Two simple approximations to the distributions of quadratic forms

Ke-Hai Yuan, Peter M Bentler
British Journal of Mathematical and Statistical Psychology 2010, 63 (Pt 2): 273-91
Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to T = sigma(d)(i=1) lambda(i)z(i)(2) with z(i) being independent and following N(0,1). Two approximations to the distribution of T have been implemented in popular software and are widely used in evaluating various models. It is important to know how accurate these approximations are when compared to each other and to the exact distribution of T. The paper systematically studies the quality of the two approximations and examines the effect of the lambda(i) and the degrees of freedom d by analysis and Monte Carlo. The results imply that the adjusted distribution for T can be as good as knowing its exact distribution. When the coefficient of variation of the lambda(i) is small, the rescaled statistic T(R) = dT/(sigma(d)(i=1) lambda(i)) is also adequate for practical model inference. But comparing T(R) against chi2(d) will inflate type I errors when substantial differences exist among the lambda(i), especially, when d is also large.

Full Text Links

Find Full Text Links for this Article


You are not logged in. Sign Up or Log In to join the discussion.

Related Papers

Remove bar
Read by QxMD icon Read

Save your favorite articles in one place with a free QxMD account.


Search Tips

Use Boolean operators: AND/OR

diabetic AND foot
diabetes OR diabetic

Exclude a word using the 'minus' sign

Virchow -triad

Use Parentheses

water AND (cup OR glass)

Add an asterisk (*) at end of a word to include word stems

Neuro* will search for Neurology, Neuroscientist, Neurological, and so on

Use quotes to search for an exact phrase

"primary prevention of cancer"
(heart or cardiac or cardio*) AND arrest -"American Heart Association"