Shared Concepts and Topics

Joint Hypothesis Tests with Complex Alternatives, the Chi-Bar-Square Statistic

Silvapulle and Sen (2004) propose a test statistic for testing hypotheses where the null or the alternative hypothesis or both involve inequalities. You can test special cases of these hypotheses with the JOINT option in the ESTIMATE and the LSMESTIMATE statement. Consider the k estimable functions bold upper L bold-italic beta and the hypotheses upper H 0 colon bold upper L bold-italic beta equals bold 0 and upper H Subscript a Baseline colon bold upper L bold-italic beta greater-than-or-equal-to bold 0. The alternative hypothesis defines a convex cone script upper C at the origin. Suppose that under the null hypothesis bold upper L ModifyingAbove bold-italic beta With caret follows a multivariate normal distribution with mean bold 0 and variance bold upper V. The restricted alternative prevents you from using the usual F or chi-square test machinery, since the distribution of the test statistic under the alternative might not follow the usual rules. Silvapulle and Sen (2004) coined a statistic that takes into account the projection of the observed estimate onto the convex cone formed by the alternative parameter space. This test statistic is called the chi-bar-square statistic, and p-values are obtained by simulation; see, in particular, Chapter 3.4 in Silvapulle and Sen (2004).

Briefly, let bold upper U be a multivariate normal random variable with mean bold 0 and variance matrix bold upper V. The chi-bar-square statistic is the random variable

StartLayout 1st Row 1st Column chi overbar squared 2nd Column equals 3rd Column bold upper U prime bold upper V Superscript negative 1 Baseline bold upper U minus upper Q 2nd Row 1st Column upper Q 2nd Column equals 3rd Column min Underscript bold-italic theta element-of upper C Endscripts left-parenthesis bold upper U minus bold-italic theta right-parenthesis prime bold upper V Superscript negative 1 Baseline left-parenthesis bold upper U minus bold-italic theta right-parenthesis EndLayout

and it can be motivated by a geometric argument. The quadratic form in Q is the bold upper V-projection of bold upper U onto the cone script upper C. Suppose that this projected point is bold upper U overTilde. If bold upper U element-of script upper C, then Q = 0 and bold upper U overTilde equals bold upper U. If bold upper U is completely outside of the cone script upper C, then bold upper U overTilde is a point on the surface of the cone. Similarly, bold upper U prime bold upper V Superscript negative 1 Baseline bold upper U is the length of the segment from the origin to bold upper U in the bold upper V-space with norm StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue equals left-parenthesis bold x prime bold upper V Superscript negative 1 Baseline bold x right-parenthesis Superscript 1 slash 2. If you apply the Pythagorean theorem, you can see that the chi-bar-square statistic measures the length of the segment from the origin to the projected point bold upper U overTilde in script upper C.

To calculate p-values for chi-bar-square statistics, a simulation-based approach is taken. Consider again the set of k estimable functions bold upper L bold-italic beta with estimate bold upper L ModifyingAbove bold-italic beta With caret equals bold upper U and variance bold upper L normal upper V normal a normal r left-bracket ModifyingAbove bold-italic beta With caret right-bracket bold upper L Superscript prime Baseline equals bold upper V.

First, the observed value of the statistic is computed as

chi overbar Subscript o b s Superscript 2 Baseline equals bold upper U prime bold upper V Superscript negative 1 Baseline bold upper U minus upper Q

Then, n independent random samples bold upper Z 1 comma ellipsis comma bold upper Z Subscript n Baseline are drawn from an upper N left-parenthesis bold 0 comma bold upper V right-parenthesis distribution and the following chi-bar-statistics are computed for the sample:

StartLayout 1st Row 1st Column chi overbar Subscript 1 Superscript 2 2nd Column equals 3rd Column bold upper Z prime 1 bold upper V Superscript negative 1 Baseline bold upper Z 1 minus min Underscript bold-italic theta element-of upper C Endscripts left-parenthesis bold upper Z 1 minus bold-italic theta right-parenthesis prime bold upper V Superscript negative 1 Baseline left-parenthesis bold upper Z 1 minus bold-italic theta right-parenthesis 2nd Row 1st Column Blank 2nd Column vertical-ellipsis 3rd Column Blank 3rd Row 1st Column chi overbar Subscript n Superscript 2 2nd Column equals 3rd Column bold upper Z prime Subscript n Baseline bold upper V Superscript negative 1 Baseline bold upper Z Subscript n minus min Underscript bold-italic theta element-of upper C Endscripts left-parenthesis bold upper Z Subscript n Baseline minus bold-italic theta right-parenthesis prime bold upper V Superscript negative 1 Baseline left-parenthesis bold upper Z Subscript n Baseline minus bold-italic theta right-parenthesis EndLayout

The p-value is estimated by the fraction of simulated statistics that are greater than or equal to the observed value chi overbar Subscript o b s Superscript 2.

Notice that unless bold upper U is interior to the cone script upper C, finding the value of Q requires the solution to a quadratic optimization problem. When k is large, or when many simulations are requested, the computation of p-values for chi-bar-square statistics might require considerable computing time.

Last updated: December 09, 2022