gkgcon: Kernel Density Estimate and GPD Both Upper and Lower Tails...
In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation

Description Usage Arguments Details Value Acknowledgments Note Author(s) References See Also Examples

Density, cumulative distribution function, quantile function and random number generation for the extreme value mixture model with kernel density estimate for bulk distribution between thresholds and conditional GPD beyond thresholds and continuity at both of them. The parameters are the kernel bandwidth lambda, lower tail (threshold ul, GPD shape xil and tail fraction phiul) and upper tail (threshold ur, GPD shape xiR and tail fraction phiur).

dgkgcon(x, kerncentres, lambda = NULL,
  ul = as.vector(quantile(kerncentres, 0.1)), xil = 0, phiul = TRUE,
  ur = as.vector(quantile(kerncentres, 0.9)), xir = 0, phiur = TRUE,
  bw = NULL, kernel = "gaussian", log = FALSE)

pgkgcon(q, kerncentres, lambda = NULL,
  ul = as.vector(quantile(kerncentres, 0.1)), xil = 0, phiul = TRUE,
  ur = as.vector(quantile(kerncentres, 0.9)), xir = 0, phiur = TRUE,
  bw = NULL, kernel = "gaussian", lower.tail = TRUE)

qgkgcon(p, kerncentres, lambda = NULL,
  ul = as.vector(quantile(kerncentres, 0.1)), xil = 0, phiul = TRUE,
  ur = as.vector(quantile(kerncentres, 0.9)), xir = 0, phiur = TRUE,
  bw = NULL, kernel = "gaussian", lower.tail = TRUE)

rgkgcon(n = 1, kerncentres, lambda = NULL,
  ul = as.vector(quantile(kerncentres, 0.1)), xil = 0, phiul = TRUE,
  ur = as.vector(quantile(kerncentres, 0.9)), xir = 0, phiur = TRUE,
  bw = NULL, kernel = "gaussian")

`x`	quantiles
`kerncentres`	kernel centres (typically sample data vector or scalar)
`lambda`	bandwidth for kernel (as half-width of kernel) or `NULL`
`ul`	lower tail threshold
`xil`	lower tail GPD shape parameter
`phiul`	probability of being below lower threshold [0, 1] or `TRUE`
`ur`	upper tail threshold
`xir`	upper tail GPD shape parameter
`phiur`	probability of being above upper threshold [0, 1] or `TRUE`
`bw`	bandwidth for kernel (as standard deviations of kernel) or `NULL`
`kernel`	kernel name (`default = "gaussian"`)
`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)

Extreme value mixture model combining kernel density estimate (KDE) for the bulk between thresholds and GPD beyond thresholds and continuity at both of them.

The user can pre-specify phiul and phiur permitting a parameterised value for the tail fractions φ_ul and φ_ur. Alternatively, when phiul=TRUE and phiur=TRUE the tail fractions are estimated as the tail fractions from the KDE bulk model.

The alternate bandwidth definitions are discussed in the kernels, with the lambda as the default. The bw specification is the same as used in the density function.

The possible kernels are also defined in kernels with the "gaussian" as the default choice.

Notice that the tail fraction cannot be 0 or 1, and the sum of upper and lower tail fractions phiul + phiur < 1, so the lower threshold must be less than the upper, ul < ur.

The cumulative distribution function has three components. The lower tail with tail fraction φ_{ul} defined by the KDE bulk model (phiul=TRUE) upto the lower threshold x < u_l:

F(x) = H(u_l) [1 - G_l(x)].

where H(x) is the kernel density estimator cumulative distribution function (i.e. mean(pnorm(x, kerncentres, bw)) and G_l(X) is the conditional GPD cumulative distribution function with negated x value and threshold, i.e. pgpd(-x, -ul, sigmaul, xil, phiul). The KDE bulk model between the thresholds u_l ≤ x ≤ u_r given by:

F(x) = H(x).

Above the threshold x > u_r the usual conditional GPD:

F(x) = H(u_r) + [1 - H(u_r)] G_r(x)

where G_r(X) is the GPD cumulative distribution function, i.e. pgpd(x, ur, sigmaur, xir, phiur).

The cumulative distribution function for the pre-specified tail fractions φ_{ul} and φ_{ur} is more complicated. The unconditional GPD is used for the lower tail x < u_l:

F(x) = φ_{ul} [1 - G_l(x)].

The KDE bulk model between the thresholds u_l ≤ x ≤ u_r given by:

F(x) = φ_{ul}+ (1-φ_{ul}-φ_{ur}) (H(x) - H(u_l)) / (H(u_r) - H(u_l)).

Above the threshold x > u_r the usual conditional GPD:

F(x) = (1-φ_{ur}) + φ_{ur} G(x)

Notice that these definitions are equivalent when φ_{ul} = H(u_l) and φ_{ur} = 1 - H(u_r).

The continuity constraint at ur means that:

φ_{ur} g_r(x) = (1-φ_{ul}-φ_{ur}) h(u_r)/ (H(u_r) - H(u_l)).

