{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q1" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "restart:M:=.4*exp(-t)+.2+.3*exp(t) +.1*exp(2*t): for i to 5 do k[i]:=eval(diff(ln(M),t$i),t=0)/n^(i-1) \+ end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "H:=n->simplify(d iff(exp(-z^2/2),z$n)*exp(z^2/2)*(-1)^n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "f:=subs(\{w=1,t^3=H(3),t^4=H(4),t^5=H(5),t^6=H(6),t^ 7=H(7),t^9=H(9),g[3]=k[3]/k[2]^(3/2),g[4]=k[4]/k[2]^2,g[5]=k[5]/k[2]^( 5/2)\},\nmtaylor(exp(add(w^(i-2)*g[i]*t^i/i!,i=3..5)),w,4))*exp(-z^2/2 )/sqrt(2*Pi)/n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "n:=20: \+ f:=subs(z=(x/n-k[1])/sqrt(k[2]),f)/sqrt(k[2]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 98 "h:=(.4/z+.2+.3*z+.1*z^2)^n: with(plots): listp lot([seq([i,-coeff(h,z,i)+eval(f,x=i)],i=-n..2*n)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 1016 254 254 {PLOTDATA 2 "6#-%'CURVESG6#7in7$$!#?\"\"!$!+ $Ga**G\"!#;7$$!#>F*$!+f(y/w$F-7$$!#=F*$!+^H(f$yF-7$$!#@$=\"FB7$$FBF*$\"($emF!#87$$FKF*$\")K\"[$H FK7$$!#7F*$\")zw@hFK7$$!#6F*$\"(DgN)FR7$$!#5F*$\"([B\"yFR7$$!\"*F*$\"( ,JU$FR7$$!\")F*$!(c&oSFR7$$!\"(F*$!(sc=\"FW7$$!\"'F*$!($H?;FW7$$!\"&F* $!(l]X\"FW7$$!\"%F*$!'vAtFW7$$!\"$F*$\"'p!)>FW7$$!\"#F*$\"'0l()FW7$$! \"\"F*$\"(*o?5FW7$$F*F*$\"'[(*pFW7$$\"\"\"F*$\"'(fk#FW7$$\"\"#F*$\"&qB )FW7$$\"\"$F*$\"'dDFFW7$$\"\"%F*$\"'lyjFW7$$\"\"&F*$\"'X%H)FW7$$\"\"'F *$\"'M'4'FW7$$\"\"(F*$\"&!f;FW7$$\"\")F*$!'fGnFW7$$\"\"*F*$!(<78\"FW7$ $\"#5F*$!(mR<\"FW7$$\"#6F*$!'qr$)FW7$$\"#7F*$!(_A<$FR7$$\"#8F*$\"(PXf \"FR7$$\"#9F*$\"(6@Y%FR7$$\"#:F*$\"(y-9&FR7$$\"#;F*$\"(=dA%FR7$$\"#F*$\"'$[u%FK7$$\"#?F*$!)q:5\\FB7 $$\"#@F*$!)%zHN'FB7$$\"#AF*$!*TD&4cF=7$$\"#BF*$!*2)[!4%F=7$$\"#CF*$!*F H\\h#F=7$$\"#DF*$!+Bi'e]\"F-7$$\"#EF*$!+^?]GzF:7$$\"#FF*$!+4Mw]QF:7$$ \"#GF*$!+W:-N " 0 "" {MPLTEXT 1 0 90 "restart :restart: assume(a>0): f:=a/(x^2+a^2)/Pi: h:=ln(a)-ln(x^2+a^2): G :=diff(h,a): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "u[0]:=G: \+ for i to 2 do g[i]:=factor(diff(G,a$i)/i!): m[i]:=simplify(int(f*g[i], x=-infinity..infinity)): u[i]:=factor(g[i]-m[i]) end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "M:=a+simplify(int(f*(u[0]*u[1]/m[1] ^2-u[0]^2*m[2]/m[1]^3),x=-infinity..infinity))/n; # mean" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#a|irG\"\"\"*&F$F%%\"nG!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "V:=simplify(int(f*(-u[0]/m[1])^2,x= -infinity..infinity))/n; # variance" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#a|irGF%%\"nG!\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "K3:=simplify(int(f*(-u[0]/m[1])^3,x=-infinity..infin ity)-6*m[2]/m[1]*int(f*(-u[0]/m[1])^2,x=-infinity..infinity)^2\n+6*int (f*(u[0]/m[1])^2,x=-infinity..infinity)*int(f*u[0]*u[1]/m[1]^2,x=-infi nity..infinity))/n^2; # skewness" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*(\"#7\"\"\"%#a|irG\"\"$%\"nG!\"#F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "result:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "exp(-z^2/2)/sqrt(2*Pi)*(1+K3/V^1.5/6*H[3](z))/sqrt(V) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "where z = ( aHAT - M )/sqrt(V)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q3" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "restart: k(0,0):=0: k(1,0 ):=0: k(0,1):=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "res:=e val(diff(exp(add(add(k(i,j)*t^i*u^j/i!