On Mon, 25 Mar 2002, Rolf Turner wrote:
> 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,
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?
G?ran
> 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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> 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
>
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--
G?ran Brostr?m tel: +46 90 786 5223
professor fax: +46 90 786 6614
Department of Statistics http://www.stat.umu.se/egna/gb/
Ume? University
SE-90187 Ume?, Sweden e-mail: gb at stat.umu.se
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