
Problem 1. Show that the sample variance covariance matrix S = { Sij}, i=1,..., p ,...
Exercise 2. Given a permutation o E S. define a matrix P, E M. (F) by setting P.(1,j) = P.(.) = 806) 1 if i=0G) ifi 00) for all 1 Sij Sn. For example, ifo is the identity permutation, then P, (1) Show that det(P.) gn(a) for all o ES.. Deduce that the matrix P, is invertible, for all o ES (ii) Show that P.P. - Por for all 9,TES. Deduce that the matrix P, is orthogonal, for all o...
linear statistics modeling and regression
2) Suppose you have multiple regression set up Ynxi XnxpBpxi Sxl and f ~ N(0nx1, σ21.). P Po X(X,X)-X, be the projection matrix on the column space of X. a) Show residual vector, e = (1,-P)Y. Here e is the vector of residuals ei S. b) Show that the variance of e, is 1 - Pi, where P is the i, j th entry of the matrix P c) Show that the sample covariance of...
4 Variance-Covariance Matrix of the OLS Estima- tor The fourth classical assumption is about the covariance matrix of the random error term: V(u X) = E uu x)= oʻI, How'we derived it?(uu^T) This assumption implies that the error terms are incorrelated and homoscedastic. Let us take a closer look at the covariance matrix. Note that all expectations are conditional on X: E(uu) = E 1 [u U2 ... Un u Juu = ET uu ... un u ... Uzun Why...
Let X=(X1,…,Xn)′ be the n×p data matrix, where Xi=(Xi1,…,Xip)′ is the ith observation. Let X¯=n−1∑ni=1Xi be the sample mean. Let sj1j2=1/n∑ni=1(Xij1−X¯j1)(Xij2−X¯j2) be the sample covariance between the j1th and j2th variables. Let S=(sj1j2) be the sample covariance matrix. Show that S=1nX′X−X¯′X¯.
1: 1 131 2 Given matrix A 2 2 2. matrix P and I S set 2. a) Show that matrix P diaqonalizes A and find D(diagonal matnx) that matches. 6) Find the eigen values of A Observe that the columns of P form set S c) orthogonal Set using the inner product standard show that set S is not an Use the Gram- Schmidt process to get an orthonormal set from S using inner product standard
1: 1 131...
x={x1,x2,x3} has the 3-variate normal dustribution with mean 0 and variance covariance matrix=(3 1 1 1 3 1 1 1 4) find PDF of x in full
and show work
betP^ I Find the stationary matrix S and limiting matrix P for: 6. A [.6 4
betP^ I Find the stationary matrix S and limiting matrix P for: 6. A [.6 4
Q3. Assume that X- (X1, X2) is multivariate normal with mean zero and the variance-covariance matrix Let λ-(A1, λ2), Ai, A2 Value-at-Risk for Y 0, Ai + λ2-1. Let Y-Aix, + λ2Xy. Find the weights λι, λ2 that minimize
Problem 2: Given a collection of data { zNJS R" we define 1. The sample mean of the points is given by 2. The sample variance of the points is given by N 2 3. The covariance matrix of the points is given by Suppose that (N) S R is a collection of data points. Using Lagrange Multipliers, show that the unit vector w for which the set (i.N), where wy, has maximum variance is the normalized eigenvector of Cov(ia)...
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Question 4 Consider a p-variate sample with size T1.,n For some CE RxP and a E R9, consider the linear transformation Furthermore, for some D E R* and bER, consider the linear transformation For any j -1.....q and k-1.....r, denote by sy,z the sample covariance between (yjl-1 and (-k1 Define the matrix Y,Z Show that Y,Z where Sx is the sample covariance matrix of ...fn
Question 4 Consider a p-variate sample...