Package 'LocalCop' reference manual

Title:	Local Likelihood Inference for Conditional Copula Models
Description:	Implements a local likelihood estimator for the dependence parameter in bivariate conditional copula models. Copula family and local likelihood bandwidth parameters are selected by leave-one-out cross-validation. The models are implemented in 'TMB', meaning that the local score function is efficiently calculated via automated differentiation (AD), such that quasi-Newton algorithms may be used for parameter estimation.
Authors:	Elif Fidan Acar [aut], Martin Lysy [aut, cre], Alan Kuchinsky [ctb]
Maintainer:	Martin Lysy <[email protected]>
License:	GPL-3
Version:	0.0.2
Built:	2024-10-24 06:16:44 UTC
Source:	https://github.com/mlysy/localcop

Local likelihood inference for conditional copula models.

Description

Fits a bivariate conditional copula $C(u_1, u_2 | \theta_x)$ , where $\theta_x$ is a variable dependence parameter, nonparametrically estimated from a single covariate $x$ via local likelihood.

Author(s)

Maintainer: Martin Lysy [email protected]

Authors:

Elif Acar [email protected]

Other contributors:

Alan Kuchinsky [contributor]

Examples

# simulate data
set.seed(123)
family <- 5 # Frank copula
n <- 1000
x <- runif(n) # covariate values
eta_fun <- function(x) 2*cos(12*pi*x) # copula dependence parameter
eta_true <- eta_fun(x)
par_true <- BiCopEta2Par(family, eta = eta_true)
udata <- VineCopula::BiCopSim(n, family=family,
                              par = par_true$par)

# bandwidth and family selection
bandset <- c(.01, .04, .1) # bandwidth set
famset <- c(2, 5) # family set
n_loo <- 100 # number of leave-one-out observations in CV likelihood calculation
system.time({
  cvsel <- CondiCopSelect(u1= udata[,1], u2 = udata[,2],
                          x = x, family = famset, band = bandset,
                          xind = n_loo)
})

# compare estimates to true value
xseq <- cvsel$x
famsel <- cvsel$cv$family
bandsel <- cvsel$cv$band
etasel <- cvsel$eta
clrs <- c("red", "blue", "green4")
names(clrs) <- bandset

plot_fun <- function(fam) {
  nband <- length(bandset)
  if(fam == 2) {
    famind <- 1:nband
    main <- "Student-t Copula"
  } else {
    famind <- nband+1:nband
    main <- "Frank Copula"
  }
  plot(xseq, BiCopEta2Tau(family, eta = eta_fun(xseq)),
       type = "l", lwd = 2, ylim = c(-.5, .5),
       xlab = expression(x), ylab = expression(tau(x)),
       main = main)
  for(ii in famind) {
    lines(xseq, BiCopEta2Tau(fam, eta = etasel[,ii]),
          col = clrs[as.character(bandsel[ii])], lwd = 1)
  }
  legend("bottomright", fill = clrs,
         legend = paste0("band_", bandsel[famind],
                         " = ", signif(cvsel$cv$cv[famind], 3)))
}

oldpar <- par(mfrow = c(1,2))
plot_fun(2)
plot_fun(5)
par(oldpar)
# simulate data
set.seed(123)
family <- 5 # Frank copula
n <- 1000
x <- runif(n) # covariate values
eta_fun <- function(x) 2*cos(12*pi*x) # copula dependence parameter
eta_true <- eta_fun(x)
par_true <- BiCopEta2Par(family, eta = eta_true)
udata <- VineCopula::BiCopSim(n, family=family,
                              par = par_true$par)

# bandwidth and family selection
bandset <- c(.01, .04, .1) # bandwidth set
famset <- c(2, 5) # family set
n_loo <- 100 # number of leave-one-out observations in CV likelihood calculation
system.time({
  cvsel <- CondiCopSelect(u1= udata[,1], u2 = udata[,2],
                          x = x, family = famset, band = bandset,
                          xind = n_loo)
})

