# [R] Extreme value distributions (Long.)

Göran Broström gb at stat.umu.se
Mon Mar 25 21:58:16 CET 2002

```On Mon, 25 Mar 2002, Göran Broström wrote:

> On Mon, 25 Mar 2002, Rolf Turner wrote:
>
> > This may not actually be an R/Splus problem, but it started
> > off that way .....
> >
> > ===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===
> > Executive summary:
> > ==================
> >
> > Simulations involving extreme value distributions seem to ``work''
> > when the underlying distribution is exponential(1) or exponential(2)
> > == chi-squared_2,
>
> This is _not correct_, exp(2) == chisq(1), in fact. Maybe you have made a
> simple error of this kind in your simulations and calculations?

This is obviously false. Another try: exp(1/2) == chisq(2), in the
parametrisation of R.

>
> Göran
>
> > but NOT when the underlying distribution is
> > chi-squared_1.
> >
> > Can anyone make an educated conjecture as to why?
> > ===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===
> >
> > More (much more!) detail:
> > =========================
> >
> > I have recently been doing some simulations which relate to extreme
> > value distributions.  I have observed a phenomenon which puzzles me,
> > and I would appreciate it if anyone could shed some light on the
> > puzzle.  The phenomenon occurs in both R and Splus.  (Also, I have
> > now discovered, in stand-alone Fortran.)
> >
> > The phenomenon boils down to this:
> >
> > 	I generate ``nsam'' samples of chi-squared_1 iid random
> > 	variables, each sample being of size ``n''.
> >
> > 	For each sample, let M be the maximum of the sample,
> > 	and let the statistic S = (M - d_n)/2, where
> >
> > 		d_n = 2*ln(n) - ln(ln(n)) - ln(pi).
> >
> > 	Count the number K of times that G(S) < 0.05 where G(x) is
> > 	the cdf of the Gumbel distribution, G(x) = exp(-exp(-x)).
> >
> > 	Then form alpha-hat = K/nsam.
> >
> > 	According to theory, alpha-hat should ---> 0.05 as n ---> infinity.
> >
> > 	(The chi-squared_1 distribution is a special case of the
> > 	Gamma distribution, which is in the domain of attraction of
> > 	the Gumbel distribution.  The normalizing constants for
> > 	a Gamma(a,b) distribution are
> >
> > 		d_n = b*[ln(n) + (a-1)*ln(ln(n)) - ln(Gamma(a))]
> > 		c_n = b
> >
> > 	and for the chi-squared_1 distribution, a = 1/2, b=2, giving
> > 	d_n = 2*ln(n) - ln(ln(n)) - ln(pi) and c = 2.  Note that I am
> > 	using the parameterization of the Gamma distribution such
> > 	that the mean is a*b and the variance is a*b^2.)
> >
> > 	In numerous simulations I have found that the values
> > 	of alpha-hat are generally substantially LESS than 0.05 ---
> > 	tending perhaps to hang around 0.03 or 0.04 for large n.
> >
> > 	The simulations that I have done so far are with nsam = 1000
> > 	and 10000, and with n varying from 100 to 10000 [n in
> > 	c(100*(1:10),1000*(2:10))].
> >
> > A colleague of mine suggested that perhaps rnorm() has a problem out
> > in the tails --- i.e. perhaps rnorm() works by calculating F^{-1}(U)
> > where U is Uniform[0,1], and the implementation of F^{-1}() does not
> > give quite as much weight as it ought for the tails.  This would
> > result in getting extreme values less often than we should, and hence
> > getting low values of alpha-hat.
> >
> > BUT I tried the simulations using a ``roll your own'' normal random
> > number generator (``myrnorm()''; see below) --- which does NOT depend
> > on approximating F^{-1} for the normal distribution --- and got the
> > same phenomenon.
> >
> > Another colleague suggested that perhaps 10000 simply isn't large
> > enough --- that at 10000, the asymptotics haven't really kicked in
> > yet, and perhaps we need n = 100000 or n = 1000000 before the
> > asymptotic result gives a good approximation to reality.  If this
> > were so it would be very disappointing; if the asymptotics are
> > no good at n = 10000, then they are not of much use in practice.
