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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 27 "Example 9: Order Statisti
cs" }}{PARA 19 "" 0 "" {TEXT -1 15 "by Zavan Karian" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 82 "restart: with(plots, display): libname:=
\"C:/mylib/statistics\",libname: with(stat):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 42 "A study of the p.d.f.s of order statistics" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 26 "First, take example 10.1-3" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f := x -> 1
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG\"\"\"" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "F := x -> int(f(t), t=0..x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-
%\"fG6#%\"tG/F2;\"\"!9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 72 "g := (n, r, y) -> (n!/((r-1)!*(n-r)!)) * F(y)^(r-1)*(1-F(y))^(n-
r)*f(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6%%\"nG%\"rG%\"yG6
\"6$%)operatorG%&arrowGF**&**-%*factorialG6#9$\"\"\")-%\"FG6#9&,&9%F4F
4!\"\"F4),&F4F4F6F<,&F3F4F;F<F4-%\"fGF8F4F4*&-F16#F:F4-F16#F?F4F<F*F*F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "G:= (n,r,y) -> sum(n!/
(k!*(n-k)!)*F(y)^k*(1-F(y))^(n-k), k=r..n);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"GGR6%%\"nG%\"rG%\"yG6\"6$%)operatorG%&arrowGF*-%$su
mG6$*&*(-%*factorialG6#9$\"\"\")-%\"FG6#9&%\"kGF7),&F7F7F9!\"\",&F6F7F
=F@F7F7*&-F46#F=F7-F46#FAF7F@/F=;9%F6F*F*F*" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "pdf := g(4,3,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%$pdfG,$*&)%\"yG\"\"#\"\"\",&F*F*F(!\"\"F*\"#7" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "int(pdf, y=1/3..2/3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"#8\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"G(4,3,2/3)-G(4,3,1/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#8\"#F" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 
"Note that this \"hides the expressions for g and G." }}{PARA 0 "" 0 "
" {TEXT -1 36 "We get around this by the following." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 62 "Can this be done if we assume that the underlying di
stribution" }}{PARA 0 "" 0 "" {TEXT -1 51 "is more complicated? Assume
 we have an exponential." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(theta > 0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x-> ExponentialPDF(theta, x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(-_F(%/ExponentialPDFG6$%&thetaG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "F := x-> int(f(t), t=0..x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"FGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%$intG6$-%\"fG
6#%\"tG/F2;\"\"!9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 
"g := (n, r, y) -> (n!/((r-1)!*(n-r)!)) * F(y)^(r-1)*(1-F(y))^(n-r)*f(
y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGR6%%\"nG%\"rG%\"yG6\"6$%)
operatorG%&arrowGF**&**-%*factorialG6#9$\"\"\")-%\"FG6#9&,&9%F4F4!\"\"
F4),&F4F4F6F<,&F3F4F;F<F4-%\"fGF8F4F4*&-F16#F:F4-F16#F?F4F<F*F*F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "G:= (n,r,y) -> sum(n!/(k!*(n
-k)!)*F(y)^k*(1-F(y))^(n-k), k=r..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"GGR6%%\"nG%\"rG%\"yG6\"6$%)operatorG%&arrowGF*-%$sumG6$*&*(-%*f
actorialG6#9$\"\"\")-%\"FG6#9&%\"kGF7),&F7F7F9!\"\",&F6F7F=F@F7F7*&-F4
6#F=F7-F46#FAF7F@/F=;9%F6F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 55 "We can now compute probabilities. Fo
r example, if n=4, " }}{PARA 0 "" 0 "" {TEXT -1 57 "what is the probab
ility that the third order statistic is" }}{PARA 0 "" 0 "" {TEXT -1 
18 "between 1/2 and 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P := simplify(G(4,3,1)-G(4,3,1/2));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,,-%$expG6#,$*&\"\"\"F+%'the
ta|irG!\"\"!\"%!\"$*&\"\")F+-F'6#,$F*F/F+F+*&\"\"$F+-F'6#,$F*!\"#F+F-*
&\"\"'F+-F'6#,$F*F-F+F+*&F1F+-F'6#,$F*#F/\"\"#F+F-" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "If we assume a sp
ecific theta such as theta=1, we get" }}{PARA 0 "" 0 "" {TEXT -1 17 "a
 numeric answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 23 "evalf(subs(theta=1,P));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"*Z\"z&f$!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can now study the effect of theta \+
on an order" }}{PARA 0 "" 0 "" {TEXT -1 48 "statistic for specific val
ues of n and r. Let's " }}{PARA 0 "" 0 "" {TEXT -1 14 "take n=4, r=3.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "pdf := simplify(g(4,3,y));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$pdfG,$*&*&),&-%$expG6#,$*&%\"yG\"\"\"%'theta|irG!\"
\"F2F0F0F2\"\"#F0-F+6#,$F.!\"#F0F0F1F2\"#7" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "N := 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"
\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to N do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gg := simplify(subs(theta=i, g(
4,3,y)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "P||i := plot(gg, y=0..
