AsyK

Description

A collection of functions related to density estimation by using Chen's (2000) idea. For observing estimated values see Laplace and RIG. Plots by using these kernels can be drawn by plot.Laplace and plot.RIG. Mean squared errors (MSE) can be calculated by mse. Here we also present a normal scale rule bandwidth which is given by Silverman (1986) for non-normal data.

Details

Kernel Density Estimation

Author(s)

Javaria Ahmad Khan, Atif Akbar.

See Also

Useful links:

• https://CRAN.R-project.org/package=AsyK

Estimate Density Values by Laplace kernel

Description

Estimated Kernel density values by using Laplace Kernel.

Usage

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

Arguments

Details

Laplace kernel is developed by Khan and Akbar. Kernel is developed by using Chen's idea. Laplace kernel is;

K_{Laplace\left(x,h^{\frac{1}{2}}\right)} (u)=\frac{1}{2\sqrt h}exp \left(-\frac{|{u-x}|}{\sqrt h}\right)

Value

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Khan, J. A.; Akbar, A. Density Estimation by Laplace Kernel. Working paper, Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan.

See Also

To examine Laplace density plot see plot.Laplace and for Mean Squared Error mse. Similarly, for RIG kernel RIG.

Examples

#Data can be simulated or real data
## Number of grid points "k" should be at least equal to the data size.
### If user define the generating scheme of gridpoints than number of gridpoints should
####be equal or greater than "k"
###### otherwise NA will be produced.
y <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 100)
h <- 2
den <- Laplace(x = xx, y = y, k = 200, h = h)

##If scheme for generating gridpoints is unknown
y <- rexp(50, 1)
h <- 3
den <- Laplace(y = y, k = 90, h = h)

##If user do not mention the number of grid points
y <- rexp(23, 1)
xx <- seq(min(y) + 0.05, max(y), length = 90)

## Not run: 
#any bandwidth can be used
require(KernSmooth)
h <- dpik(y)
den <- Laplace(x = xx, y = y, h = h)

## End(Not run)

#if bandwidth is missing
y <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 100)
den <- Laplace(x = xx, y = y, k = 90)

Calculate Bandwidth proposed by Silverman for non-normal data.

Description

Calculate Bandwidth proposed by Silverman for non-normal data.

Usage

NSR(y)

Arguments

Value

h

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Silverman, B. W. 1986. Density Estimation. Chapman & Hall/ CRC, London.

Examples

y <- rexp(10, 1)
 NSR(y)

Estimated Density Values by Reciprocal Inverse Gaussian kernel

Description

Estimated Kernel density values by using Reciprocal Inverse Gaussian Kernel.

Usage

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

Arguments

Details

Scaillet 2003. proposed Reciprocal Inverse Gaussian kerenl. He claimed that his proposed kernel share the same properties as those of gamma kernel estimator.

K_{RIG \left( \ln{ax}4\ln {(\frac{1}{h})} \right)}(y)=\frac{1}{\sqrt {2\pi y}} exp\left[-\frac{x-h}{2h} \left(\frac{y}{x-h}-2+\frac{x-h}{y}\right)\right]

Value

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Scaillet, O. 2004. Density estimation using inverse and reciprocal inverse Gaussian kernels. Nonparametric Statistics, 16, 217-226.

See Also

To examine RIG density plot see plot.RIG and for Mean Squared Error mse. Similarly, for Laplace kernel Laplace.

Examples

#Data can be simulated or real data
## Number of grid points "k" should be at least equal to the data size.
### If user define the generating scheme of gridpoints than number of gridpoints should
####be equal or greater than "k"
###### otherwise NA will be produced.
y <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 100)
h <- 2
den <- RIG(x = xx, y = y, k = 200, h = h)

##If scheme for generating gridpoints is unknown
y <- rexp(50, 1)
h <- 3
den <- RIG(y = y, k = 90, h = h)

## Not run: 
##If user do not mention the number of grid points
y <- rexp(23, 1)
xx <- seq(min(y) + 0.05, max(y), length = 90)
#any bandwidth can be used
require(KernSmooth)
h <- dpik(y)
den <- RIG(x = xx, y = y, h = h)

## End(Not run)
#if bandwidth is missing
y <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 100)
den <- RIG(x = xx, y = y, k = 90)

Calculate Mean Squared Error( MSE) by using different Kernels

Description

This function calculates the mean squared error (MSE) by using user specified kernel.This function is same as provided in package "DELTD". For details see https://CRAN.R-project.org/package=DELTD.

Usage

mse(kernel, type)

Arguments

Value

Mean Squared Error (MSE)

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

https://CRAN.R-project.org/package=DELTD

See Also

This is also available in DELTD

Examples

y <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 500)
h <- 2
gr <- Laplace(x = xx, y = y, k = 200, h = h)
mse(kernel = gr, type = "Exp")
## if distribution is other than mentioned \code{type} is used then NaN will be produced.
## Not run: 
mse(kernel = gr, type ="Beta")

## End(Not run)

Density Plot by Laplace kernel

Description

Plot density by using Laplace Kernel.

Usage

## S3 method for class 'Laplace'
plot(x, ...)

Arguments

Value

nothing

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Khan, J. A.; Akbar, A. Density Estimation by Laplace Kernel. Working paper, Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan.

See Also

To examine Laplace estimated values for density see Laplace and for Mean Squared Error mse. Similarly, for plot of Laplace kernel plot.RIG.

Examples

y <- rexp(100, 1)
h <- 0.79 * IQR(y) * length(y) ^ (-1/5)
xx <- seq(min(y) + 0.05, max(y), length = 100)
den <- Laplace(x = xx, y = y, k = 100, h = h)
plot(den, type = "l")


##other details can also be added
y <- rexp(100, 1)
h <- 0.79 * IQR(y) * length(y) ^ (-1/5)
den <- Laplace(x = xx, y = y, k = 100, h = h)
plot(den, type = "s", ylab = "Density Function", lty = 1, xlab = "Time")

## To add true density along with estimated
d1 <- density(y, bw = h)
lines(d1, type = "p", col = "red")
legend("topright", c("Real Density", "Density by RIG Kernel"),
col = c("red", "black"), lty = c(1, 2))

Density Plot by Reciprocal Inverse Gaussian kernel

Description

Plot density by using Reciprocal Inverse Gaussian Kernel.

Usage

## S3 method for class 'RIG'
plot(x, ...)

Arguments

Value

nothing

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Scaillet, O. 2004. Density estimation using inverse and reciprocal inverse Gaussian kernels. Nonparametric Statistics, 16, 217-226.

See Also

To examine RIG estimated values for density see RIG and for Mean Squared Error mse. Similarly, for plot of Laplace kernel plot.Laplace.

Examples

y <- rexp(200, 1)
h <- 0.79 * IQR(y) * length(y) ^ (-1/5)
xx <- seq(min(y) + 0.05, max(y), length = 200)
den <- RIG(x = xx, y = y, k = 200, h = h)
plot(den, type = "l")


##other details can also be added
y <- rexp(200, 1)
h <- 0.79 * IQR(y) * length(y) ^ (-1/5)
den <- RIG(x = xx, y = y, k = 200, h = h)
plot(den, type = "s", ylab = "Density Function", lty = 1, xlab = "Time")

## To add true density along with estimated
d1 <- density(y, bw = h)
lines(d1, type = "p", col = "red")
legend("topright", c("Real Density", "Density by RIG Kernel"),
col = c("red", "black"), lty = c(1, 2))