> >
> > I tried a simulation --- computations done entirely in Fortran;
> > completely independent of R or Splus --- with n in
> > c(100*(1:10),10000*(1:10)).  I got the following values of
> > alpha-hat:
> >
> > 0.0290 0.0190 0.0390 0.0260 0.0260 0.0280 0.0260 0.0390 0.0250 0.0280
> > 0.0470 0.0360 0.0340 0.0300 0.0400 0.0300 0.0370 0.0450 0.0410 0.0310
> >
> > For this simulation ``nsam'' was 1000.  The chi-squared variates were
> > formed by squaring N(0,1) variates which were in turn generated using
> > the same procedure as in ``myrnorm()''.
> >
> > I also tried simulations with the ``standard'' exponential(1)
> > distribution --- pdf = f(x) = exp(-x), and the exponential(2)
> > distributions.  For these distributions (also special cases of the
> > Gamma family of course) d_n = log(n); c_n = 1, and d_n = 2*log(n);
> > c_n = 2, respectively.  I used nsam = 1000 and n varying from 100 to
> > 10000 as before.  For these distributions, the alpha-hat values hung
> > around 0.05 just about as they should.
> >
> > So I'm totally stumped as to what's going on.  Has anyone any
> >
> > I enclose, at the end of this email, a page of graphs (in PostScript
> > form) of alpha-hat versus n, for
> >
> > 	(1) The exponential(1) distribution,
> >
> > 	(2) The chisquared_1 distribution, with the simulation
> > 	done by generating N(0,1) variables from rnorm() and
> > 	squaring them.
> >
> > 	(3) The chisquared_1 distribution, with the simulation done
> > 	by generating N(0,1) variables from ``myrnorm()'' and
> > 	squaring them.
> >
> > 	(Note: myrnorm() produces Z = R*cos(2*pi*theta) where
> > 	theta ~ U[0,1] and R = sqrt(-2*(log(U))) where U ~ U[0,1],
> > 	theta and U independent.)
> >
> > 	(The ``whiskers'' on the plotted points give approximate
> > 	95% confidence intervals for the ``true'' alpha.)
> >
> > I'd really appreciate any hint anyone can give me as to why I'm
> > not getting 0.05 when I should be getting 0.05!!!  It's driving
> > me crazy!
> >
> > 					cheers,
> >
> > 						Rolf Turner
> > 						rolf at math.unb.ca
> >
> > ===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===+===
> > %%DocumentNeededResources: font Helvetica
> > %%+ font Helvetica-Bold
> > %%+ font Helvetica-Oblique
> > %%+ font Helvetica-BoldOblique
> > %%+ font Symbol
> > %%DocumentMedia: a4 595 841 0 () ()
> > %%Title: R Graphics Output
> > %%Creator: R Software
> > %%Pages: (atend)
> > %%Orientation: Portrait
> > %%BoundingBox: 18 61 577 781
> > %%BeginProlog
> > /bp  { gs gs } def
> > % begin .ps.prolog
> > /gs  { gsave } def
> > /gr  { grestore } def
> > /ep  { showpage gr gr } def
> > /m   { moveto } def
> > /l   { lineto } def
> > /np  { newpath } def
> > /cp  { closepath } def
> > /f   { fill } def
> > /o   { stroke } def
> > /c   { newpath 0 360 arc } def
> > /r   { 3 index 3 index moveto 1 index 4 -1 roll
