psdengpd function

P-Splines Density Estimate and GPD Tail Extreme Value Mixture Model

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.

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)

Returns

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.

Note

Unlike most of the other extreme value mixture model functions the psdengpd functions have not been vectorised as this is not appropriate. The main inputs (x, p or q) must be either a scalar or a vector, which also define the output length. The B-splines coefficients beta and knots design.knots are vectors.

Default values are provided for P-spline inputs of degree and nseg only, but all others must be provided by the user. The default sample size for rpsdengpd 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 permitted for the parameters/B-spline criteria.

Due to symmetry, the lower tail can be described by GPD by negating the quantiles.

Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.

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)
## End(Not run)

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

Author(s)

Alfadino Akbar and Carl Scarrott carl.scarrott@canterbury.ac.nz .