# compare estimates to true value
xseq <- cvsel$x
famsel <- cvsel$cv$family
bandsel <- cvsel$cv$band
etasel <- cvsel$eta
clrs <- c("red", "blue", "green4")
names(clrs) <- bandset

plot_fun <- function(fam) {
  nband <- length(bandset)
  if(fam == 2) {
    famind <- 1:nband
    main <- "Student-t Copula"
  } else {
    famind <- nband+1:nband
    main <- "Frank Copula"
  }
  plot(xseq, BiCopEta2Tau(family, eta = eta_fun(xseq)),
       type = "l", lwd = 2, ylim = c(-.5, .5),
       xlab = expression(x), ylab = expression(tau(x)),
       main = main)
  for(ii in famind) {
    lines(xseq, BiCopEta2Tau(fam, eta = etasel[,ii]),
          col = clrs[as.character(bandsel[ii])], lwd = 1)
  }
  legend("bottomright", fill = clrs,
         legend = paste0("band_", bandsel[famind],
                         " = ", signif(cvsel$cv$cv[famind], 3)))
}

oldpar <- par(mfrow = c(1,2))
plot_fun(2)
plot_fun(5)
par(oldpar)

Cross-validated likelihood.

Description

Leave-one-out local likelihood copula parameter estimates are interpolated, then used to calculate the conditional copula likelihood function.

Usage

CondiCopLikCV(
  u1,
  u2,
  family,
  x,
  xind = 100,
  degree = 1,
  eta,
  nu,
  kernel = KernEpa,
  band,
  optim_fun,
  cveta_out = FALSE,
  cv_all = FALSE,
  cl = NA
)
CondiCopLikCV(
  u1,
  u2,
  family,
  x,
  xind = 100,
  degree = 1,
  eta,
  nu,
  kernel = KernEpa,
  band,
  optim_fun,
  cveta_out = FALSE,
  cv_all = FALSE,
  cl = NA
)

Arguments

`u1`	Vector of first uniform response.
`u2`	Vector of second uniform response.
`family`	An integer defining the bivariate copula family to use. See `ConvertPar()`.
`x`	Vector of observed covariate values.
`xind`	Vector of indices in `sort(x)` at which to calculate leave-one-out parameter estimates. Can also be supplied as a single integer, in which case `xind` equally spaced observations are taken from `x`.
`degree`	Integer specifying the polynomial order of the local likelihood function. Currently only 0 and 1 are supported.
`eta`, `nu`, `kernel`, `band`, `optim_fun`, `cl`	See `CondiCopLocFit()`.
`cveta_out`	If `TRUE`, return the CV estimate of eta at each point in `x` in addition to the CV log-likelihood.
`cv_all`	If `FALSE`, evaluate the CV likelihood at only the leave-one-out observations specified by `xind`. Otherwise, interpolate the leave-one-out estimates of eta to all values in `x`, and evaluate the CV likelihood at all observations.

Value

If cveta_out = FALSE, scalar value of the cross-validated log-likelihood. Otherwise, a list with elements:

x: The sorted values of x.
eta: The leave-one-out estimates interpolated from the values in xind to all of those in x.
nu: The scalar value of the estimated (or provided) second copula parameter.
loglik: The cross-validated log-likelihood.

Local likelihood estimation.

Description

Estimate the bivariate copula dependence parameter eta at multiple covariate values.

Usage

CondiCopLocFit(
  u1,
  u2,
  family,
  x,
  x0,
  nx = 100,
  degree = 1,
  eta,
  nu,
  kernel = KernEpa,
  band,
  optim_fun,
  cl = NA
)
CondiCopLocFit(
  u1,
  u2,
  family,
  x,
  x0,
  nx = 100,
  degree = 1,
  eta,
  nu,
  kernel = KernEpa,
  band,
  optim_fun,
  cl = NA
)

