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A Course in Mathematical Statistics phần 10

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 | I.4 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 II.1 Noncentral t-Distribution It was seen in Chapter 9 Application 2 that if the independent r.v. s X and Y were distributed as N Q 1 and x2 respectively then the distribution of the r.v. T x y r was the Student s t-distribution with r d.f. Now let X and Y be independent r.v. s distributed as N 8 1 and x2 respectively and set T x ẬY r . The r.v. T is said to have the noncentral t-distribution with r d.f. and noncentrality parameter 8. This distribution as well as an r.v. having this distribution if often denoted by t r8. Using the definition of a t r8 r.v. it can be found by well known methods that its p.d.f. 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 II.2 Noncentral -Distribution It was seen in Chapter 7 see corollary to Theorem 5 that if X1 . Xr were independent normally distributed r.v. s with variance 1 and mean Q then the r.v. X ĩj 1Xj was distributed as X Let now the r.v. s X1 . Xr be independent normally distributed with variance 1 but means ft . pr respectively. Then the distribution of the r.v. X EJ 1X2 is