BS {DELTD}

Estimate Density Values by Birnbaum-Saunders kernel

Description

This function calculates the estimated Values by using Birnbaum-Saunders Kernel. The Birnbaum-Saunders kernel is developed by Jin and Kawczak (2003). They claimed that performance of their developed kernel is better near the boundary points in terms of boundary reduction.

K_{BS(h^\frac{1}{2},x)} (y)=\frac{1}{2\sqrt 2 \pi h} \left(\sqrt \frac{1}{xy} +\sqrt\frac{x}{y^3}\right)exp\left(-\frac{1}{2h}\left(\frac{y}{x}-2+\frac{x}{y}\right)\right)

Usage

BS(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

In this function, choice of bandwidth, number of grid points and scheme that how these grid points are generated are user based. If any parameter(s) is missing then function used default parameters. But at least x or k should be specified otherwise NA will be produced. If x is missing then function will generate k grid points between minimum and maximum values of vector. Similarly, if k is missing then function consider it same to length of main vector. In case if h is missing then function used normal scale rule bandwidth for non-normal data and described in Silverman (1986).

Value

x

grid points

y

estimated values of density

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Jin, X.; Kawczak, J. 2003. Birnbaum-Saunders & Lognormal kernel estimators for modeling durations in high frequency financial data. Annals of Economics and Finance 4, 103-124.

See Also

For further kernels see Erlang, Gamma and LogN. To plot the density by using BS kernel plot.BS and to calculate MSE by 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.
alpha = 10
theta = 15 / 60

y <- rgamma(n = 1000, shape = alpha, scale = theta)
xx <- seq(min(y) + 0.05, max(y), length =200)
h <- 1.1
den <- BS(x = xx, y = y, k = 200, h = h)

##If scheme for generating grid points is unknown
y <- rgamma(n = 1000, shape = alpha, scale = theta)
h <- 3
BS(y = y, k = 90, h = h)

## Not run: 
##If user do not mention the number of grid points
y <- rgamma(n = 1000, shape = alpha, scale = theta)
xx <- seq(0.001, 1000, length = 1000)

#any bandwidth can be used
require(KernSmooth)
h <- dpik(y)     #Direct Plug-In Bandwidth
BS(x = xx, y = y, h = h)

## End(Not run)

## Not run: 
#if both generating scheme and number of grid points are missing then function generate NA
y <- rgamma(n = 1000, shape = alpha, scale = theta)
band = 3
BS(y = y, h = band)

## End(Not run)

#if bandwidth is missing
y <- rgamma(n = 1000, shape = alpha, scale = theta)
xx <- seq(0.001, 100, length = 1000)
BS(x = xx, y = y, k = 900)