
A movie to illustrate the extremal types theorem, that is, convergence
of the distribution of the maximum of a random sample of size
ett(
n = 20,
distn,
params = list(),
panel_plot = TRUE,
hscale = NA,
vscale = hscale,
n_add = 1,
delta_n = 1,
arrow = TRUE,
leg_cex = 1.25,
...
)
An integer scalar. The size of the samples drawn from the
distribution chosen using distn
. n
must be no smaller
than 2.
A character scalar specifying the distribution from which
observations are sampled. Distributions "beta"
,
"cauchy"
, "chisq"
, "chi-squared"
,
"exponential"
, "f"
, "gamma"
, "gp"
,
"lognormal"
, "log-normal"
, "ngev"
, "normal"
,
"t"
, "uniform"
and "weibull"
are recognised, case
being ignored.
If distn
is not supplied then distn = "exponential"
is used.
The "gp"
case uses the gp
distributional functions in the
revdbayes
package.
The "ngev"
case is a negated GEV(1 / gev
distributional functions in the
revdbayes
package.
If
The other cases use the distributional functions in the
stats-package
.
If distn = "gamma"
then the (shape, rate)
parameterisation is used. If scale
is supplied via params
then rate
is inferred from this.
If distn = "beta"
then ncp
is forced to be zero.
A named list of additional arguments to be passed to the
density function associated with distribution distn
.
The (shape, rate)
parameterisation is used for the gamma
distribution (see GammaDist
) even if the value of
the scale
parameter is set using params
.
If a parameter value is not supplied then the default values in the
relevant distributional function set using distn
are used,
except for
"beta"
(shape1 = 2, shape2 = 2
),
"chisq"
(df = 4
),
"f"
(df1 = 4, df2 = 8
),
"ngev"
(shape = 0.2
).
"gamma"
(shape = 2
,
"gp"
(shape = 0.1
),
"t"
(df = 4
) and
"weibull"
(shape = 2
).
A logical parameter that determines whether the plot
is placed inside the panel (TRUE
) or in the standard graphics
window (FALSE
). If the plot is to be placed inside the panel
then the tkrplot library is required.
Numeric scalars. Scaling parameters for the size
of the plot when panel_plot = TRUE
. The default values are 1.4 on
Unix platforms and 2 on Windows platforms.
An integer scalar. The number of simulated datasets to add to each new frame of the movie.
A numeric scalar. The amount by which n is increased (or decreased) after one click of the + (or -) button in the parameter window.
A logical scalar. Should an arrow be included to show the simulated sample maximum from the top plot being placed into the bottom plot?
The argument cex
to legend
.
Allows the size of the legend to be controlled manually.
Additional arguments to the rpanel functions
rp.button
and
rp.doublebutton
, not including panel
,
variable
, title
, step
, action
, initval
,
range
.
Nothing is returned, only the animation is produced.
Loosely speaking, a consequence of the
Extremal Types Theorem
is that, in many situations, the maximum of a large number
Samples of size n
are repeatedly simulated from the distribution
chosen using distn
. These samples are summarized using a histogram
that appears at the top of the movie screen. For each sample the maximum
of these n
values is calculated, stored and added to another plot,
situated below the first plot.
A rug
is added to a histogram provided that it
contains no more than 1000 points.
This plot is either a histogram or an empirical c.d.f., chosen using a
radio button.
The probability density function (p.d.f.) of the original
variables is superimposed on the top histogram.
There is a checkbox to add to the bottom plot the exact p.d.f./c.d.f. of
the sample maxima and an approximate (large n
) GEV p.d.f./c.d.f.
implied by the ETT.
The GEV shape parameter distn
and distn
.
Once it starts, four aspects of this movie are controlled by the user.
There are buttons to increase (+) or decrease (-) the sample size, that is, the number of values over which a maximum is calculated.
Each time the button labelled "simulate another n_add
samples of size n" is clicked n_add
new samples are simulated
and their sample maxima are added to the bottom histogram.
There is a button to switch the bottom plot from displaying a histogram of the simulated maxima, the exact p.d.f. and the limiting GEV p.d.f. to the empirical c.d.f. of the simulated data, the exact c.d.f. and the limiting GEV c.d.f.
There is a box that can be used to display only the bottom
plot. This option is selected automatically if the sample size
There is a box that can be used to display only the bottom
plot. This option is selected automatically if the sample size
For further detail about the examples specified by distn
see Chapter 1 of Leadbetter et al. (1983) and Chapter 3 of
Coles (2001). In many of these examples
("exponential"
, "normal"
, "gamma"
,
"lognormal"
, "chi-squared"
, "weibull"
, "ngev"
)
the limiting GEV distribution has a shape
parameter that is equal to 0. In the "uniform"
case the limiting
shape parameter is -1 and in the "beta"
case it is
-1 / shape2
, where shape2
is the
second parameter of the Beta
distribution.
In the other cases the limiting shape parameter is positive,
with respective values shape
("gp"
, see gp
),
1 / df
("t"
, see TDist
),
1 ("cauchy"
, see Cauchy
),
2 / df2
("f"
, see FDist
).
Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. http://dx.doi.org/10.1007/978-1-4471-3675-0_3
Leadbetter, M., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4612-5449-2
movies
: a user-friendly menu panel.
smovie
: general information about smovie.
# NOT RUN {
# Exponential data: xi = 0
ett()
# Uniform data: xi =-1
ett(distn = "uniform")
# Student t data: xi = 1 / df
ett(distn = "t", params = list(df = 5))
# }
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