最尤法(回帰分析の例) 数値計算(BFGS法、導関数不要)
プログラム
// Maximum Likelihood Estimation // Numerical Result using BFGS #include<oxstd.h> #include<oxfloat.h> // to use M_2PI #import<maximize> // to use maximization package static decl s_mX, s_vY; // static variable to use in the functions // Loglikelihood Function fLoglik(const vP, const adFunc, const avScore, const amHess) { decl ck,cnobs;decl dsum,dsig2;decl vb; cnobs=rows(s_vY);ck=rows(vP); vb=vP[0:ck-2];dsig2=vP[ck-1]; dsum=(sumsqrc(s_vY-s_mX*vb))/(2*dsig2); adFunc[0]=-0.5*cnobs*log(M_2PI*dsig2)-dsum; return 1; } main() { decl ck,cnobs; // scalar (count, number of ...) decl dfunc,ds2,dssr; // scalar (double) decl mhess,mi,msigma; // matrix decl vb,ve,vp,vpvalue,vse,vyhat,vzvalue; // vector println("Maximum Likelihood Esimation using BFGS method."); s_mX=<0;1;2;3;4>; // X matrix: independent variable s_mX=ones(5,1)~s_mX; // + constant term s_vY=<10;20;20;30;40>; // Y vector: dependent variable cnobs=rows(s_mX); // number of observations ck=columns(s_mX)+1; // number of parameters (including constant term & sigma^2) vp=<0;0;10>; // initial values // println("initial values:",vp'); MaxBFGS(fLoglik,&vp,&dfunc,0,TRUE); Num2Derivative(fLoglik,vp,&mhess); // Numerical Hessian Matrix mi=-mhess;msigma=invert(mi); vse=sqrt(diagonal(msigma))'; vzvalue=vp./vse; // z-value for H0: parameter=0 vpvalue=tailn(fabs(vzvalue)); println("%r",{"beta0","beta1","sigma^2"}, "%c",{"est. coeff.", "std. err.","z-value", "p-value"}, "%10.6",vp~vse~vzvalue~vpvalue); }
計算結果
Maximum Likelihood Esimation using BFGS method. initial values: 0.00000 0.00000 10.000 est. coeff. std. err. z-value p-value beta0 10.000 1.8974 5.2705 6.8037e-008 beta1 7.0000 0.77459 9.0370 8.0519e-020 sigma^2 6.0000 3.7947 1.5811 0.056922