Arguments

`u1`	Vector of first uniform response.
`u2`	Vector of second uniform response.
`family`	An integer defining the bivariate copula family to use. See `ConvertPar()`.
`x`	Vector of observed covariate values.
`x0`	Vector of covariate values within `range(x)` at which to fit the local likelihood. Does not have to be a subset of `x`.
`nx`	If `x0` is missing, defaults to `nx` equally spaced values in `range(x)`.
`degree`	Integer specifying the polynomial order of the local likelihood function. Currently only 0 and 1 are supported.
`eta`	Optional initial value of the copula dependence parameter (scalar). If missing will be estimated unconditionally by `VineCopula::BiCopEst()`.
`nu`	Optional initial value of second copula parameter, if it exists. If missing and required, will be estimated unconditionally by `VineCopula::BiCopEst()`. If provided and required, will not be estimated.
`kernel`	Kernel function to use. Should accept a numeric vector parameter and return a non-negative numeric vector of the same length. See `KernFun()`.
`band`	Kernal bandwidth parameter (positive scalar). See `KernWeight()`.
`optim_fun`	Optional specification of local likelihood optimization algorithm. See Details.
`cl`	Optional parallel cluster created with `parallel::makeCluster()`, in which case optimization for each element of `x0` will be done in parallel on separate cores. If `cl == NA`, computations are run serially.

Details

By default, optimization is performed with the quasi-Newton algorithm provided by stats::nlminb(), which uses gradient information provided by automatic differentiation (AD) as implemented by TMB.

If the default method is to be overridden, optim_fun should be provided as a function taking a single argument corresponding to the output of CondiCopLocFun(), and return a scalar value corresponding to the estimate of eta at a given covariate value in x0. Note that TMB calculates the negative local (log)likelihood, such that the objective function is to be minimized. See Examples.

Value

List with the following elements:

x: The vector of covariate values x0 at which the local likelihood is fit.
eta: The vector of estimated dependence parameters of the same length as x0.
nu: The scalar value of the estimated (or provided) second copula parameter.

Examples

# simulate data
family <- 5 # Frank copula
n <- 1000
x <- runif(n) # covariate values
eta_fun <- function(x) 2*cos(12*pi*x) # copula dependence parameter
eta_true <- eta_fun(x)
par_true <- BiCopEta2Par(family, eta = eta_true)
udata <- VineCopula::BiCopSim(n, family=family,
                              par = par_true$par)

# local likelihood estimation
x0 <- seq(min(x), max(x), len = 100)
band <- .02
system.time({
  eta_hat <- CondiCopLocFit(u1 = udata[,1], u2 = udata[,2],
                            family = family, x = x, x0 = x0, band = band)
})

# custom optimization routine using stats::optim (gradient-free)
my_optim <- function(obj) {
  opt <- stats::optim(par = obj$par, fn = obj$fn, method = "Nelder-Mead")
  return(opt$par[1]) # always return constant term, even if degree > 0
}
system.time({
  eta_hat2 <- CondiCopLocFit(u1 = udata[,1], u2 = udata[,2],
                             family = family, x = x, x0 = x0, band = band,
                             optim_fun = my_optim)
})

plot(x0, BiCopEta2Tau(family, eta = eta_fun(x0)), type = "l",
     xlab = expression(x), ylab = expression(tau(x)))
lines(x0, BiCopEta2Tau(family, eta = eta_hat$eta), col = "red")
lines(x0, BiCopEta2Tau(family, eta = eta_hat2$eta), col = "blue")
legend("bottomright", fill = c("black", "red", "blue"),
       legend = c("True", "optim_default", "Nelder-Mead"))
# simulate data
family <- 5 # Frank copula
n <- 1000
x <- runif(n) # covariate values
eta_fun <- function(x) 2*cos(12*pi*x) # copula dependence parameter
eta_true <- eta_fun(x)
par_true <- BiCopEta2Par(family, eta = eta_true)
udata <- VineCopula::BiCopSim(n, family=family,
                              par = par_true$par)

# local likelihood estimation
x0 <- seq(min(x), max(x), len = 100)
band <- .02
system.time({
  eta_hat <- CondiCopLocFit(u1 = udata[,1], u2 = udata[,2],
                            family = family, x = x, x0 = x0, band = band)
})