> >        lineto exch 1 index lineto lineto closepath } def
> > /p1  { stroke } def
> > /p2  { gsave bg setrgbcolor fill grestore newpath } def
> > /p3  { gsave bg setrgbcolor fill grestore stroke } def
> > /t   { 6 -2 roll moveto gsave rotate
> >        ps mul neg 0 2 1 roll rmoveto
> >        1 index stringwidth pop
> >        mul neg 0 rmoveto show grestore } def
> > /cl  { grestore gsave newpath 3 index 3 index moveto 1 index
> >        4 -1 roll lineto  exch 1 index lineto lineto
> >        closepath clip newpath } def
> > /rgb { setrgbcolor } def
> > /s   { scalefont setfont } def
> > /R   { /Font1 findfont } def
> > /B   { /Font2 findfont } def
> > /I   { /Font3 findfont } def
> > /BI  { /Font4 findfont } def
> > /S   { /Font5 findfont } def
> > 1 setlinecap 1 setlinejoin
> > % end   .ps.prolog
> > %%IncludeResource: font Helvetica
> > /Helvetica findfont
> > dup length dict begin
> >   {1 index /FID ne {def} {pop pop} ifelse} forall
> >   /Encoding ISOLatin1Encoding def
> >   currentdict
> >   end
> > /Font1 exch definefont pop
> > %%IncludeResource: font Helvetica-Bold
> > /Helvetica-Bold findfont
> > dup length dict begin
> >   {1 index /FID ne {def} {pop pop} ifelse} forall
> >   /Encoding ISOLatin1Encoding def
> >   currentdict
> >   end
> > /Font2 exch definefont pop
> > %%IncludeResource: font Helvetica-Oblique
> > /Helvetica-Oblique findfont
> > dup length dict begin
> >   {1 index /FID ne {def} {pop pop} ifelse} forall
> >   /Encoding ISOLatin1Encoding def
> >   currentdict
> >   end
> > /Font3 exch definefont pop
> > %%IncludeResource: font Helvetica-BoldOblique
> > /Helvetica-BoldOblique findfont
> > dup length dict begin
> >   {1 index /FID ne {def} {pop pop} ifelse} forall
> >   /Encoding ISOLatin1Encoding def
> >   currentdict
> >   end
> > /Font4 exch definefont pop
> > %%IncludeResource: font Symbol
> > /Symbol findfont
> > dup length dict begin
> >   {1 index /FID ne {def} {pop pop} ifelse} forall
> >   currentdict
> >   end
> > /Font5 exch definefont pop
> > %%EndProlog
> > %%Page: 1 1
> > bp
> > 56.97 589.42 557.32 741.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > 75.50 655.81 1.78 c p1
> > 80.18 665.70 1.78 c p1
> > 84.86 668.52 1.78 c p1
> > 89.54 661.46 1.78 c p1
> > 94.22 671.35 1.78 c p1
> > 98.90 668.52 1.78 c p1
> > 103.58 679.82 1.78 c p1
> > 108.26 661.46 1.78 c p1
> > 112.94 679.82 1.78 c p1
> > 117.61 651.57 1.78 c p1
> > 164.41 665.70 1.78 c p1
> > 211.21 658.63 1.78 c p1
> > 258.01 672.76 1.78 c p1
> > 304.80 686.89 1.78 c p1
> > 351.60 668.52 1.78 c p1
> > 398.40 660.05 1.78 c p1
> > 445.19 668.52 1.78 c p1
> > 491.99 669.93 1.78 c p1
> > 538.79 668.52 1.78 c p1
> > 18.00 60.94 577.28 780.94 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > np
> > 70.82 589.42 m
> > 538.79 589.42 l
> > o
> > np
> > 70.82 589.42 m
> > 70.82 584.66 l
> > o
> > np
> > 164.41 589.42 m
