| Title: | Density Estimation by Extreme Value Distributions |
|---|---|
| Description: | Provides mean squared error (MSE) and plot the kernel densities related to extreme value distributions with their estimated values. By using Gumbel and Weibull Kernel. See Salha et al. (2014) <doi:10.4236/ojs.2014.48061> and Khan and Akbar (2021) <doi:10.4236/ojs.2021.112018 >. |
| Authors: | Javaria Ahmad Khan, Atif Akbar. |
| Maintainer: | Javaria Ahmad Khan <[email protected]> |
| License: | GPL-2 |
| Version: | 1.2.3 |
| Built: | 2025-03-12 02:42:25 UTC |
| Source: | https://github.com/cran/DEEVD |
Two extreme value distributions; Weibull and Gumbel kernel related functions are presented. Weibull and Gumbel present estimated values and plot.Weibull and plot.Gumbel plot the densities. While mean squared error can be calculated by using mse. Further, some related data sets are also presented.
Density Estimation by Extreme Value Distributions
Javaria Ahmad Khan, Atif Akbar.
Salha, R. B., El Shekh Ahmed, H. I., & Alhoubi, I. M. 2014. Hazard Rate Function Estimation Using Weibull Kernel. Open Journal of Statistics 4 (08), 650-661.
Khan, J. A., & Akbar, A. 2021. Density Estimation Using Gumbel Kernel Estimator. Open Journal of Statistics 11 (2), 319-328.
Useful links:
The Gumbel kernel is developed by Khan and Akbar (2020). They provided evidence that performance of their proposed is better then Weibull kernel especially when data belongs to family of extreme distributions. Gumbel Kernel is
Gumbel(x = NULL, y, k = NULL, h = NULL)Gumbel(x = NULL, y, k = NULL, h = NULL)
x |
scheme for generating grid points |
y |
a numeric vector of positive values. |
k |
gird points |
h |
the bandwidth |
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).
x |
grid points |
y |
estimated values of density |
Javaria Ahmad Khan, Atif Akbar.
Khan, J. A., & Akbar, A. 2021. Density Estimation Using Gumbel Kernel Estimator. Open Journal of Statistics 11 (2), 319-328.
For other kernel see Weibull. To plot the density by using Gumbel kernel plot.Gumbel and to calculate MSE use mse.
#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 <- Gumbel(x = xx, y = y, k = 200, h = h) ##If scheme for generating gridpoints is unknown y <- rexp(50, 1) h <- 3 den <- Gumbel(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) #Direct Plug-In Bandwidth den <- Gumbel(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 <- Gumbel(x = xx, y = y, k = 90)#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 <- Gumbel(x = xx, y = y, k = 200, h = h) ##If scheme for generating gridpoints is unknown y <- rexp(50, 1) h <- 3 den <- Gumbel(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) #Direct Plug-In Bandwidth den <- Gumbel(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 <- Gumbel(x = xx, y = y, k = 90)
A dataset containing the flood discharge in per second in cubic meter from Mississippi river.
MississippiMississippi
A vector with 23 observations
Gumbel, E. J. 1941. The return period of flood flows. The Annals of Mathematical Statistics. 12, 163-190.
This function calculates the mean squared error (MSE) by using user specified kernel. But distribution of vector should be Exponential, Gamma, Gumbel, Frechet or Weibull. Any other choice of distribution will result NaN. This function is simillar to function mse in DELTD, but here more distributions are available for distribution vector.
mse(kernel, type)mse(kernel, type)
kernel |
type of kernel which is to be used |
type |
mention distribution of vector.If exponential distribution then use |
Mean Squared Error (MSE)
Javaria Ahmad Khan, Atif Akbar
Salha, R. B., El Shekh Ahmed, H. I., & Alhoubi, I. M. 2014. Hazard Rate Function Estimation Using Weibull Kernel. Open Journal of Statistics 4 (08), 650-661.
Khan, J. A., & Akbar, A. 2021. Density Estimation Using Gumbel Kernel Estimator. Open Journal of Statistics 11 (2), 319-328.
y <- rexp(100, 1) xx <- seq(min(y) + 0.05, max(y), length = 500) h <- 2 gr <- Gumbel(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)y <- rexp(100, 1) xx <- seq(min(y) + 0.05, max(y), length = 500) h <- 2 gr <- Gumbel(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)
Plot kernel density by using Gumbel Kernel.
## S3 method for class 'Gumbel' plot(x, ...)## S3 method for class 'Gumbel' plot(x, ...)
x |
an object of class "Gumbel" |
... |
Not presently used in this implementation |
nothing
Javaria Ahmad Khan, Atif Akbar.
Khan, J. A., & Akbar, A. 2021. Density Estimation Using Gumbel Kernel Estimator. Open Journal of Statistics 11 (2), 319-328.