# custom optimization routine using stats::optim (gradient-free)
my_optim <- function(obj) {
  opt <- stats::optim(par = obj$par, fn = obj$fn, method = "Nelder-Mead")
  return(opt$par[1]) # always return constant term, even if degree > 0
}
system.time({
  eta_hat2 <- CondiCopLocFit(u1 = udata[,1], u2 = udata[,2],
                             family = family, x = x, x0 = x0, band = band,
                             optim_fun = my_optim)
})

plot(x0, BiCopEta2Tau(family, eta = eta_fun(x0)), type = "l",
     xlab = expression(x), ylab = expression(tau(x)))
lines(x0, BiCopEta2Tau(family, eta = eta_hat$eta), col = "red")
lines(x0, BiCopEta2Tau(family, eta = eta_hat2$eta), col = "blue")
legend("bottomright", fill = c("black", "red", "blue"),
       legend = c("True", "optim_default", "Nelder-Mead"))

Create a TMB local likelihood function.

Description

Wraps a call to TMB::MakeADFun().

Usage

CondiCopLocFun(u1, u2, family, x, x0, wgt, degree = 1, eta, nu)
CondiCopLocFun(u1, u2, family, x, x0, wgt, degree = 1, eta, nu)

Arguments

`u1`	Vector of first uniform response.
`u2`	Vector of second uniform response.
`family`	An integer defining the bivariate copula family to use. See `ConvertPar()`.
`x`	Vector of observed covariate values.
`x0`	Scalar covariate value at which to evaluate the local likelihood. Does not have to be a subset of `x`.
`wgt`	Vector of positive kernel weights.
`degree`	Integer specifying the polynomial order of the local likelihood function. Currently only 0 and 1 are supported.
`eta`	Value of the copula dependence parameter. Scalar or vector of length two, depending on whether `degree` is 0 or 1.
`nu`	Value of the other copula parameter. Scalar or vector of same length as `u1`. Ignored if `family != 2`.

Value

A list as returned by a call to TMB::MakeADFun(). In particular, this contains elements fun and gr for the negative local likelihood and its gradient with respect to eta.

Examples

# the following example shows how to create
# an unconditional copula likelihood function

# simulate data
n <- 1000 # sample size
family <- 2 # Student-t copula
rho <- runif(1, -1, 1) # unconditional dependence parameter
nu <- runif(1, 4, 20)# degrees of freedom parameter
udata <- VineCopula::BiCopSim(n, family = family, par = rho, par2 = nu)

# create likelihood function

# parameter conversion: equivalent to BiCopPar2Eta(family = 2, ...)
rho2eta <- function(rho) .5 * log((1+rho)/(1-rho))
nll_obj <- CondiCopLocFun(u1 = udata[,1], u2 = udata[,2], family = family,
                          x = rep(0, n), x0 = 0, # centered covariate x - x0 == 0
                          wgt = rep(1, n), # unweighted
                          degree = 0, # zero-order fit
                          eta = c(rho2eta(rho), 0),
                          nu = nu)

# likelihood function: recall that TMB requires a _negative_ ll
stucop_lik <- function(rho) {
  -nll_obj$fn(c(rho2eta(rho), 0))
}

# compare to VineCopula.
rhovec <- runif(50, -1, 1)
system.time({
  ll1 <- sapply(rhovec, stucop_lik) # LocalCop
})
system.time({
  ll2 <- sapply(rhovec, function(rho) {
    # VineCopula
    sum(log(VineCopula::BiCopPDF(u1 = udata[,1], u2 = udata[,2],
                                 family = family,
                                 par = rho, par2 = nu)))
  })
})

# difference between the two
range(ll1 - ll2)
# the following example shows how to create
# an unconditional copula likelihood function

# simulate data
n <- 1000 # sample size
family <- 2 # Student-t copula
rho <- runif(1, -1, 1) # unconditional dependence parameter
nu <- runif(1, 4, 20)# degrees of freedom parameter
udata <- VineCopula::BiCopSim(n, family = family, par = rho, par2 = nu)

