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#

See Silvapulle and Sen, Constrained Statistical Inference, page 75 for further details about this function. Let

\[Z_{px1} N(0, V)\]

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.

Source#

hypotest.src

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