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 Constrained Maximum Likelihood - 制約条件付き最尤法

Constrained Maximum Likelihood(CML)は、線形または非線形、等式条件または不等式パラメータ条件

のもとで一般の最尤法問題を解きます。すなわち次のように定式化される問題を解きます。

*

例）

次の例は、線形等式条件、非線形不等式条件、およびパラメーター範囲の制約を持つ

tobitモデルの推定の例です。

library cml;

#include cml.ext;

cmlset;

proc lpr(x,z);

local t,s,m,u;

s = x[4];

m = z[.,2:4]*x[1:3,.];

u = z[.,1] ./= 0;

t = z[.,1]-m;

retp(u.*(-(t.*t)./(2*s)-.5*ln(2*s*pi)) + (1-u).*(ln(cdfnc(m/sqrt(s)))));

endp;

x0 = { 1, 1, 1, 1 };

_cml_A = { 1 -1 0 0 };

_cml_B = { 0 };

proc ineq(x);

local b;

b = x[1:3];

retp(b'b - 3);

endp;

_cml_IneqProc = &ineq;

_cml_Bounds = { -10 10,

-10 10,

-10 10,

.01 10 };

{ x,f,g,cov,ret } = CMLPrt(CML("tobit",0,&lpr,x0));

print "linear equality Lagrangeans";

print vread(_cml_Lagrange,"lineq");

print;

print "nonlinear inequality Lagrangeans";

print vread(_cml_Lagrange,"nlinineq");

print;

print "bounds Lagangreans";

print vread(_cml_Lagrange,"bounds");

--------------------------------------------------------------------

結果の出力

=====================================================================

CML Version 2.0.0 02/08/2001 9:51 am

=====================================================================

Data Set: tobit

---------------------------------------------------------------------

return code = 0

normal convergence

Mean log-likelihood -1.34034

Number of cases 100

Covariance of the parameters computed by the following method:

Inverse of computed Hessian

Parameters Estimates Std. err. Gradient

------------------------------------------------------------------

P01 -0.1832 0.0710 -0.2073

P02 -0.1832 0.0710 0.1682

P03 1.7126 0.0152 0.1825

P04 1.0718 0.1589 -0.0000

Number of iterations 8

Minutes to convergence 0.06683

linear equality Lagrangeans

-0.1877

nonlinear inequality Lagrangeans

0.0533

bounds Lagangreans

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