By rearrangement, the GPD scale parameter sigmaur is then:

σ_ur = φ_{ur} (H(u_r) - H(u_l))/ h(u_r) (1-φ_{ul}-φ_{ur}).

where h(x), g_l(x) and g_r(x) are the KDE and conditional GPD density functions for lower and upper tail respectively. In the special case of where the tail fraction is defined by the bulk model this reduces to

σ_ur = [1-H(u_r)] / h(u_r)

The continuity constraint at ul means that:

φ_{ul} g_l(x) = (1-φ_{ul}-φ_{ur}) h(u_l)/ (H(u_r) - H(u_l)).

The GPD scale parameter sigmaul is replaced by:

σ_ul = φ_{ul} (H(u_r) - H(u_l))/ h(u_l) (1-φ_{ul}-φ_{ur}).

In the special case of where the tail fraction is defined by the bulk model this reduces to

σ_ul = H(u_l)/ h(u_l)

If no bandwidth is provided lambda=NULL and bw=NULL then the normal reference rule is used, using the bw.nrd0 function, which is consistent with the density function. At least two kernel centres must be provided as the variance needs to be estimated.

See gpd for details of GPD upper tail component and dkden for details of KDE bulk component.

dgkgcon gives the density, pgkgcon gives the cumulative distribution function, qgkgcon gives the quantile function and rgkgcon gives a random sample.

Based on code by Anna MacDonald produced for MATLAB.

Unlike most of the other extreme value mixture model functions the gkgcon 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 kerncentres can also be a scalar or vector.

The kernel centres kerncentres can either be a single datapoint or a vector of data. The kernel centres (kerncentres) and locations to evaluate density (x) and cumulative distribution function (q) would usually be different.

Default values are provided for all inputs, except for the fundamentals kerncentres, x, q and p. The default sample size for rgkgcon 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 or kernel centres.

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.

Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz.

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

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

Bowman, A.W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71(2), 353-360.

Duin, R.P.W. (1976). On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Transactions on Computers C25(11), 1175-1179.

MacDonald, A., Scarrott, C.J., Lee, D., Darlow, B., Reale, M. and Russell, G. (2011). A flexible extreme value mixture model. Computational Statistics and Data Analysis 55(6), 2137-2157.

Wand, M. and Jones, M.C. (1995). Kernel Smoothing. Chapman && Hall.

kernels, kfun, density, bw.nrd0 and dkde in ks package.

Other kdengpdcon: bckdengpdcon, fbckdengpdcon, fgkgcon, fkdengpdcon, fkdengpd, kdengpdcon, kdengpd

Other gkg: fgkgcon, fgkg, fkdengpd, gkg, kdengpd, kden

Other gkgcon: fgkgcon, fgkg, fkdengpdcon, gkg, kdengpdcon

Other bckdengpdcon: bckdengpdcon, bckdengpd, bckden, fbckdengpdcon, fbckdengpd, fbckden, fkdengpdcon, kdengpdcon

Other fgkgcon: fgkgcon

## Not run: 
set.seed(1)
par(mfrow = c(2, 2))

kerncentres=rnorm(1000,0,1)
x = rgkgcon(1000, kerncentres, phiul = 0.15, phiur = 0.15)
xx = seq(-6, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-6, 6))
lines(xx, dgkgcon(xx, kerncentres, phiul = 0.15, phiur = 0.15))

# three tail behaviours
plot(xx, pgkgcon(xx, kerncentres), type = "l")
lines(xx, pgkgcon(xx, kerncentres,xil = 0.3, xir = 0.3), col = "red")
lines(xx, pgkgcon(xx, kerncentres,xil = -0.3, xir = -0.3), col = "blue")
legend("topleft", paste("Symmetric xil=xir=",c(0, 0.3, -0.3)),
  col=c("black", "red", "blue"), lty = 1)

# asymmetric tail behaviours
x = rgkgcon(1000, kerncentres, xil = -0.3, phiul = 0.1, xir = 0.3, phiur = 0.1)
xx = seq(-6, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-6, 6))
lines(xx, dgkgcon(xx, kerncentres, xil = -0.3, phiul = 0.1, xir = 0.3, phiur = 0.1))

plot(xx, dgkgcon(xx, kerncentres, xil = -0.3, phiul = 0.2, xir = 0.3, phiur = 0.2),
  type = "l", ylim = c(0, 0.4))
lines(xx, dgkgcon(xx, kerncentres, xil = -0.3, phiul = 0.3, xir = 0.3, phiur = 0.3),
  col = "red")
lines(xx, dgkgcon(xx, kerncentres, xil = -0.3, phiul = TRUE, xir = 0.3, phiur = TRUE),
  col = "blue")
legend("topleft", c("phiul = phiur = 0.2", "phiul = phiur = 0.3", "Bulk Tail Fraction"),
  col=c("black", "red", "blue"), lty = 1)

## End(Not run)