chiBarSquare¶

Purpose¶

Compute the probability for a chi-bar square statistic from an hypothesis involving parameters under constraints.

Format¶

SLprob = chiBarSquare(SL, cov, a, b, c, d, bounds)
Parameters
• SL (scalar) – chi-bar square statistic

• cov (KxK matrix) – positive covariance matrix

• a (MxK matrix) – linear equality constraint coefficients

• b (Mx1 vector) –

linear equality constraint constants. These arguments specify the linear equality constraints of the following type:

$a * X = b$

where x is the $$Kx1$$ parameter vector.

• c (MxK matrix) – linear inequality constraint coefficients.

• d (Mx1 vector) –

linear inequality constraint constants. These arguments specify the linear inequality constraints of the following type:

$c * X \leq d$

where x is the $$Kx1$$ parameter vector.

• bounds (Kx2 matrix) – bounds on parameters. The first column contains the lower bounds, and the second column the upper bounds.

Returns

SLprob (scalar) – probability of SL.

Examples¶

// Covariance matrix
V = { 0.0005255598 -0.0006871606 -0.0003191342,
-0.0006871606 0.0037466205 0.0012285813,
-0.0003191342 0.0012285813 0.0009081412 };

// Chi-bar square statistic
SL = 3.860509;

// Bounds on parameters
bounds = { 0 200, 0 200, 0 200 };

// Covariance
vi = invpd(V);

SLprob = chiBarSquare(SL, vi, 0, 0, 0, 0, bounds);


After running above code,

SLprob = 0.10885000


Remarks¶

$Z_{px1} N(0, V)$

where V is a positive definite covariance matrix. Define

$x^{-2}(V, C)=Z′V^{-1}Z−\min_{\theta \epsilon C}(Z - \theta)′ V^{-1}(Z - \theta)$

C is a closed convex cone describing a set of constraints. ChiBarSquare() computes the probability of this statistic given V and C.

hypotest.src