# create likelihood function

# parameter conversion: equivalent to BiCopPar2Eta(family = 2, ...)
rho2eta <- function(rho) .5 * log((1+rho)/(1-rho))
nll_obj <- CondiCopLocFun(u1 = udata[,1], u2 = udata[,2], family = family,
                          x = rep(0, n), x0 = 0, # centered covariate x - x0 == 0
                          wgt = rep(1, n), # unweighted
                          degree = 0, # zero-order fit
                          eta = c(rho2eta(rho), 0),
                          nu = nu)

# likelihood function: recall that TMB requires a _negative_ ll
stucop_lik <- function(rho) {
  -nll_obj$fn(c(rho2eta(rho), 0))
}

# compare to VineCopula.
rhovec <- runif(50, -1, 1)
system.time({
  ll1 <- sapply(rhovec, stucop_lik) # LocalCop
})
system.time({
  ll2 <- sapply(rhovec, function(rho) {
    # VineCopula
    sum(log(VineCopula::BiCopPDF(u1 = udata[,1], u2 = udata[,2],
                                 family = family,
                                 par = rho, par2 = nu)))
  })
})

# difference between the two
range(ll1 - ll2)

Local likelihood bandwidth and/or family selection.

Description

Selects among a set of bandwidths and/or copula families the one which maximizes the cross-validated local likelihood. See CondiCopLikCV() for details.

Usage

CondiCopSelect(
  u1,
  u2,
  family,
  x,
  xind = 100,
  degree = 1,
  nu,
  kernel = KernEpa,
  band,
  nband = 6,
  optim_fun,
  cv_all = FALSE,
  full_out = TRUE,
  cl = NA
)
CondiCopSelect(
  u1,
  u2,
  family,
  x,
  xind = 100,
  degree = 1,
  nu,
  kernel = KernEpa,
  band,
  nband = 6,
  optim_fun,
  cv_all = FALSE,
  full_out = TRUE,
  cl = NA
)

Arguments

`u1`	Vector of first uniform response.
`u2`	Vector of second uniform response.
`family`	Vector of integers specifying the family set. See `ConvertPar()`.
`x`	Vector of observed covariate values.
`xind`	Specification of `xind` for each bandwidth. Can be a scalar integer, a vector of `nband` integers, or a list of `nband` vectors of integers.
`degree`	Integer specifying the polynomial order of the local likelihood function. Currently only 0 and 1 are supported.
`nu`	Optional vector of fixed `nu` parameter for each family. If missing or `NA` get estimated from the data (if required)
`kernel`, `optim_fun`, `cl`	See `CondiCopLocFit()`.
`band`	Vector of positive numbers specifying the bandwidth value set.
`nband`	If `band` is missing, automatically choose `nband` bandwidth values spanning the range of `x`.
`cv_all`	If `FALSE`, evaluate the CV likelihood at only the leave-one-out observations specified by `xind`. Otherwise, interpolate the leave-one-out estimates of eta to all values in `x`, and evaluate the CV likelihood at all observations.
`full_out`	Logical; whether or not to output all fitted models or just the selected family/bandwidth combination. See Value.

Value

If full_out = FALSE, a list with elements family and bandwidth containing the selected value of each. Otherwise, a list with the following elements:

cv: A data frame with ⁠nBF = length(band) x length(family)⁠ rows and columns named family, band, and cv containing the cross-validated likelihood evaluated at each combination of bandwidth and family values.
x: The sorted values of x.
eta: A ⁠length(x) x nBF⁠ matrix of eta estimates, the columns of which are in the same order as the rows of cv.
nu: A vector of length nBF second copula parameters, with zero if they don't exist.

