View source: R/extremal_types_theorem.R
ett  R Documentation 
A movie to illustrate the extremal types theorem, that is, convergence
of the distribution of the maximum of a random sample of size n
from certain distributions to a member of the Generalized Extreme Value
(GEV) family, as n
tends to infinity.
Samples of size n
are simulated repeatedly from the chosen
distribution. The distributions (simulated empirical and true) of the
sample maxima are compared to the relevant GEV limit.
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,
...
)
n 
An integer scalar. The size of the samples drawn from the
distribution chosen using 
distn 
A character scalar specifying the distribution from which
observations are sampled. Distributions If The The The other cases use the distributional functions in the

params 
A named list of additional arguments to be passed to the
density function associated with distribution If a parameter value is not supplied then the default values in the
relevant distributional function set using 
panel_plot 
A logical parameter that determines whether the plot
is placed inside the panel ( 
hscale , vscale 
Numeric scalars. Scaling parameters for the size
of the plot when 
n_add 
An integer scalar. The number of simulated datasets to add to each new frame of the movie. 
delta_n 
A numeric scalar. The amount by which n is increased (or decreased) after one click of the + (or ) button in the parameter window. 
arrow 
A logical scalar. Should an arrow be included to show the simulated sample maximum from the top plot being placed into the bottom plot? 
leg_cex 
The argument 
... 
Additional arguments to the rpanel functions

Loosely speaking, a consequence of the
Extremal Types Theorem
is that, in many situations, the maximum of a large number
n
of independent random variables has approximately a
GEV(\mu, \sigma, \xi)
) distribution, where \mu
is a location
parameter, \sigma
is a scale parameter and \xi
is a shape
parameter. See Coles (2001) for an introductory account and
Leadbetter et al (1983) for greater detail and more examples.
The Extremal Types Theorem is an asymptotic result that considers the
possible limiting distribution of linearly normalised maxima
as n
tends to infinity.
This movie considers examples where this limiting result holds and
illustrates graphically the closeness of the limiting approximation
provided by the relevant GEV limit to the true finiten
distribution.
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 \xi
that applies in the limiting
case is used. The GEV location \mu
and scale
\sigma
are set based on constants used to normalise the maxima
to achieve the GEV limit.
Specifically, \mu
is set at the 100(11/n
)% quantile of the
distribution distn
and \sigma
at
(1 / n
) / f(\mu)
, where f
is the
density function of the distribution 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
n
exceeds 100000.
There is a box that can be used to display only the bottom
plot. This option is selected automatically if the sample size
n
exceeds 100000.
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"
, "chisquared"
, "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
).
Nothing is returned, only the animation is produced.
Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, SpringerVerlag, London. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/9781447136750_3")}
Leadbetter, M., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. SpringerVerlag, New York. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/9781461254492")}
movies
: a userfriendly menu panel.
smovie
: general information about smovie.
# 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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