/j!/n^(i+j-1),i=0..4),j=0..3)),t $4,u$3),\{t=0,u=0\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG,<*&-% \"kG6$\"\"%\"\"$\"\"\"%\"nG!\"'F,**\"#7F,-F(6$F+\"\"#F,F-!\"&-F(6$F,F, F,F,**F0F,-F(6$F+F,F,F-F4-F(6$F,F3F,F,**F*F,-F(6$F+\"\"!F,F-F4-F(6$F,F +F,F,**\"\"'F,-F(6$F3F?F,F-F4-F(6$F3F+F,F,**\"#=F,-F(6$F3F,F,F-F4-F(6$ F3F3F,F,**F+F,FDF3F-!\"%-F(6$F?F+F,F,*,FIF,FDF,F-FO-F(6$F?F3F,FJF,F,** \"#OF,FJF,F-FOF5F3F,*,FVF,FDF,F-FOF5F,F:F,F,**F+F,-F(6$F*F,F,F-F4FSF,F ,*(-F(6$F*F?F,F-F4FPF,F,*,F0F,F=F,F-FOF5F,FSF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "k:=(i,j)->evalf(eval(diff(ln(Int(exp(-x+t*ln( x)+u*ln(x)^2),x=0..infinity)),t$i,u$j),\{t=0,u=0\})):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "res;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&$\"+&4&o,?!\"$\"\"\"%\"nG!\"'F(*&$\"+zO4[k!\"%F(F)!\"&F(*&$ \"+(4d&yZF/F(F)F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "rest art: k(0,0,0):=0: k(1,0,0):=0: k(0,1,0):=0: k(0,0,1):=0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "res:=eval(diff(exp(add(add( add(k(i,j,L)*t1^i*t2^j*t3^L/i!/j!/L!/n^(i+j+L-1),i=0..2),j=0..2),L=0.. 2)),t1$2,t2$2,t3$2),\{t1=0,t2=0,t3=0\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG,B*&-%\"kG6%\"\"#F*F*\"\"\"%\"nG!\"&F+*(-F(6%F*\"\"!F1F+F ,!\"%-F(6%F1F*F*F+F+**F*F+-F(6%F*F+F1F+F,F2-F(6%F1F+F*F+F+**F*F+-F(6%F *F1F+F+F,F2-F(6%F1F*F+F+F+*(-F(6%F*F*F1F+F,F2-F(6%F1F1F*F+F+**\"\"%F+- F(6%F+F1F+F+F,F2-F(6%F+F*F+F+F+**F*F+-F(6%F+F+F1F*F,!\"$FBF+F+**FEF+FK F+F,F2-F(6%F+F+F*F+F+**F*F+-F(6%F+F1F*F+F,F2-F(6%F+F*F1F+F+*(-F(6%F*F1 F*F+F,F2-F(6%F1F*F1F+F+**F*F+FFF*F,FMFYF+F+*(FEF+-F(6%F+F+F+F*F,F2F+** FEF+-F(6%F*F+F+F+F,F2-F(6%F1F+F+F+F+**F*F+F/F+F,FMF\\oF*F+*,\"\")F+FFF +F,FMF\\oF+FKF+F+**F/F+F,FMFYF+FBF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "k:=(i,j,L)->evalf(eval(diff(ln(Int(exp(-x+t1*ln(x)+t 2*ln(x)^2+t3*ln(x)^3),x=0..infinity)),t1$i,t2$j,t3$L),\{t1=0,t2=0,t3=0 \})):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "res;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&$\"+G.^yR!\"\"\"\"\"%\"nG!\"&F(*&$\"+1rm " 0 "" {MPLTEXT 1 0 118 "restart: k(0,0,0,0,0):=0: k(1,0,0,0,0):=0: k(0,1,0,0,0):=0: k(0,0,1,0,0):=0: k(0,0,0,1,0):=0: k(0,0,0,0,1):=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "res:=eval(diff(exp(add(add(add(add (add(k(i1,i2,i3,i4,i5)*t1^i1*t2^i2*t3^i3*t4^i4*t5^i5/i1!/i2!/i3!/i4!/i 5!/n^(i1+i2+i3+i4+i5-1),i1=0..1)\n,i2=0..1),i3=0..1),i4=0..1),i5=0..1) ),t1,t2,t3,t4,t5),\{t1=0,t2=0,t3=0,t4=0,t5=0\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG,8*&-%\"kG6'\"\"\"F*F*F*F*F*%\"nG!\"%F**(-F(6'F* F*F*\"\"!F0F*F+!\"$-F(6'F0F0F0F*F*F*F**(-F(6'F*F*F0F*F0F*F+F1-F(6'F0F0 F*F0F*F*F**(-F(6'F*F*F0F0F*F*F+F1-F(6'F0F0F*F*F0F*F**(-F(6'F*F*F0F0F0F *F+F1-F(6'F0F0F*F*F*F*F**(-F(6'F*F0F0F*F0F*F+F1-F(6'F0F*F*F0F*F*F**(-F (6'F*F0F0F0F*F*F+F1-F(6'F0F*F*F*F0F*F**(-F(6'F*F0F*F0F0F*F+F1-F(6'F0F* F0F*F*F*F**(-F(6'F*F0F0F*F*F*F+F1-F(6'F0F*F*F0F0F*F**(-F(6'F*F0F*F*F0F *F+F1-F(6'F0F*F0F0F*F*F**(-F(6'F*F0F*F0F*F*F+F1-F(6'F0F*F0F*F0F*F*" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "k:=(i1,i2,i3,i4,i5)->evalf (eval(diff(ln(Int(exp(-x+t1*ln(x)+t2*ln(x)^2+t3*ln(x)^3+t4*ln(x)^4+t5* ln(x)^5),x=0..infinity)),\nt1$i1,t2$i2,t3$i3,t4$i4,t5$i5),\{t1=0,t2=0, t3=0,t4=0,t5=0\})):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "res;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+XP6P7\"\"$\"\"\"%\"nG!