R: Mixture of Gammas Bulk and GPD Tail Extreme Value Mixture...
mgammagpd
R Documentation
Mixture of Gammas Bulk and GPD Tail Extreme Value Mixture Model
Description
Density, cumulative distribution function, quantile function and
random number generation for the extreme value mixture model with mixture of gammas for bulk
distribution upto the threshold and conditional GPD above threshold. The parameters
are the multiple gamma shapes mgshape, scales mgscale and mgweights, threshold u
GPD scale sigmau and shape xi and tail fraction phiu.
mgamma weights (positive) as list or vector (NULL for equi-weighted)
u
threshold
sigmau
scale parameter (positive)
xi
shape parameter
phiu
probability of being above threshold [0, 1] or TRUE
log
logical, if TRUE then log density
q
quantiles
lower.tail
logical, if FALSE then upper tail probabilities
p
cumulative probabilities
n
sample size (positive integer)
Details
Extreme value mixture model combining mixture of gammas for the bulk
below the threshold and GPD for upper tail.
The user can pre-specify phiu permitting a parameterised value for the tail
fraction φ_u. Alternatively, when phiu=TRUE the tail fraction is
estimated as the tail fraction from the mixture of gammas bulk model.
Suppose there are M>=1 gamma components in the mixture model. If you
wish to have a single (scalar) value for each parameter within each of the
M components then these can be input as a vector of length M. If
you wish to input a vector of values for each parameter within each of the
M components, then they are input as a list with each entry the
parameter object for each component (which can either be a scalar or
vector as usual). No matter whether they are input as a vector or list there
must be M elements in mgshape and mgscale, one for each
gamma mixture component. Further, any vectors in the list of parameters must
of the same length of the x, q, p or equal to the sample size n, where
relevant.
If mgweight=NULL then equal weights for each component are assumed. Otherwise,
mgweight must be a list of the same length as mgshape and
mgscale, filled with positive values. In the latter case, the weights are rescaled
to sum to unity.
The cumulative distribution function with tail fraction φ_u defined by the
upper tail fraction of the mixture of gammas bulk model (phiu=TRUE), upto the
threshold 0 < x ≤ u, given by:
F(x) = H(x)
and above the threshold x > u:
F(x) = H(u) + [1 - H(u)] G(x)
where H(x) and G(X) are the mixture of gammas and conditional GPD
cumulative distribution functions.
The cumulative distribution function for pre-specified φ_u, upto the
threshold 0 < x ≤ u, is given by:
F(x) = (1 - φ_u) H(x)/H(u)
and above the threshold x > u:
F(x) = φ_u + [1 - φ_u] G(x)
Notice that these definitions are equivalent when φ_u = 1 - H(u).
The gamma is defined on the non-negative reals, so the threshold must be positive.
Though behaviour at zero depends on the shape (α):
f(0+)=∞ for 0<α<1;
f(0+)=1/β for α=1 (exponential);
f(0+)=0 for α>1;
where β is the scale parameter.
See gammagpd for details of simpler parametric mixture model
with single gamma for bulk component and GPD for upper tail.
Value
dmgammagpd gives the density,
pmgammagpd gives the cumulative distribution function,
qmgammagpd gives the quantile function and
rmgammagpd gives a random sample.
Acknowledgments
Thanks to Daniela Laas, University of St Gallen, Switzerland for reporting various bugs in these functions.
Note
All inputs are vectorised except log and lower.tail, and the gamma mixture
parameters can be vectorised within the list. The main inputs (x, p or q)
and parameters must be either a scalar or a vector. If vectors are provided they must all be
of the same length, and the function will be evaluated for each element of vector. In the case of
rmgammagpd any input vector must be of length n.
Default values are provided for all inputs, except for the fundamentals
x, q and p. The default sample size for
rmgammagpd is 1.
Missing (NA) and Not-a-Number (NaN) values in x,
p and q are passed through as is and infinite values are set to
NA. None of these are not permitted for the parameters.
Error checking of the inputs (e.g. invalid probabilities) is carried out and
will either stop or give warning message as appropriate.
McLachlan, G.J. and Peel, D. (2000). Finite Mixture Models. Wiley.
Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value
threshold estimation and uncertainty quantification. REVSTAT - Statistical
Journal 10(1), 33-59. Available from http://www.ine.pt/revstat/pdf/rs120102.pdf
do Nascimento, F.F., Gamerman, D. and Lopes, H.F. (2011). A semiparametric
Bayesian approach to extreme value estimation. Statistical Computing, 22(2), 661-675.