> > 164.41 584.66 l
> > o
> > np
> > 258.01 589.42 m
> > 258.01 584.66 l
> > o
> > np
> > 351.60 589.42 m
> > 351.60 584.66 l
> > o
> > np
> > 445.19 589.42 m
> > 445.19 584.66 l
> > o
> > np
> > 538.79 589.42 m
> > 538.79 584.66 l
> > o
> > /ps 8 def R 8 s
> > 70.82 572.31 (0) 0.50 0.00 0.00 t
> > 164.41 572.31 (2000) 0.50 0.00 0.00 t
> > 258.01 572.31 (4000) 0.50 0.00 0.00 t
> > 351.60 572.31 (6000) 0.50 0.00 0.00 t
> > 445.19 572.31 (8000) 0.50 0.00 0.00 t
> > 538.79 572.31 (10000) 0.50 0.00 0.00 t
> > np
> > 56.97 595.07 m
> > 56.97 736.33 l
> > o
> > np
> > 56.97 595.07 m
> > 52.21 595.07 l
> > o
> > np
> > 56.97 623.32 m
> > 52.21 623.32 l
> > o
> > np
> > 56.97 651.57 m
> > 52.21 651.57 l
> > o
> > np
> > 56.97 679.82 m
> > 52.21 679.82 l
> > o
> > np
> > 56.97 708.08 m
> > 52.21 708.08 l
> > o
> > np
> > 56.97 736.33 m
> > 52.21 736.33 l
> > o
> > 45.56 595.07 (0.00) 0.50 0.00 90.00 t
> > 45.56 623.32 (0.02) 0.50 0.00 90.00 t
> > 45.56 651.57 (0.04) 0.50 0.00 90.00 t
> > 45.56 679.82 (0.06) 0.50 0.00 90.00 t
> > 45.56 708.08 (0.08) 0.50 0.00 90.00 t
> > 45.56 736.33 (0.10) 0.50 0.00 90.00 t
> > np
> > 56.97 589.42 m
> > 557.32 589.42 l
> > 557.32 741.98 l
> > 56.97 741.98 l
> > 56.97 589.42 l
> > o
> > 18.00 540.94 577.28 780.94 cl
> > /ps 10 def B 10 s
> > 0.0000 0.0000 0.0000 rgb
> > 307.14 757.87 (Empirical sig. level, exponential distribution) 0.50 0.00 0.00 t
> > /ps 8 def R 8 s
> > 307.14 553.30 (series length) 0.50 0.00 0.00 t
> > 56.97 589.42 557.32 741.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [ 3.00 5.00] 0 setdash
> > np
> > 56.97 665.70 m
> > 557.32 665.70 l
> > o
> > 18.00 60.94 577.28 780.94 cl
> > /ps 12 def S 12 s
> > 0.0000 0.0000 0.0000 rgb
> > 20.88 659.83 (a) 0.00 0.00 0.00 t
> > /ps 12 def R 12 s
> > 21.85 663.52 (^) 0.00 0.00 0.00 t
> > 56.97 589.42 557.32 741.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > np
> > 75.50 655.81 m
> > 75.50 665.54 l
> > o
> > np
> > 80.18 665.70 m
> > 80.18 675.43 l
> > o
> > np
> > 84.86 668.52 m
> > 84.86 678.26 l
> > o
> > np
> > 89.54 661.46 m
> > 89.54 671.19 l
> > o
> > np
> > 94.22 671.35 m
> > 94.22 681.08 l
> > o
> > np
> > 98.90 668.52 m
> > 98.90 678.26 l
> > o
> > np
> > 103.58 679.82 m
> > 103.58 689.56 l
> > o
> > np
> > 108.26 661.46 m
> > 108.26 671.19 l
> > o
> > np
> > 112.94 679.82 m
> > 112.94 689.56 l
> > o
> > np
> > 117.61 651.57 m
> > 117.61 661.31 l
> > o
> > np
> > 164.41 665.70 m
> > 164.41 675.43 l
> > o
> > np
> > 211.21 658.63 m
> > 211.21 668.37 l
> > o
> > np
> > 258.01 672.76 m
> > 258.01 682.50 l
> > o
> > np
> > 304.80 686.89 m
> > 304.80 696.62 l
> > o
> > np
> > 351.60 668.52 m
> > 351.60 678.26 l
> > o
> > np
> > 398.40 660.05 m
> > 398.40 669.78 l
> > o
> > np
> > 445.19 668.52 m
> > 445.19 678.26 l
> > o
> > np
> > 491.99 669.93 m
> > 491.99 679.67 l
> > o
> > np
> > 538.79 668.52 m
> > 538.79 678.26 l
