计算统计-2.2答案

(a.)Sample-c(3.91,4.85,2.28,4.06,3.70,4.04,5.46,3.53,2.28,1.96,2.53,3.88,2.22,3.47,4.82,2.46,2.99,2.54,0.52,2.50)#The log-likelihood function of the parameter thetalogL-function(theta) L=0 for(x in Sample) L=L+log(1-cos(x-theta)/(2*pi) return(L) #Then plot the curve of the log-likelihood functioncurve(logL,-pi,pi)(b.)The method of moment for theta:xbar-mean(Sample)theta theta1 0.05844061(c.)#The derivative of the log-lokelihood function with respect to thetadlogL-function(theta) dL-0 for(x in Sample) dL=dL-sin(x-theta)/(1-cos(x-theta) return(dL)#The Newton methodNewton - function(f, tol=1E-12,x0=1,N=100) h - 0.001 i - 1 x1 - x0 while (i=N) df_dx - (f(x0+h)-f(x0)/h x1 - (x0 - (f(x0)/df_dx) root - x1 i - i + 1 if (abs(x1-x0) tol) break x0 Newton(dlogL,x0=-2.7)1 -2.6667 Newton(dlogL,x0=2.7)1 2.873095(d.)startpoints roots for(i in 1:200)+ root-Newton(dlogL,x0=startpointsi)+ roots roots unique(roots) 1 -3.0930 -2.7860 -2.6670 -2.5080 -2.3880 -2.2970 -2.2320 -1.6580 -1.4470 -0.9533 -0.0120 0.7906 2.0040 2.2360 2.3610 2.4750 2.5140 2.8730 3.1900 根据不同的迭代初值设置，共收敛到19个不同的根。说明对数似然方程的导数在所在区间内有不止一个零点。 distinstart for(i in 1:length(unique(roots)+ n-which(roots=unique(roots)i)+ distinstart distinstart 1 -3.1415927 -2.7942809 -2.7311333 -2.5732643 -2.3838216 -2.3522478 -2.2259526 -2.1943788 -1.4366077 -1.4050339 -0.8051318 0.5209676 1.9733622 2.225952615 2.2891002 2.4785429 2.5101167 2.5416905 3.015297519个不同的根将迭代初值分为19个小区间，这19个小区间的起始位置如上所示。(e.) min(diff(distinstart)1 0.0315738 which.min(diff(distinstart)1 16 distinstar