For Weibull kernel see plot.Weibull. To calculate Gumbel estimated values see Gumbel and for MSE mse.
y <- rlnorm(100, meanlog = 0, sdlog = 1) h <- 1.5 xx <- seq(min(y) + 0.05, max(y), length = 200) den <- Gumbel(x = xx, y = y, k = 200, h = h) plot(den, type = "l") ##other details can also be added y <- rlnorm(100, meanlog = 0, sdlog = 1) grid <- seq(min(y) + 0.05, max(y), length = 200) h <- 0.79 * IQR(y) * length(y) ^ (-1/5) gr <- Gumbel(x = grid, y = y, k = 200, h = h) plot(gr, 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 Gumbel Kernel"), col=c("red", "black"), lty=c(1,2))y <- rlnorm(100, meanlog = 0, sdlog = 1) h <- 1.5 xx <- seq(min(y) + 0.05, max(y), length = 200) den <- Gumbel(x = xx, y = y, k = 200, h = h) plot(den, type = "l") ##other details can also be added y <- rlnorm(100, meanlog = 0, sdlog = 1) grid <- seq(min(y) + 0.05, max(y), length = 200) h <- 0.79 * IQR(y) * length(y) ^ (-1/5) gr <- Gumbel(x = grid, y = y, k = 200, h = h) plot(gr, 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 Gumbel Kernel"), col=c("red", "black"), lty=c(1,2))
Plot density by using Weibull Kernel.
## S3 method for class 'Weibull' plot(x, ...)## S3 method for class 'Weibull' plot(x, ...)
x |
an object of class "Weibull" |
... |
Not presently used in this implementation |
nothing
Javaria Ahmad Khan, Atif Akbar.
Salha, R. B., El Shekh Ahmed, H. I., & Alhoubi, I. M. 2014. Hazard Rate Function Estimation Using Weibull Kernel. Open Journal of Statistics 4 (08), 650-661.
For Gumbel kernel see plot.Gumbel. To calculate Weibull estimated values see Weibull and for
MSE use mse.
y <- rlnorm(100, meanlog = 0, sdlog = 1) h <- 1.5 xx <- seq(min(y) + 0.05, max(y), length = 200) den <- Weibull(x = xx, y = y, k = 200, h = h) plot(den, type = "l") ##other details can also be added y <- rlnorm(100, meanlog = 0, sdlog = 1) grid <- seq(min(y) + 0.05, max(y), length = 200) h <- 0.79 * IQR(y) * length(y) ^ (-1/5) gr <- Weibull(x = grid, y = y, k = 200, h = h) plot(gr, 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 = "green") legend("topright", c("Real Density", "Density by Weibull Kernel"), col=c("green", "black"), lty=c(1,2))y <- rlnorm(100, meanlog = 0, sdlog = 1) h <- 1.5 xx <- seq(min(y) + 0.05, max(y), length = 200) den <- Weibull(x = xx, y = y, k = 200, h = h) plot(den, type = "l") ##other details can also be added y <- rlnorm(100, meanlog = 0, sdlog = 1) grid <- seq(min(y) + 0.05, max(y), length = 200) h <- 0.79 * IQR(y) * length(y) ^ (-1/5) gr <- Weibull(x = grid, y = y, k = 200, h = h) plot(gr, 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 = "green") legend("topright", c("Real Density", "Density by Weibull Kernel"), col=c("green", "black"), lty=c(1,2))
A dataset containing the flood discharge in per second in cubic meter from Rhone river.
RhoneRhone
A vector with 23 observations
Gumbel, E. J. 1941. The return period of flood flows. The Annals of Mathematical Statistics. 12, 163-190.
A dataset which contains a length of treatment spells (in days) of control patients in suicide study.
SuicideSuicide
A vector with 86 observations
Silverman, B. W. 1986. Density Estimation. Chapman & Hall/ CRC, London.
The Weibull kernel is developed by Salha et al. (2014). They used it to nonparametric estimation of the probability density function (pdf) and the hazard rate function for independent and identically distributed (iid) data. Weibull Kernel is
Weibull(x = NULL, y, k = NULL, h = NULL)Weibull(x = NULL, y, k = NULL, h = NULL)
x |
scheme for generating grid points |
y |
a numeric vector of positive values |
k |
number of gird points |
h |
the bandwidth |
see the details in the Gumbel
x |
grid points |
y |
estimated values of density |
Javaria Ahmad Khan, Atif Akbar.
Salha, R. B., El Shekh Ahmed, H. I., & Alhoubi, I. M. 2014. Hazard Rate Function Estimation Using Weibull Kernel. Open Journal of Statistics 4 (08), 650-661. Silverman, B. W. 1986. Density Estimation. Chapman & Hall/ CRC, London.
For Gumbel kernel see Gumbel. To plot its density see plot.Weibull and to calculate MSE mse.
#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 <- Weibull(x = xx, y = y, k = 200, h = h) ##If scheme for generating gridpoints is unknown y <- rexp(50, 1) h <- 3 den <- Weibull(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 <- Weibull(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 <- Weibull(x = xx, y = y, k = 90)#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 <- Weibull(x = xx, y = y, k = 200, h = h) ##If scheme for generating gridpoints is unknown y <- rexp(50, 1) h <- 3 den <- Weibull(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 <- Weibull(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 <- Weibull(x = xx, y = y, k = 90)