> > o
> > np
> > 75.50 655.81 m
> > 75.50 646.07 l
> > o
> > np
> > 80.18 665.70 m
> > 80.18 655.96 l
> > o
> > np
> > 84.86 668.52 m
> > 84.86 658.79 l
> > o
> > np
> > 89.54 661.46 m
> > 89.54 651.72 l
> > o
> > np
> > 94.22 671.35 m
> > 94.22 661.61 l
> > o
> > np
> > 98.90 668.52 m
> > 98.90 658.79 l
> > o
> > np
> > 103.58 679.82 m
> > 103.58 670.09 l
> > o
> > np
> > 108.26 661.46 m
> > 108.26 651.72 l
> > o
> > np
> > 112.94 679.82 m
> > 112.94 670.09 l
> > o
> > np
> > 117.61 651.57 m
> > 117.61 641.83 l
> > o
> > np
> > 164.41 665.70 m
> > 164.41 655.96 l
> > o
> > np
> > 211.21 658.63 m
> > 211.21 648.90 l
> > o
> > np
> > 258.01 672.76 m
> > 258.01 663.02 l
> > o
> > np
> > 304.80 686.89 m
> > 304.80 677.15 l
> > o
> > np
> > 351.60 668.52 m
> > 351.60 658.79 l
> > o
> > np
> > 398.40 660.05 m
> > 398.40 650.31 l
> > o
> > np
> > 445.19 668.52 m
> > 445.19 658.79 l
> > o
> > np
> > 491.99 669.93 m
> > 491.99 660.20 l
> > o
> > np
> > 538.79 668.52 m
> > 538.79 658.79 l
> > o
> > 56.97 557.32 349.42 501.98 cl
> > 56.97 349.42 557.32 501.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > 75.50 386.14 1.78 c p1
> > 80.18 386.14 1.78 c p1
> > 84.86 386.14 1.78 c p1
> > 89.54 403.09 1.78 c p1
> > 94.22 400.27 1.78 c p1
> > 98.90 386.14 1.78 c p1
> > 103.58 393.21 1.78 c p1
> > 108.26 391.79 1.78 c p1
> > 112.94 401.68 1.78 c p1
> > 117.61 388.97 1.78 c p1
> > 164.41 391.79 1.78 c p1
> > 211.21 417.22 1.78 c p1
> > 258.01 403.09 1.78 c p1
> > 304.80 393.21 1.78 c p1
> > 351.60 420.05 1.78 c p1
> > 398.40 417.22 1.78 c p1
> > 445.19 407.33 1.78 c p1
> > 491.99 407.33 1.78 c p1
> > 538.79 393.21 1.78 c p1
> > 18.00 60.94 577.28 780.94 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > np
> > 70.82 349.42 m
> > 538.79 349.42 l
> > o
> > np
> > 70.82 349.42 m
> > 70.82 344.66 l
> > o
> > np
> > 164.41 349.42 m
> > 164.41 344.66 l
> > o
> > np
> > 258.01 349.42 m
> > 258.01 344.66 l
> > o
> > np
> > 351.60 349.42 m
> > 351.60 344.66 l
> > o
> > np
> > 445.19 349.42 m
> > 445.19 344.66 l
> > o
> > np
> > 538.79 349.42 m
> > 538.79 344.66 l
> > o
> > /ps 8 def R 8 s
> > 70.82 332.31 (0) 0.50 0.00 0.00 t
> > 164.41 332.31 (2000) 0.50 0.00 0.00 t
> > 258.01 332.31 (4000) 0.50 0.00 0.00 t
> > 351.60 332.31 (6000) 0.50 0.00 0.00 t
> > 445.19 332.31 (8000) 0.50 0.00 0.00 t
> > 538.79 332.31 (10000) 0.50 0.00 0.00 t
> > np
> > 56.97 355.07 m
> > 56.97 496.33 l
> > o
> > np
> > 56.97 355.07 m
> > 52.21 355.07 l
> > o
> > np
> > 56.97 383.32 m
> > 52.21 383.32 l
> > o
> > np
> > 56.97 411.57 m
> > 52.21 411.57 l
> > o
> > np
> > 56.97 439.82 m
> > 52.21 439.82 l
> > o
> > np
> > 56.97 468.08 m
> > 52.21 468.08 l
> > o
> > np
> > 56.97 496.33 m
> > 52.21 496.33 l
> > o
> > 45.56 355.07 (0.00) 0.50 0.00 90.00 t
> > 45.56 383.32 (0.02) 0.50 0.00 90.00 t
> > 45.56 411.57 (0.04) 0.50 0.00 90.00 t
