Matrix Concepts
Matrix X = rectangular array of numbers (n p)
Vector x = single column = n 1 matrix
Length, direction, angle
Linear dependence of vectors x1, x2, . . . , xk : c1x1 + c2x2 + . . . ck xk = 0 for some c1, c2, . . . , ck not all zero.
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T
Random Vectors and Matrices
Elements of a random matrix/vector X are jointly distributed random variables.
Mean E(X) of a random matrix/vector X is the matrix/vector of elementwise means: E(X)i,j = E Xi,j Covariance matrix of a random vector X is the ma
Confidence Statements About the Mean
Suppose as before that X1, X2, . . . , Xn is a random sample from Np(, ).
A confidence ellipsoid for is the set of satisfying n X - S-1 X - p(n - 1) Fp,n-p(). (n - p)
That is, the probability that this (random) set
Large Sample Inference The T 2 and Bonferroni procedures are valid for all sample sizes n and dimensions p, provided the data are sampled from a multivariate normal population.
If n is large enough and p not too large, the t and F distributions used for
Inference About the Mean
Suppose that X1, X2, . . . , Xn is a random sample from Np(, ).
Point estimates for (primary interest): X; for (secondary interest): S; both unbiased.
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Sampling Distributions
1 X Np , n .
(n - 1)S Wp(, n - 1), the Wishart dist