TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 32

CHAPTER 63 Independent Observations from the Same Multivariate Population. This Chapter discusses a model that is a special case of the model in Chapter , but it goes into more depth towards the end. | CHAPTER 63 Independent Observations from the Same Multivariate Population This Chapter discusses a model that is a special case of the model in Chapter but it goes into more depth towards the end. . Notation and Basic Statistics Notational conventions are not uniform among the different books about multivariate statistic. Johnson and Wichern arrange the data in a r x n matrix X. Each column is a separate independent observation of a q vector with mean p. and dispersion matrix S. There are n observations. 1333 1334 63. INDEPENDENT OBSERVATIONS FROM SAME POPULATION We will choose an alternative notation which is also found in the literature and write the matrix as a n x r matrix Y . As before each column represents a variable and each row a usually independent observation. Decompose Y into its row vectors as follows y Ly-J Each row written as a column vector y4 has mean p. and dispersion matrix S and different rows are independent of each other. In other words E Y . V Y is an array of rank 4 not a matrix. In terms of Kronecker products one can write V vec Y S 0 I. One can form the following descriptive statistics y - yj is the vector of sample means W 2i yj y y4 y T is matrix of corrected squares and cross products the sample covariance matrix is S n - W with divisor n and R is the matrix of sample correlation coefficients. Notation the ith sample variance is called s not s2 as one might perhaps expect . The sample means indicate location the sample standard deviations dispersion and the sample correlation coefficients linear relationship. . NOTATION AND BASIC STATISTICS 1335 How do we get these descriptive statistics from the data Y through a matrix manipulation yT 1 iT Y now Y isf I Y is the matrix of observations with the appropriate sample mean taken out of each element therefore W yi - y yn y yi - y T . . . _ y - y T. Y T I - T I - Y Y T I - Y. nn n Then S n n W and in order to get the sample correlation matrix R use S11 0 0 22 0 0

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