N 3 ( µ Σ) ρ ρ Σ= ρ ρ ; < ρ < ρ ρ [6] Fid the joit probability desity fuctio of ( X X X X ) [7] Let ( ) + 3 X ~ N µ Σ with µ ad Σ= Fid the distributio of 3 Y= X + X XX [8] Suppose Y ~ N( X µ I) where X is a p atrix of costats ad µ is a p vector of costats Fid the distributio of ( ) Y I X XX XY µ = ad I B = Verify whether AX ad BX are idepedet [9] Suppose X ~ N ( µ Σ) ( ) with ( ) Σ= Cosider ( ) A = ad [0] Let X X X be a rado saple fro a populatio which is N ( µ Σ) (a) Derive the sufficiet statistic for µ whe Σ=Σ 0 is kow (b) Derive the sufficiet statistic for Σ whe µ = µ 0 (c) Check whether the derived sufficiet statistic are ubiased estiators for the correspodig ukow paraeters [] Suppose that the distributio of the Σ> 0 Let A & Σ be partitioed as A A = A A A Σ & Σ = Σ rado atrix A is Wishart ( ) A & Σ are k x k A & Σ are k x -k A & Σ are -k x k ad A Σ are -k x - k Fid the distributios of & A & A Σ Σ W Σ

6 [] Let X X X be a rado saple fro a populatio which is N ( µ Σ) Defie the data atrix as rove that S = ( X X ) ( X X ) X X = X [3] Suppose that the distributio of the rado atrix A is Wishart ( ) Σ> 0 Let Φ be a syetric atrix of full rak rove that ( j ) i E exp tr A i λ Φ = where λ λ are the eige values of Σ ΦΣ W Σ [4] Let A be a Wishart ( ) fid the characteristic fuctio of W Σ For a k o rado atrix of full row rak M M AM [5] Let X X X be a rado saple fro a populatio which is N ( µ Σ) Derive the distributio of X SX X Σ X Σ Fid a ubiased estiator of Σ [7] Let X X X be a rado saple fro N ( µ Σ) Σ> 0 Defie a trasforatio X Y= AX+ β A is a o sigular atrix of costats [6] Let A be a Wishart W ( ) ad β is a vector of costats Show that Hotellig s T statistic for testig H 0: µ = µ 0 ( ) agaist H A : µ µ 0 Y = Y Y Y are the sae based o X ( X X X ) = ad that based o

7 () () [8] Suppose X X () () () N µ Σ ad X X be a rado saple fro N ( µ ) () Σ Σ> 0 () () (a)uder the coditio that µ = µ fid the distributio of ( X X) Sp ( X X) Where X ad X deote the saple ea vectors ad S is the pooled saple covariace atrix p be a rado saple fro ( ) (b) Derive the appropriate test statistic based o Hotellig s () () () () H 0 : µ = µ agaist H A : µ µ 00 α % cofidece regios for (c) Obtai ( ) () µ () µ ad µ µ T statistic for testig () () [9] Let X X X N µ Σ Σ> 0 Derive the testig procedure for testig H0 : µ i= µ i ; i () + = agaist H A : at least oe such relatio does ot hold be a rado saple fro a populatio which is ( ) [30] Let X X X N µ Σ Σ= diag σ σ Obtai a siultaeous cofidece iterval for µ µ ad µ + µ such that the 00 α % joit cofidece is exactly ( ) be a rado saple fro ( ) ( ) [3] Let X X X N µ Σ Σ> 0 Usig Boferroi s approach costruct siultaeous cofidece itervals of cofidece level at least 90% for µ µ ad µ µ uder the followig scearios; (a) the two cotrasts are give equal iportace ad (b) iportace of the cotrast µ µ is three ties that of the cotrast µ µ be a rado saple fro ( )