Examples

# simulate data
set.seed(123)
family <- 5 # Frank copula
n <- 1000
x <- runif(n) # covariate values
eta_fun <- function(x) 2*cos(12*pi*x) # copula dependence parameter
eta_true <- eta_fun(x)
par_true <- BiCopEta2Par(family, eta = eta_true)
udata <- VineCopula::BiCopSim(n, family=family,
                              par = par_true$par)

# bandwidth and family selection
bandset <- c(.01, .04, .1) # bandwidth set
famset <- c(2, 5) # family set
n_loo <- 100 # number of leave-one-out observations in CV likelihood calculation
system.time({
  cvsel <- CondiCopSelect(u1= udata[,1], u2 = udata[,2],
                          x = x, family = famset, band = bandset,
                          xind = n_loo)
})

# compare estimates to true value
xseq <- cvsel$x
famsel <- cvsel$cv$family
bandsel <- cvsel$cv$band
etasel <- cvsel$eta
clrs <- c("red", "blue", "green4")
names(clrs) <- bandset

plot_fun <- function(fam) {
  nband <- length(bandset)
  if(fam == 2) {
    famind <- 1:nband
    main <- "Student-t Copula"
  } else {
    famind <- nband+1:nband
    main <- "Frank Copula"
  }
  plot(xseq, BiCopEta2Tau(family, eta = eta_fun(xseq)),
       type = "l", lwd = 2, ylim = c(-.5, .5),
       xlab = expression(x), ylab = expression(tau(x)),
       main = main)
  for(ii in famind) {
    lines(xseq, BiCopEta2Tau(fam, eta = etasel[,ii]),
          col = clrs[as.character(bandsel[ii])], lwd = 1)
  }
  legend("bottomright", fill = clrs,
         legend = paste0("band_", bandsel[famind],
                         " = ", signif(cvsel$cv$cv[famind], 3)))
}

oldpar <- par(mfrow = c(1,2))
plot_fun(2)
plot_fun(5)
par(oldpar)
# simulate data
set.seed(123)
family <- 5 # Frank copula
n <- 1000
x <- runif(n) # covariate values
eta_fun <- function(x) 2*cos(12*pi*x) # copula dependence parameter
eta_true <- eta_fun(x)
par_true <- BiCopEta2Par(family, eta = eta_true)
udata <- VineCopula::BiCopSim(n, family=family,
                              par = par_true$par)

# bandwidth and family selection
bandset <- c(.01, .04, .1) # bandwidth set
famset <- c(2, 5) # family set
n_loo <- 100 # number of leave-one-out observations in CV likelihood calculation
system.time({
  cvsel <- CondiCopSelect(u1= udata[,1], u2 = udata[,2],
                          x = x, family = famset, band = bandset,
                          xind = n_loo)
})

# compare estimates to true value
xseq <- cvsel$x
famsel <- cvsel$cv$family
bandsel <- cvsel$cv$band
etasel <- cvsel$eta
clrs <- c("red", "blue", "green4")
names(clrs) <- bandset

plot_fun <- function(fam) {
  nband <- length(bandset)
  if(fam == 2) {
    famind <- 1:nband
    main <- "Student-t Copula"
  } else {
    famind <- nband+1:nband
    main <- "Frank Copula"
  }
  plot(xseq, BiCopEta2Tau(family, eta = eta_fun(xseq)),
       type = "l", lwd = 2, ylim = c(-.5, .5),
       xlab = expression(x), ylab = expression(tau(x)),
       main = main)
  for(ii in famind) {
    lines(xseq, BiCopEta2Tau(fam, eta = etasel[,ii]),
          col = clrs[as.character(bandsel[ii])], lwd = 1)
  }
  legend("bottomright", fill = clrs,
         legend = paste0("band_", bandsel[famind],
                         " = ", signif(cvsel$cv$cv[famind], 3)))
}

oldpar <- par(mfrow = c(1,2))
plot_fun(2)
plot_fun(5)
par(oldpar)

Conversions between various bivariate copula parametrizations.

Description

Conversions between various bivariate copula parametrizations.

