evmix (version 2.12)

psdengpd: P-Splines Density Estimate 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 P-splines density estimate for bulk distribution upto the threshold and conditional GPD above threshold. The parameters are the B-spline coefficients beta (and associated features), threshold u GPD scale sigmau and shape xi and tail fraction phiu.

Usage

dpsdengpd(x, beta = NULL, nbinwidth = NULL, xrange = NULL,
  nseg = 10, degree = 3, u = NULL, sigmau = NULL, xi = 0,
  phiu = TRUE, design.knots = NULL, log = FALSE)
ppsdengpd(q, beta = NULL, nbinwidth = NULL, xrange = NULL,
  nseg = 10, degree = 3, u = NULL, sigmau = NULL, xi = 0,
  phiu = TRUE, design.knots = NULL, lower.tail = TRUE)
qpsdengpd(p, beta = NULL, nbinwidth = NULL, xrange = NULL,
  nseg = 10, degree = 3, u = NULL, sigmau = NULL, xi = 0,
  phiu = TRUE, design.knots = NULL, lower.tail = TRUE)
rpsdengpd(n = 1, beta = NULL, nbinwidth = NULL, xrange = NULL,
  nseg = 10, degree = 3, u = NULL, sigmau = NULL, xi = 0,
  phiu = TRUE, design.knots = NULL)

Arguments

x

quantiles

beta

vector of B-spline coefficients (required)

nbinwidth

scaling to convert count frequency into proper density

xrange

vector of minimum and maximum of B-spline (support of density)

nseg

number of segments between knots

degree

degree of B-splines (0 is constant, 1 is linear, etc.)

u

threshold

sigmau

scale parameter (positive)

xi

shape parameter

phiu

probability of being above threshold \([0, 1]\) or TRUE

design.knots

spline knots for splineDesign function

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)

Value

dpsdengpd gives the density, ppsdengpd gives the cumulative distribution function, qpsdengpd gives the quantile function and rpsdengpd gives a random sample.

Details

Extreme value mixture model combining P-splines density estimate 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 \(\phi_u\). Alternatively, when phiu=TRUE the tail fraction is estimated as the tail fraction from the KDE bulk model.

The cumulative distribution function with tail fraction \(\phi_u\) defined by the upper tail fraction of the P-splines density estimate (phiu=TRUE), upto the threshold \(x \le u\), 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 P-splines density estimate and conditional GPD cumulative distribution functions respectively.

The cumulative distribution function for pre-specified \(\phi_u\), upto the threshold \(x \le u\), is given by: $$F(x) = (1 - \phi_u) H(x)/H(u)$$ and above the threshold \(x > u\): $$F(x) = \phi_u + [1 - \phi_u] G(x)$$ Notice that these definitions are equivalent when \(\phi_u = 1 - H(u)\).

See gpd for details of GPD upper tail component. The specification of the underlying B-splines and the P-splines density estimator are discussed in the psden function help.

References

http://en.wikipedia.org/wiki/B-spline

http://statweb.lsu.edu/faculty/marx/

http://en.wikipedia.org/wiki/Generalized_Pareto_distribution

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

Eilers, P.H.C. and Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science 11(2), 89-121.

See Also

psden and fpsden.

Other psden: fpsdengpd, fpsden, psden

Other psdengpd: fpsdengpd, psden

Other fpsdengpd: fpsdengpd

Examples

# NOT RUN {
set.seed(1)
par(mfrow = c(1, 1))

x = rnorm(1000)
xx = seq(-6, 6, 0.01)
y = dnorm(xx)

# Plenty of histogram bins (100)
breaks = seq(-4, 4, length.out=101)

# P-spline fitting with cubic B-splines, 2nd order penalty and 8 internal segments
# CV search for penalty coefficient. 
fit = fpsdengpd(x, lambdaseq = 10^seq(-5, 5, 0.25), breaks = breaks,
             xrange = c(-4, 4), nseg = 10, degree = 3, ord = 2)
hist(x, freq = FALSE, breaks = seq(-4, 4, length.out=101), xlim = c(-6, 6))

# P-splines only
with(fit, lines(xx, dpsden(xx, beta, nbinwidth, design = design.knots), lwd = 2, col = "blue"))

# P-splines+GPD
with(fit, lines(xx, dpsdengpd(xx, beta, nbinwidth, design = design.knots, 
   u = u, sigmau = sigmau, xi = xi, phiu = phiu), lwd = 2, col = "red"))
abline(v = fit$u, col = "red")

legend("topleft", c("True Density","P-spline density", "P-spline+GPD"),
 col=c("black", "blue", "red"), lty = 1)
# }
# NOT RUN {
# }