\"%!\"\" *&$\"+E*\\du%\"\"!F(F)!\"$F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q4 " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "rest art: f:=a*x^(a-1)*exp(-x^a): h:=ln(a)+(a-1)*ln(x)-x^a: G:=diff(h, a):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "u[0]:=G: for i to 4 do g[i]:=diff(G,a$i)/i!: m[i]:=simplify(int(f*g[i],x=0..infinity)): u [i]:=g[i]-m[i] end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "s eq((u[i]),i=0..2); # U0, U1 and U2 (put the bars)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%,(*&\"\"\"F%%\"aG!\"\"F%-%#lnG6#%\"xGF%*&)F+F&F%F(F%F ',(*&F%F%*$)F&\"\"#F%F'F'*&F-F%)F(F2F%F'*(\"\"'F',*F6F%*$)%#PiGF2F%F%* &\"#7F%%&gammaGF%F'*&F6F%)F=F2F%F%F%F&!\"#F%,(*&F%F%*$)F&\"\"$F%F'F%*& #F%F2F%*&F-F%)F(FEF%F%F'*&#F%\"\"%F%*&,.FLF%*&FLF%-%%ZetaG6#FEF%F%F8F' *&F9F%F=F%F%*&F6F%F?F%F'*&F2F%)F=FEF%F%F%F&!\"$F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "seq(evalf(m[i]),i=1..3); # m1, m2 and m3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,$*&$\"+g1oB=!\"*\"\"\"%\"aG!\"#! \"\",$*&$\"+=Cp_v!#5F(F)!\"$F(,$*&$\"+Wg*pH\"F'F(F)!\"%F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "the solution, in terms of a1, a2 and a3 i s then given by the formulas of 'Expanding ML estimator'" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "mean" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "a+evalf(simplify(int(f*(u [0]*u[1]/m[1]^2-u[0]^2*m[2]/m[1]^3),x=0..infinity)))/n;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&%\"aG\"\"\"*($\"+t)**ep)!#5F%F$F%%\"nG!\"\"F% " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "variance" }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(simplify(int(f*(-u[0 ]/m[1])^2,x=0..infinity)))/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*($ \"+fhT$[&!#5\"\"\"%\"aG\"\"#%\"nG!\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q5" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "restart: f:=exp(-(x^2+y^2-2*rho*x*y)/2/(1-rho^2))/2/ Pi/sqrt(1-rho^2): assume(rho^2<1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "int(int(f*exp(t*x*y),x=-infinity..infinity),y=-infini ty..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$*& \"\"\"F(*$,**&)%\"tG\"\"#F()%%rho|irGF.F(F(*(F.F(F0F(F-F(!\"\"F(F(*$F, F(F2#F(F.F2/-%%csgnG6#,**&#F(F.F(F+F(F(*&F0F(F-F(F2F;F(*&#F(F.F(F3F(F2 F(7$%)infinityG%*otherwiseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "CGF:=-ln(t^2*rho^2-2*rho*t+1-t^2)/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "for i to 4 do k[i]:=factor(eval(diff(CGF,t$i),t=0)) e nd do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"kG6#\"\"\"%%rho|irG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"kG6#\"\"#,&*$)%%rho|irGF'\"\"\"F, F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"kG6#\"\"$,$*(\"\"#\"\"\"% %rho|irGF+,&*$)F,F*F+F+F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"kG6#\"\"%,(*&\"\"'\"\"\")%%rho|irGF'F+F+*&\"#OF+)F-\"\"#F+F+F*F+" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "result:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "exp(-z^2/2)/sqrt(2*Pi)*(1+k [3]/k[2]^1.5/6/sqrt(n)*H[3](z)+k[4]/k[2]^2/24/n*H[4](z)+k[3]^3/k[2]^3/ 72/n*H[6](z))/sqrt(k[2]/n):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "wh ere z = ( XYbar - k[1])/sqrt(k[2]/n)" }{MPLTEXT 1 0 0 "" }}}} {MARK "5 0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }