Tham khảo tài liệu 'a course in mathematical statistics phần 10', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Some Theorems About Matrices and Quadratic Forms 507 and rank A1 rank A 2 rank i - A1 - A 2 n. iv If A j 1 . m are symmetric idempotent matrices of the same order and m 1Aj is also idempotent the AA 0 for 1 i j m. The proof of the theorems formulated in this appendix may be found in most books of linear algebra. For example see Birkhoff and MacLane A Survey of Modern Algebra 3d ed. MacMillan 1965 S. Lang Linear Algebra Addison-Wesley 1968 D. C. Murdoch Linear Algebra for Undergraduates Wiley 1957 S. Perlis Theory of Matrices Addison-Wesley 1952. For a brief exposition of most results from linear algebra employed in statistics see also C. R. Rao Linear Statistical Inference and Its Applications Chapter 1 Wiley 1965 H. Scheffé The Analysis of Variance Appendices I and II Wiley 1959 and F. A. Graybill An Introduction to Linear Statistical Models Vol. I Chapter 1 McGraw-Hill 1961. Appendix II Noncentral t X and F Distributions Noncentral t-Distribution It was seen in Chapter 9 Application 2 that if the independent . s X and Y were distributed as N Q 1 and x2 respectively then the distribution of the . T x y r was the Student s t-distribution with r . Now let X and Y be independent . s distributed as N 8 1 and x2 respectively and set T x ẬY r . The . T is said to have the noncentral t-distribution with r . and noncentrality parameter 8. This distribution as well as an . having this distribution if often denoted by t r8. Using the definition of a t r8 . it can be found by well known methods that its . is given by 1 r JQ 2 1 1 2 A2 X exp X exp- x t. dx t efi. 2 x 8 8 7 7 Noncentral -Distribution It was seen in Chapter 7 see corollary to Theorem 5 that if X1 . Xr were independent normally distributed . s with variance 1 and mean Q then the . X ĩj 1Xj was distributed as X Let now the . s X1 . Xr be independent normally distributed with variance 1 but means ft . pr respectively. Then the distribution of the . X EJ 1X2 is