10, color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display([seq(P||i, i=1..5)], insequ
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"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "For a fixed theta
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he shapes of the order statistics for r=1, 2, 3, ..." }}{PARA 0 "" 0 "
" {TEXT -1 51 "Taking theta =1 and n=4, we look at r=1,2,3, and 4." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "for i from 1 to 4 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "gg :=
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37 "P||i := plot(gg, y=0..4, color=blue):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "disp
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Un
fortunately, the expressions for g and G in both examples" }}{PARA 0 "
" 0 "" {TEXT -1 59 "\"hide\" their actual forms. The following will pr
oduce more " }}{PARA 0 "" 0 "" {TEXT -1 58 "explicit representations o
f g and G, if that is desired.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "assume( theta > 0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := ExponentialPDF(theta, y
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&-%$expG6#,$*&%\"yG\"\"\"
%'theta|irG!\"\"F.F,F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"int(f, y=0..t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#,$*&%\"
tG\"\"\"%'theta|irG!\"\"F,F,F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "F := subs(t=y,%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&-%$expG6#,
$*&%\"yG\"\"\"%'theta|irG!\"\"F.F.F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "g := (n!/((r-1)!*(n-r)!)) * F^(r-1)*(1-F)^(n-r)*f;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG*&**-%*factorialG6#%\"nG\"\"\")
,&-%$expG6#,$*&%\"yGF+%'theta|irG!\"\"F5F5F+F+,&%\"rGF+F+F5F+)F.,&F*F+
F7F5F+F.F+F+*(-F(6#F6F+-F(6#F9F+F4F+F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "G := sum(n!/(k!*(n-k)!)*F^k*(1-F)^(n-k), k=r..n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG*&**-%*factorialG6#%\"nG\"\"\"),
&-%$expG6#,$*&%\"yGF+%'theta|irG!\"\"F5F5F+F+%\"rGF+)F.,&F*F+F6F5F+-%*
hypergeomG6%7$F+,&F*F5F6F+7#,&F+F+F6F+,&F+F+-F/6#F2F5F+F+*&-F(6#F6F+-F
(6#F8F+F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 55 "For contrasrt, we now consider order statistics from a " 
}}{PARA 0 "" 0 "" {TEXT -1 49 "symmetric, two-parameter distribution: \+
N(mu,var)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "assume(sigma > 0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "f := NormalPDF(mu, sigma^2, y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG,$*&*&-%%sqrtG6#\"\"#\"\"\"-%$expG6#,$*&*$),&%\"y
GF,%#muG!\"\"F+F,F,*$)%'sigma|irGF+F,F7#F7F+F,F,*&F:F,-F)6#%#PiGF,F7#F
,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(f, y=-infinity..
t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$erfG6#,$*&*&-%%sqrtG6#\"\"
#\"\"\",&%\"tGF.%#muG!\"\"F.F.%'sigma|irGF2#F.F-F4F4F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F := subs(t=y,%);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&-%$erfG6#,$*&*&-%%sqrtG6#\"\"#\"\"\"
,&%\"yGF0%#muG!\"\"F0F0%'sigma|irGF4#F0F/F6F6F0" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "g := (n!/((r-1)!*(n-r)!)) * F^(r-1)*(1-F)^(n-r
)*f;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,$*&*,-%*factorialG6#%\"
nG\"\"\"),&-%$erfG6#,$*&*&-%%sqrtG6#\"\"#F,,&%\"yGF,%#muG!\"\"F,F,%'si
gma|irGF<#F,F8F>F>F,,&%\"rGF,F,F<F,),&F>F,*&#F,F8F,F/F,F<,&F+F,F@F<F,F
5F,-%$expG6#,$*&*$)F9F8F,F,*$)F=F8F,F<#F<F8F,F,**-F)6#F?F,-F)6#FEF,F=F
,-F66#%#PiGF,F<F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "G := s
implify(sum(n!/(k!*(n-k)!)*F^k*(1-F)^(n-k), k=r..n));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"GG*&**-%&GAMMAG6#,&%\"nG\"\"\"F,F,F,),&-%$erfG
6#,$*&*&-%%sqrtG6#\"\"#F,,&%\"yGF,%#muG!\"\"F,F,%'sigma|irGF<#F,F8F>F>
F,%\"rGF,),&F>F,*&#F,F8F,F/F,F<,&F+F,F?F<F,-%*hypergeomG6%7$F,,&F+F<F?
F,7#,&F,F,F?F,*&,&F/F,F,F,F,,&F<F,F/F,F<F,F,*&-F(6#FKF,-F(6#,(F+F,F,F,
F?F<F,F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 44 "As before, we can calculate probabilities in" }}{PARA 0 "
" 0 "" {TEXT -1 15 "specific cases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "GG := simplify(subs(\{n=4,
 r=3, mu=0, sigma=1\}, G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GGG,
$*&,&!\"&\"\"\"*&\"\"$F)-%$erfG6#,$*&-%%sqrtG6#\"\"#F)%\"yGF)#F)F4F)F)
F)),&F,F)F)F)F+F)#!\"\"\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "evalf(subs(y=1/2, GG)- subs(y=-1/2, GG));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+e&HJY&!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 44 "And we can obtain expressions and graph
s for" }}{PARA 0 "" 0 "" {TEXT -1 47 "specific p.d.f.s of order statis
tics or through" }}{PARA 0 "" 0 "" {TEXT -1 44 "animation observe the \+
effect of the symmetry" }}{PARA 0 "" 0 "" {TEXT -1 44 "of the underlyi
ng distribution by comparing " }}{PARA 0 "" 0 "" {TEXT -1 19 "g_i with
 g_(n-i+1)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 51 "gg := simplify(subs(\{n=4, r=3, mu=0, sigma=1\}, g
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ggG,$*&**),&-%$erfG6#,$*&-%%
sqrtG6#\"\"#\"\"\"%\"yGF3#F3F2F3F3F3F2F3,&!\"\"F3F*F3F3F/F3-%$expG6#,$
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0 "" {MPLTEXT 1 0 2 "F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$erfG6#
,$*&*&-%%sqrtG6#\"\"#\"\"\",&%\"yGF.%#muG!\"\"F.F.%'sigma|irGF2#F.F-F4
F4F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(gg, y=-3.5..3.
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "N := 10;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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