Notes for SI 422_up to March 23.pdf - Indian Institute of Technology Bombay SI 422 Regression Analysis January 10 2 Notation 1 A0 denotes transpose of

Notes for SI 422_up to March 23.pdf - Indian Institute of...

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Indian Institute of Technology Bombay SI 422: Regression Analysis January 10 - March 23, 2020
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2 Notation 1. A 0 denotes transpose of the matrix A . 2. 1 n is a n × 1 vector with all entries equal to 1. 3. 0 n is a n × 1 vector with all entries equal to 0. 4. O n × m is a n × m matrix with all entries equal to 0. 5. I n is the n × n identity matrix whose diagonal entries are 1 and off-diagonal entries are 0. 6. A diagonal matrix of order p with diagonal entries a 1 , a 2 , . . . , a p is denoted by Diag { a 1 , a 2 , . . . , a p } . 7. Ordered eigenvalues of p × p real symmetric matrix A is denoted by { λ 1 ( A ) , λ 2 ( A ) , . . . λ p ( A ) } . Therefore, λ 1 ( A ) λ 2 ( A ) ≥ · · · ≥ λ p ( A ) and λ j ( A ) denotes the j -th largest eigenvalue of A . 8. For p × p real symmetric matrix A , define Λ A := Diag( λ 1 ( A ) , λ 2 ( A ) , . . . λ p ( A )). 9. v j ( A ) denotes the eigenvector of A corresponding to the eigenvalue λ j ( A ). 10. For p × p real symmetric matrix A , define V A := [ v 1 ( A ) v 2 ( A ) · · · v p ( A )] = (( V ij, A )) 1 i,j p . Note that V A is also a p × p matrix. 11. For p × p real symmetric matrix A , define V A ,k := [ v 1 ( A ) v 2 ( A ) · · · v k ( A )] 1 k p . Note that V A is a p × k matrix. 12. Det( A ) denotes determinant of the square matrix A . 13. Z is always a N (0 , 1) random variable. 14. τ α is the upper α -point or 100(1 - α )-th percentile of the standard normal distribution. 15. χ 2 ν is a χ 2 -distributed random variable or χ 2 -distribution with degrees of freedom (DF) ν . 16. χ 2 0 ν,λ is a non-central χ 2 -distribution with DF ν and non-centrality parameter λ . 17. ( χ 2 ν ) α is the upper α -point or 100(1 - α )-th percentile of the χ 2 ν distribution. 18. t ν is a t-distributed random variable or t-distribution with degrees of freedom (DF) ν . 19. t 0 ν,δ is a non-central t-distribution with DF ν and non-centrality parameter δ . 20. ( t ν ) α is the upper α -point or 100(1 - α )-th percentile of the t ν distribution. 21. F ν,η is a F-distributed random variable or F-distribution with DF ν, η . 22. F 0 ν,η,λ is a non-central F-distribution with DF ν, η and non-centrality parameter λ . 23. ( F ν,η ) α is the upper α -point or 100(1 - α )-th percentile of the F ν,η distribution. 24. Let X be a random variable and f ( · ) be a measurable function. Then f ( X ) observed is the observed value of f ( X ). 25. For a scaler a , a 0 indicates that a is close to zero. 26. For a set A , | A | denotes cardinality of A . 27. For two sets A and B , A × B = { ( a, b ) : a A, b B } .
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Chapter 0 Prerequisites Some basic results in probability: (BP1) If X ∼ N (0 , 1), then X 2 χ 2 1 . (BP2) If X i ∼ N (0 , 1) independently across i = 1 , 2 , . . . , n . Then n i =1 X 2 i χ 2 n . (BP3) E ( χ 2 ν ) = ν . (BP4) If X ∼ N ( δ, 1), then X 2 χ 2 0 1 2 . (BP5) If X i ∼ N ( δ i , 1) independently across i = 1 , 2 , . . . , n . Then n X i =1 X 2 i χ 2 0 n,λ where λ = n X i =1 δ 2 i . (BP6) E ( χ 2 0 ν,λ ) = ν + λ . (BP7) Suppose X ∼ N (0 , 1) and V χ 2 ν are two independent random variables. Then X p V/ν t ν . (BP8) E ( t ν ) = 0. (BP9) Suppose X ∼ N ( δ, 1) and V χ 2 ν are two independent random variables. Then X p V/ν t 0 ν,δ . (BP10) E ( t 0 ν,δ ) = δ p ν 2 Γ ( ν - 1 2 ) Γ ( ν 2 ) if ν > 1.
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