[R] Extreme value distributions (Long.)

Rolf Turner rolf at math.unb.ca
Mon Mar 25 20:41:01 CET 2002


This may not actually be an R/Splus problem, but it started
off that way .....

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Executive summary:
==================

Simulations involving extreme value distributions seem to ``work''
when the underlying distribution is exponential(1) or exponential(2)
== chi-squared_2, but NOT when the underlying distribution is
chi-squared_1.

Can anyone make an educated conjecture as to why?
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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
comments or suggestions?

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

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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
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