> > 45.56 439.82 (0.06) 0.50 0.00 90.00 t
> > 45.56 468.08 (0.08) 0.50 0.00 90.00 t
> > 45.56 496.33 (0.10) 0.50 0.00 90.00 t
> > np
> > 56.97 349.42 m
> > 557.32 349.42 l
> > 557.32 501.98 l
> > 56.97 501.98 l
> > 56.97 349.42 l
> > o
> > 18.00 300.94 577.28 540.94 cl
> > /ps 10 def B 10 s
> > 0.0000 0.0000 0.0000 rgb
> > 307.14 517.87 (Empirical sig. level, chisquared_1 distribution, built-in rng) 0.50 0.00 0.00 t
> > /ps 8 def R 8 s
> > 307.14 313.30 (series length) 0.50 0.00 0.00 t
> > 56.97 349.42 557.32 501.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [ 3.00 5.00] 0 setdash
> > np
> > 56.97 425.70 m
> > 557.32 425.70 l
> > o
> > 18.00 60.94 577.28 780.94 cl
> > /ps 12 def S 12 s
> > 0.0000 0.0000 0.0000 rgb
> > 20.88 419.83 (a) 0.00 0.00 0.00 t
> > /ps 12 def R 12 s
> > 21.85 423.52 (^) 0.00 0.00 0.00 t
> > 56.97 349.42 557.32 501.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > np
> > 75.50 386.14 m
> > 75.50 395.88 l
> > o
> > np
> > 80.18 386.14 m
> > 80.18 395.88 l
> > o
> > np
> > 84.86 386.14 m
> > 84.86 395.88 l
> > o
> > np
> > 89.54 403.09 m
> > 89.54 412.83 l
> > o
> > np
> > 94.22 400.27 m
> > 94.22 410.01 l
> > o
> > np
> > 98.90 386.14 m
> > 98.90 395.88 l
> > o
> > np
> > 103.58 393.21 m
> > 103.58 402.94 l
> > o
> > np
> > 108.26 391.79 m
> > 108.26 401.53 l
> > o
> > np
> > 112.94 401.68 m
> > 112.94 411.42 l
> > o
> > np
> > 117.61 388.97 m
> > 117.61 398.70 l
> > o
> > np
> > 164.41 391.79 m
> > 164.41 401.53 l
> > o
> > np
> > 211.21 417.22 m
> > 211.21 426.96 l
> > o
> > np
> > 258.01 403.09 m
> > 258.01 412.83 l
> > o
> > np
> > 304.80 393.21 m
> > 304.80 402.94 l
> > o
> > np
> > 351.60 420.05 m
> > 351.60 429.78 l
> > o
> > np
> > 398.40 417.22 m
> > 398.40 426.96 l
> > o
> > np
> > 445.19 407.33 m
> > 445.19 417.07 l
> > o
> > np
> > 491.99 407.33 m
> > 491.99 417.07 l
> > o
> > np
> > 538.79 393.21 m
> > 538.79 402.94 l
> > o
> > np
> > 75.50 386.14 m
> > 75.50 376.41 l
> > o
> > np
> > 80.18 386.14 m
> > 80.18 376.41 l
> > o
> > np
> > 84.86 386.14 m
> > 84.86 376.41 l
> > o
> > np
> > 89.54 403.09 m
> > 89.54 393.36 l
> > o
> > np
> > 94.22 400.27 m
> > 94.22 390.53 l
> > o
> > np
> > 98.90 386.14 m
> > 98.90 376.41 l
> > o
> > np
> > 103.58 393.21 m
> > 103.58 383.47 l
> > o
> > np
> > 108.26 391.79 m
> > 108.26 382.06 l
> > o
> > np
> > 112.94 401.68 m
> > 112.94 391.95 l
> > o
> > np
> > 117.61 388.97 m
> > 117.61 379.23 l
> > o
> > np
> > 164.41 391.79 m
> > 164.41 382.06 l
> > o
> > np
> > 211.21 417.22 m
> > 211.21 407.49 l
> > o
> > np
> > 258.01 403.09 m
> > 258.01 393.36 l
> > o
> > np
> > 304.80 393.21 m
> > 304.80 383.47 l
> > o
> > np
> > 351.60 420.05 m
> > 351.60 410.31 l
> > o
> > np
> > 398.40 417.22 m
> > 398.40 407.49 l
> > o
> > np
> > 445.19 407.33 m
> > 445.19 397.60 l
> > o
> > np
> > 491.99 407.33 m
> > 491.99 397.60 l
> > o
> > np
> > 538.79 393.21 m
