DELTD (version 2.6.8)

Erlang: Estimate Density Values by Erlang kernel

Description

This function provide the estimated values for density by using Erlang Kernel. Erlang kernel is developed by Salha et al. (2014). They developed this asymmetrical kernal with its hazard function and also proved its asymtotic normality. $$K_{E(x,\frac{1}{h})} (y)=\frac{1}{\Gamma (1+\frac{1}{h})} \left[\frac{1}{x} (1+\frac{1}{h}) \right]^\frac{h+1}{h} y^\frac{1}{h} exp\left(-\frac{y}{x} (1+\frac{1}{h}) \right)$$

Usage

Erlang(x = NULL, y, k = NULL, h = NULL)

Value

x

grid points

y

estimated values of density

Arguments

x

scheme for generating grid points

y

a numeric vector of positive values.

k

gird points.

h

the bandwidth

Author

Javaria Ahmad Khan, Atif Akbar.

Details

see the details in the BS.

References

Salha, R. B.; Ahmed, E. S.; Alhoubi, I. M. 2014. Hazard rate function estimation using Erlang Kernel. Pure Mathematical Sciences 3 (4), 141-152.

See Also

For further MSE by using other kernels see Beta, BS, Gamma and LogN. For plotting these estimated values plot.Erlang and for calculating MSE use mse.

Examples

## Data: Simulated or real data can be used
## Number of grid points "k" should be at least equal to the data size.
## If user defines the generating scheme of grid points then length
## of grid points should be equal or greater than "k", Otherwise NA will be produced.
y <- rlnorm(100, meanlog = 0, sdlog = 1)
xx <- seq(min(y) + 0.05, max(y), length = 500)
h <-2
den <- Erlang(x = xx, y = y, k = 200, h = h)

##If scheme for generating grid points is unknown
y <- rlnorm(1000, meanlog = 0, sdlog = 1)
h <- 3
Erlang(y = y, k = 90, h = h)

if (FALSE) {
##If user do not mention the number of grid points
 y <- rlnorm(100, meanlog = 0, sdlog = 1)
xx <- seq(0.001, 1000, length = 1000)

#any bandwidth can be used
require(kedd)
h <- h.ucv(y)     #Unbaised cross validation bandwidth
Erlang(x = xx, y = y, h = h)
}

if (FALSE) {
#if generating scheme and number of grid points are missing then function generate NA
y <- rlnorm(100, meanlog = 0, sdlog = 1)
band = 3
Erlang(y = y, h = band)
}

#if bandwidth is missing
 y <- rlnorm(100, meanlog = 0, sdlog = 1)
xx <- seq(0.001, 100, length = 100)
Erlang(x = xx, y = y, k = 90)