Erlang {DELTD}

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)

Arguments

x

scheme for generating grid points

y

a numeric vector of positive values.

k

gird points.

h

the bandwidth

Details

see the details in the BS.

Value

x

grid points

y

estimated values of density

Author(s)

Javaria Ahmad Khan, Atif Akbar.

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)

## Not run: 
##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)

## End(Not run)

## Not run: 
#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)

## End(Not run)

#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)