> > 538.79 383.47 l
> > o
> > 56.97 557.32 109.42 261.98 cl
> > 56.97 109.42 557.32 261.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > 75.50 153.21 1.78 c p1
> > 80.18 161.68 1.78 c p1
> > 84.86 153.21 1.78 c p1
> > 89.54 156.03 1.78 c p1
> > 94.22 151.79 1.78 c p1
> > 98.90 161.68 1.78 c p1
> > 103.58 160.27 1.78 c p1
> > 108.26 163.09 1.78 c p1
> > 112.94 141.91 1.78 c p1
> > 117.61 151.79 1.78 c p1
> > 164.41 147.56 1.78 c p1
> > 211.21 164.51 1.78 c p1
> > 258.01 168.75 1.78 c p1
> > 304.80 163.09 1.78 c p1
> > 351.60 177.22 1.78 c p1
> > 398.40 181.46 1.78 c p1
> > 445.19 165.92 1.78 c p1
> > 491.99 164.51 1.78 c p1
> > 538.79 158.86 1.78 c p1
> > 18.00 60.94 577.28 780.94 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > np
> > 70.82 109.42 m
> > 538.79 109.42 l
> > o
> > np
> > 70.82 109.42 m
> > 70.82 104.66 l
> > o
> > np
> > 164.41 109.42 m
> > 164.41 104.66 l
> > o
> > np
> > 258.01 109.42 m
> > 258.01 104.66 l
> > o
> > np
> > 351.60 109.42 m
> > 351.60 104.66 l
> > o
> > np
> > 445.19 109.42 m
> > 445.19 104.66 l
> > o
> > np
> > 538.79 109.42 m
> > 538.79 104.66 l
> > o
> > /ps 8 def R 8 s
> > 70.82 92.31 (0) 0.50 0.00 0.00 t
> > 164.41 92.31 (2000) 0.50 0.00 0.00 t
> > 258.01 92.31 (4000) 0.50 0.00 0.00 t
> > 351.60 92.31 (6000) 0.50 0.00 0.00 t
> > 445.19 92.31 (8000) 0.50 0.00 0.00 t
> > 538.79 92.31 (10000) 0.50 0.00 0.00 t
> > np
> > 56.97 115.07 m
> > 56.97 256.33 l
> > o
> > np
> > 56.97 115.07 m
> > 52.21 115.07 l
> > o
> > np
> > 56.97 143.32 m
> > 52.21 143.32 l
> > o
> > np
> > 56.97 171.57 m
> > 52.21 171.57 l
> > o
> > np
> > 56.97 199.82 m
> > 52.21 199.82 l
> > o
> > np
> > 56.97 228.08 m
> > 52.21 228.08 l
> > o
> > np
> > 56.97 256.33 m
> > 52.21 256.33 l
> > o
> > 45.56 115.07 (0.00) 0.50 0.00 90.00 t
> > 45.56 143.32 (0.02) 0.50 0.00 90.00 t
> > 45.56 171.57 (0.04) 0.50 0.00 90.00 t
> > 45.56 199.82 (0.06) 0.50 0.00 90.00 t
> > 45.56 228.08 (0.08) 0.50 0.00 90.00 t
> > 45.56 256.33 (0.10) 0.50 0.00 90.00 t
> > np
> > 56.97 109.42 m
> > 557.32 109.42 l
> > 557.32 261.98 l
> > 56.97 261.98 l
> > 56.97 109.42 l
> > o
> > 18.00 60.94 577.28 300.94 cl
> > /ps 10 def B 10 s
> > 0.0000 0.0000 0.0000 rgb
> > 307.14 277.87 (Empirical sig. level, chisquared_1 distribution, home-made rng) 0.50 0.00 0.00 t
> > /ps 8 def R 8 s
> > 307.14 73.30 (series length) 0.50 0.00 0.00 t
> > 56.97 109.42 557.32 261.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [ 3.00 5.00] 0 setdash
> > np
> > 56.97 185.70 m
> > 557.32 185.70 l
> > o
> > 18.00 60.94 577.28 780.94 cl
> > /ps 12 def S 12 s
> > 0.0000 0.0000 0.0000 rgb
> > 20.88 179.83 (a) 0.00 0.00 0.00 t
> > /ps 12 def R 12 s
> > 21.85 183.52 (^) 0.00 0.00 0.00 t
> > 56.97 109.42 557.32 261.98 cl
> > 0.0000 0.0000 0.0000 rgb
> > 0.75 setlinewidth
> > [] 0 setdash
> > np
> > 75.50 153.21 m
> > 75.50 162.94 l
> > o
> > np
> > 80.18 161.68 m
