maxlikmtProfileLimits

Purpose

Computes confidence limits by inversion of the likelihood ratio statistic.

Format

out1 = maxlikmtProfileLimits(&logl, out0[, ...., c1])

Parameters:

• &logl (pointer) – Pointer to log-likelihood function used to generate results of an estimation by a call to maxlikmt().
• out0 (struct) – Instance of maxlikmtResults structure containing results of an estimation generated by a call to maxlikmt().
• .. (Various) – Optional input arguments. They can be any set of structures, matrices, arrays, strings, required to compute the log-likelihood function.
• c1 – The set of optional input arguments must contain the instance of the maxlikmtResults structure used in the call to

maxlikmt() that produced the results in out0.

c1.bayesAlpha	Exponent of the Dirichlet random variates used in the weighted bootstrap. Default = 1.4.
c1.priorProc	Pointer to a procedure for computing the prior. Assumes a uniform prior if not provided.
c1.numSamples	Number of re-samples in the weighted likelihood bootstrap.
c1.BayesFname	Filename for the simulated posterior parameters dataset. Defaults to a unique “BAYESxxxx” pattern.
c1.maxBootTime	Maximum time allowed for resampling.
c1.Bounds	Bounds on parameters, either 1x2 for all parameters or Kx2 for individual parameter bounds. Default = {-1e256, 1e256}.
c1.algorithm	Descent algorithm for optimization, includes BFGS, DFP, Newton, and BHHH.
c1.switch	Controls algorithm switching based on various performance metrics.
c1.lineSearch	Method for line search in optimization, includes augmented trust region method and others. Default varies based on constraints.
c1.active	Kx1 vector to control which parameters are active or fixed at start value.
c1.numObs	Number of observations, required if the log-likelihood function returns a scalar.
c1.maxIters	Maximum number of iterations for the optimization process. Default = 10000.
c1.tol	Convergence tolerance, optimization stops when all elements of the direction vector are below this value. Default = 1e-5.
c1.weights	Vector of weights for the objective function returning a vector. Default = 1.
c1.covParType	Determines the type of covariance matrix computed, QML or ML. Default = 1.
c1.alpha	Probability level for statistical tests. Default = .05.
c1.feasibleTest	If nonzero, parameters are tested for feasibility before computing the function in line search. Default = 1.
c1.maxTries	Maximum number of attempts in random search. Default = 100.
c1.randRadius	Radius of the random search, if attempted. Default = .001.
c1.gradMethod	Method for computing numerical gradient, includes central, forward, and backward difference.
c1.hessMethod	Method for computing numerical Hessian, similar options as gradient computation.
c1.gradStep	Increment size for computing numerical gradient, can be scalar or Kx1 vector.
c1.hessStep	Increment size for computing numerical Hessian, options similar to gradStep.
c1.gradCheck	If nonzero and analytical gradients/Hessian provided, numerical versions are computed for comparison.
c1.state	Seed for random number generator, ensuring reproducibility.
c1.title	Title of the run, for identification in output.
c1.printIters	If nonzero, iteration information is printed. Default = 0.
c1.disableKey	If nonzero, keyboard input is disabled during execution.

Returns:out1 (struct) – Instance of maxlikmtResults structure that is a duplicate of out0 except that the member, out1.profileLimits, has been set to the confidence limits by inversion of the likelihood ratio statistic.

Example

library maxlikmt;

// Define the log-likelihood function
proc lpr(struct PV p, y, x, ind);
    local s2, b0, b, yh, u, res, g1, g2;

    struct modelResults mm;

    b0 = pvUnpack(p,"b0");
    b = pvUnpack(p,"b");
    s2 = pvUnpack(p,"variance");

    yh = b0 + x * b;
    res = y - yh;
    u = y[.,1] ./= 0;

    if ind[1];
        mm.function = u.*lnpdfmvn(res,s2) + (1-u).*(ln(cdfnc(yh/sqrt(s2))));
    endif;

    if ind[2];
        yh = yh/sqrt(s2);
        g1 = ((res~x.*res)/s2)~((res.*res/s2)-1)/(2*s2);
        g2 = (-(ones(rows(x),1)~x)/sqrt(s2))~(yh/(2*s2));
        g2 = (pdfn(yh)./cdfnc(yh)).*g2;
        mm.gradient = u.*g1 + (1-u).*g2;
    endif;

    retp(mm);
endp;

struct PV p0;
p0 = pvPack(pvCreate, 1, "b0");
p0 = pvPack(p0, 1|1|1, "b");
p0 = pvPack(p0, 1, "variance");

struct maxlikmtControl c0;
c0 = maxlikmtcontrolcreate;
c0.title = "Tobit Example";
c0.Bounds = {-10 10, -10 10, -10 10, -10 10, .1 10};

z = loadd(getGAUSSHome("pkgs/maxlikmt/examples/maxlikmttobit.dat"));
y = z[., 1];
x = z[., 2:4];

struct maxlikmtResults out1;
out1 = maxlikmt(&lpr, p0, y, x, c0);

// Compute limits by inversion of likelihood ratio statistic
out1 = maxlikmtProfileLimits(&lpr, out1, y, x, c0);

// Print the results
call maxlikmtPrt(out1);