Usage

BiCopEta2Par(family, eta, eta2 = 0)

BiCopPar2Eta(family, par, par2 = 0)

BiCopEta2Tau(family, eta, eta2 = 0)

BiCopTau2Eta(family, tau)
BiCopEta2Par(family, eta, eta2 = 0)

BiCopPar2Eta(family, par, par2 = 0)

BiCopEta2Tau(family, eta, eta2 = 0)

BiCopTau2Eta(family, tau)

Arguments

`family`	An integer defining the bivariate copula family to use. See Details.
`eta`, `eta2`	Vector of parameters on the `eta` scale. See Details.
`par`, `par2`	Vector of parameters on the `par` scale.
`tau`	Vector of parameters on the `tau` scale.

Details

The copula family integer codes are identical to those of the VineCopula package. Currently, the following families are implemented:

1: Gaussian copula.
2: Student-t copula.
3: Clayton copula.
4: Gumbel copula.
5: Frank copula.
13: Clayton copula – rotated 180 degrees.
14: Gumbel copula – rotated 180 degrees.
23: Clayton copula – rotated 90 degrees.
24: Gumbel copula – rotated 90 degrees.
33: Clayton copula – rotated 270 degrees.
34: Gumbel copula – rotated 270 degrees.

Value

Vector of converted parameters.

Local likelihood kernel functions.

Description

Local likelihood kernel functions.

Usage

KernEpa(t)

KernGaus(t)

KernBeta(t, par = 0.5)

KernBiQuad(t)

KernTriAng(t)
KernEpa(t)

KernGaus(t)

KernBeta(t, par = 0.5)

KernBiQuad(t)

KernTriAng(t)

Arguments

`t`	Vector of distances from mode of kernel.
`par`	Shape parameter for Beta kernel (positive scalar).

Details

Describe kernels here.

Value

Vector of kernel weights.

Calculate local likelihood kernel weights.

Description

Calculate local likelihood kernel weights.

Usage

KernWeight(x, x0, band, kernel = KernEpa, band_type = "constant")
KernWeight(x, x0, band, kernel = KernEpa, band_type = "constant")

Arguments

`x`	Vector of observed covariate values.
`x0`	Scalar covariate value at which local likelihood estimation is performed.
`band`	Kernel bandwidth parameter (positive scalar). See Details.
`kernel`	Kernel function to use. Should accept a numeric vector parameter and return a non-negative numeric vector of the same length. See `KernFun()`.
`band_type`	A character string specifying the type of bandwidth: either "constant" or "variable". See Details.

Details

For the constant bandwidth of size band = h, the weights are calculated as

wgt = kernel((x-x0) / h) / h

where kernel is the kernel function. For bandwidth type "variable", a fixed fraction band of observations is used, i.e,

h = sort( abs(x-x0) )[ floor(band*length(x)) ]

Value

A vector of nonnegative kernel weights of the same length as x.

Examples

x <- sort(runif(20))
x0 <- runif(1, min = min(x), max= max(x))
KernWeight(x, x0, band=0.3, kernel = KernEpa, band_type = "constant")
KernWeight(x, x0, band=0.3, kernel = KernEpa, band_type = "variable")
x <- sort(runif(20))
x0 <- runif(1, min = min(x), max= max(x))
KernWeight(x, x0, band=0.3, kernel = KernEpa, band_type = "constant")
KernWeight(x, x0, band=0.3, kernel = KernEpa, band_type = "variable")

Package 'LocalCop'

Help Index

Local likelihood inference for conditional copula models.

Description

Author(s)

See Also

Examples

Cross-validated likelihood.

Description

Usage

Arguments

Value

See Also

Local likelihood estimation.

Description

Usage

Arguments

Details

Value

Examples

Create a TMB local likelihood function.

Description

Usage

Arguments

Value

Examples

Local likelihood bandwidth and/or family selection.

Description

Usage

Arguments

Value

Examples

Conversions between various bivariate copula parametrizations.

Description

Usage

Arguments

Details

Value

Local likelihood kernel functions.

Description

Usage

Arguments

Details

Value

Calculate local likelihood kernel weights.

Description

Usage

Arguments

Details

Value

Examples