> > 80.18 171.42 l
> > o
> > np
> > 84.86 153.21 m
> > 84.86 162.94 l
> > o
> > np
> > 89.54 156.03 m
> > 89.54 165.77 l
> > o
> > np
> > 94.22 151.79 m
> > 94.22 161.53 l
> > o
> > np
> > 98.90 161.68 m
> > 98.90 171.42 l
> > o
> > np
> > 103.58 160.27 m
> > 103.58 170.01 l
> > o
> > np
> > 108.26 163.09 m
> > 108.26 172.83 l
> > o
> > np
> > 112.94 141.91 m
> > 112.94 151.64 l
> > o
> > np
> > 117.61 151.79 m
> > 117.61 161.53 l
> > o
> > np
> > 164.41 147.56 m
> > 164.41 157.29 l
> > o
> > np
> > 211.21 164.51 m
> > 211.21 174.24 l
> > o
> > np
> > 258.01 168.75 m
> > 258.01 178.48 l
> > o
> > np
> > 304.80 163.09 m
> > 304.80 172.83 l
> > o
> > np
> > 351.60 177.22 m
> > 351.60 186.96 l
> > o
> > np
> > 398.40 181.46 m
> > 398.40 191.19 l
> > o
> > np
> > 445.19 165.92 m
> > 445.19 175.66 l
> > o
> > np
> > 491.99 164.51 m
> > 491.99 174.24 l
> > o
> > np
> > 538.79 158.86 m
> > 538.79 168.59 l
> > o
> > np
> > 75.50 153.21 m
> > 75.50 143.47 l
> > o
> > np
> > 80.18 161.68 m
> > 80.18 151.95 l
> > o
> > np
> > 84.86 153.21 m
> > 84.86 143.47 l
> > o
> > np
> > 89.54 156.03 m
> > 89.54 146.30 l
> > o
> > np
> > 94.22 151.79 m
> > 94.22 142.06 l
> > o
> > np
> > 98.90 161.68 m
> > 98.90 151.95 l
> > o
> > np
> > 103.58 160.27 m
> > 103.58 150.53 l
> > o
> > np
> > 108.26 163.09 m
> > 108.26 153.36 l
> > o
> > np
> > 112.94 141.91 m
> > 112.94 132.17 l
> > o
> > np
> > 117.61 151.79 m
> > 117.61 142.06 l
> > o
> > np
> > 164.41 147.56 m
> > 164.41 137.82 l
> > o
> > np
> > 211.21 164.51 m
> > 211.21 154.77 l
> > o
> > np
> > 258.01 168.75 m
> > 258.01 159.01 l
> > o
> > np
> > 304.80 163.09 m
> > 304.80 153.36 l
> > o
> > np
> > 351.60 177.22 m
> > 351.60 167.49 l
> > o
> > np
> > 398.40 181.46 m
> > 398.40 171.72 l
> > o
> > np
> > 445.19 165.92 m
> > 445.19 156.18 l
> > o
> > np
> > 491.99 164.51 m
> > 491.99 154.77 l
> > o
> > np
> > 538.79 158.86 m
> > 538.79 149.12 l
> > o
> > ep
> > %%Trailer
> > %%Pages: 1
> > %%EOF
> > -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
> > r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html
> > Send "info", "help", or "[un]subscribe"
> > (in the "body", not the subject !)  To: r-help-request at stat.math.ethz.ch
> > _._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._
> >
>
>

--
Göran Broström                      tel: +46 90 786 5223
professor                           fax: +46 90 786 6614
Department of Statistics            http://www.stat.umu.se/egna/gb/
Umeå University
SE-90187 Umeå, Sweden             e-mail: gb at stat.umu.se

-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-
r-help mailing list -- Read http://www.ci.tuwien.ac.at/~hornik/R/R-FAQ.html
Send "info", "help", or "[un]subscribe"
(in the "body", not the subject !)  To: r-help-request at stat.math.ethz.ch
_._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._._

```