k1fun {PPMiss}

Constant K1

Description

Calculates an estimate for the constant K_1 given by

K_1 = \int_0^1\int_0^1\frac{1}{l_0(u)l_n(v)}\lim_{\theta\to a}\frac{\partial C_{\theta}(u,v)}{\partial\theta}\,dudv,

where l_m(x):= F_m'\big(F_m^{(-1)}(x)\big), a is such that C_a(u,v)=uv (the product copula), and \{F_n\}_{n \geq 0} is a sequence of absolutely continuous distribution functions.

Usage

k1fun(dCdtheta, fun, data, empirical, mean = 0, sd = 1)

Arguments

dCdtheta

a function providing the limit as \theta \to a of the copula derivative with respect to \theta. For the readily available copulas, namely, frank, amh, fgm and gauss, a=0.

fun

optionally, a function providing an estimator for the probability density function.

data

the observed time series. Only used to obtain the quantile function when empirical = TRUE.

empirical

logical. If TRUE, the sample estimators for the density and quantile functions are considered. Otherwise, the gaussian density and quantile functions are used instead.

mean

the mean of the gaussian distribution. Only used if empirical = FALSE

sd

the standard deviation of the gaussian distribution. Only used if empirical = FALSE

Details

Here F' and F^{(-1)} are replaced by sample estimators for these functions or the gaussian density and quantile functions are used, depending on the context.

The function kdens is used as sample estimator of F' and quantile is the sample estimator of F^{(-1)}.

Value

The value of K_1.

Examples

trunc = 50000
cks <- arfima.coefs(d = 0.25, trunc = trunc)
eps <- rnorm(trunc+1000)
x <- sapply(1:1000, function(t) sum(cks*rev(eps[t:(t+trunc)])))

# kernel density function
dfun <- kdens(x)

# calculating K1 using four copulas and empirical estimates for F' and F^{(-1)}
K1_frank_e <- k1fun(dCdtheta = dCtheta_frank, fun = dfun,
                 data = x, empirical = TRUE)
K1_amh_e <- k1fun(dCdtheta = dCtheta_amh, fun = dfun,
                 data = x, empirical = TRUE)
K1_fgm_e <- k1fun(dCdtheta = dCtheta_fgm, fun = dfun,
                 data = x, empirical = TRUE)
K1_gauss_e <- k1fun(dCdtheta = dCtheta_gauss, fun = dfun,
                 data = x, empirical = TRUE)

# calculating K1 using four copulas and gaussian marginals
K1_frank_g <- k1fun(dCdtheta = dCtheta_frank, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))
K1_amh_g <- k1fun(dCdtheta = dCtheta_amh, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))
K1_fgm_g <- k1fun(dCdtheta = dCtheta_fgm, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))
K1_gauss_g <- k1fun(dCdtheta = dCtheta_gauss, fun = NULL, data = NULL,
                  empirical = FALSE, mean = mean(x), sd = sd(x))

# comparing results
 data.frame(MARGINAL = c("Empirical", "Gaussian"),
            FRANK = c(K1_frank_e, K1_frank_g),
            AMH = c(K1_amh_e,  K1_amh_g),
            FGM = c(K1_fgm_e, K1_fgm_g),
            GAUSS = c(K1_gauss_e, K1_gauss_g))