Matrix polynomial generalizations of the sample variance-covariance matrix when pn−1 → y ∈ (0, ∞)

2017 ◽  
Vol 48 (4) ◽  
pp. 575-607
Author(s):  
Monika Bhattacharjee ◽  
Arup Bose
Keyword(s):  
Covariance Matrix ◽  
Matrix Polynomial ◽  
Sample Variance ◽  
Variance Covariance Matrix

Polynomial generalizations of the sample variance-covariance matrix when pn−1 → 0

2016 ◽  
Vol 05 (04) ◽  
pp. 1650014 ◽  
Author(s):  
Monika Bhattacharjee ◽  
Arup Bose
Keyword(s):  
Covariance Matrix ◽  
Random Matrices ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Probability Space ◽  
Sample Variance ◽  
Limiting Spectral Distribution ◽  
Matrix Polynomials ◽  
State Formula ◽  
Variance Covariance Matrix

Let [Formula: see text] be random matrices, where [Formula: see text] are independently distributed. Suppose [Formula: see text], [Formula: see text] are non-random matrices of order [Formula: see text] and [Formula: see text] respectively. Suppose [Formula: see text], [Formula: see text] and [Formula: see text]. Consider all [Formula: see text] random matrix polynomials constructed from the above matrices of the form [Formula: see text] [Formula: see text] and the corresponding centering polynomials [Formula: see text] [Formula: see text]. We show that under appropriate conditions on the above matrices, the variables in the non-commutative ∗-probability space [Formula: see text] with state [Formula: see text] converge. We also show that the limiting spectral distribution of [Formula: see text] exists almost surely whenever [Formula: see text] and [Formula: see text] are self-adjoint. The limit can be expressed in terms of, semi-circular, circular and other families and, limits of [Formula: see text], [Formula: see text] and non-commutative limit of [Formula: see text]. Our results fully generalize the results already known for [Formula: see text].

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed
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