TY - JOUR
T1 - Approximating the Distributions of Singular Quadratic Expressions and their Ratios
TT - تقریب زدنهای توزیعهای عبارتهای درجه دوم تکین و نسبتهای آنها
JF - JIRSS
JO - JIRSS
VL - 11
IS - 2
UR - http://jirss.irstat.ir/article-1-190-en.html
Y1 - 2012
SP - 147
EP - 171
KW - Burg’s estimator
KW - density approximation
KW - Durbin-Watson statistic
KW - indefinite quadratic expressions
KW - quadratic forms
KW - simulations
KW - singular Gaussian vectors
N2 - Noncentral indefinite quadratic expressions in possibly non- singular normal vectors are represented in terms of the difference of two positive definite quadratic forms and an independently distributed linear combination of standard normal random variables. This result also ap- plies to quadratic forms in singular normal vectors for which no general representation is currently available. The distribution of the positive definite quadratic forms involved in the representations is approximated by means of gamma-type distributions. We are also considering general ratios of quadratic forms, as well as ratios whose denominator involves an idempotent matrix and ratios for which the quadratic form in the denominator is positive definite. Additionally, an approximation to the density of ratios of quadratic expressions in singular normal vectors is being proposed. The results are applied to the Durbin-Watson statistic and Burg’s estimator, both of which are expressible as ratios of quadratic forms.
M3
ER -