Package 'DELTD'

DELTD

Description

A collection of asymmetrical kernels belong to lifetime distributions for kernel density estimation is presented. i.e. plot.BS, plot.Beta, plot.Erlang, plot.Gamma and plot.LogN. Estimated values can also observed by using Beta, BS, Gamma, Erlang and LogN. For calculating mean squared error by using different kernels functions are mse can be used.

A collection of asymmetrical kernels belong to lifetime distributions for kernel density estimation is presented. i.e. plot.BS, plot.Erlang, plot.Gamma and plot.LogN. Estimated values can also observed by using BS, Gamma, Erlang and LogN, where data can belong to any distribution. For calculating mean squared error by using different kernel functions mse .

Details

Kernel Density Estimation using Lifetime Distributions

Kernel Density Estimation using Lifetime Distributions

Author(s)

Javaria Ahmad Khan, Atif Akbar.

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.

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

• Chen, S. X. 2000. Probability density function estimation using Gamma kernels. Annals of the Institute of Statistical Mathematics 52 (3), 471-480.

• Chen, S. X. 2000. Beta kernel smothers for regression curves. Statistica Sinica 10, 73-91.

• Buckland, S. T.; Burnham, K. P.; Anderson, D. R.; Laake, J. L. 1993. Density Estimation using Distance Sampling. Chapman & Hall, London.

• 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.

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

• Chen, S. X. 2000. Probability density function estimation using Gamma kernels. Annals of the Institute of Statistical Mathematics 52 (3), 471-480.

• Chen, S. X. 2000.Beta kernel smoothers for regression curves. Statistica Sinica 10, 73-91.

See Also

Useful links:

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

Useful links:

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

---

Estimate Density Values by Beta kernel

Description

This function provide the estimated Kernel density values by using Beta Kernel. The Beta kernel is developed by Chen (2000) by using Beta distribution of first kind. He was first to introduce asymetrical kernels to control boundary Bias. Beta Kernel is

$K_{Beta( \frac{x}{h}+1, \frac{1-x}{h}+1)}(y)=\frac{y^ \frac{x}{h} (1-y)^{\frac{1-x}{b}}} { B \{ \frac{x}{h}+1, \frac{(1-x)}{h}+1 \}}$

Usage

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

Arguments

x	scheme for generating grid points
y	a numeric vector of positive values
k	number of 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 by using uniform distribution. 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). This function can be only used if data is between (0, 1). Similarly, x should be also lies between (0, 1).

Value

x	grid points
y	estimated values of density

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Chen, S. X. 2000. Beta kernel smothers for regression curves. Statistica Sinica 10, 73-91. Silverman, B. W. 1986. Density Estimation. Chapman & Hall/ CRC, London.

See Also

For further kernels see Erlang, BS, Gammaand LogN. To plot its density see plot.Beta and to calculate MSE 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 <- runif(50)
xx <- sample(0.00001:900, 500, replace = FALSE)/1000
h <- 0.9
Beta(x = xx, y = y, k = 500, h = h)

## If scheme for generating grid points is unknown
y <- runif(500)
h <- 0.9
Beta(x = xx, y = y, k = 500, h = h)

## Not run: 
## If user do not mention the number of grid points
y <- runif(1000)
xx <- seq(0.001, 1000, length = 2000)

## any bandwidth can be used
require(kedd)
h <- h.bcv(y) ## Biased cross validation
Beta(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 <- runif(1000)
band = 0.8
Beta(y = y, h = band)

## End(Not run)

## if bandwidth is missing
y <- runif(100)
xx <- seq(0.001, 100, length = 300)
Beta(x = xx, y = y, k = 200)

---

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)

---

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)

---

Estimate Density Values by Gamma kernel

Description

This function provide the estimated Kernel density values by using Gamma Kernel.The Gamma kernel is developed by Chen (2000). He was first to introduce asymetrical kernels to control boundary Bias. Gamma Kernel is

$K_{Gam1( \frac{x}{h}+1, h)}(y) = \frac{y^ \frac{x}{h} exp(-\frac{y}{h})}{ \Gamma (\frac{x}{h}+1)h^{ \frac{x}{h}+1}}$

Usage

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

Arguments

x	scheme for generating grid points
y	a numeric vector of positive values
k	number of 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

Chen, S. X. 2000. Probability density function estimation using Gamma kernels. Annals of the Institute of Statistical Mathematics 52 (3), 471-480. Silverman, B. W. 1986. Density Estimation. Chapman & Hall/ CRC, London.

See Also

For further kernels see Erlang, BS, Betaand LogN. To plot its density see plot.Gamma and to calculate MSE mse.

Examples

##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 <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 500)
h <- 2
den <- Gamma(x = xx, y = y, k = 200, h = h)

##If scheme for generating grid points is unknown
y <- rexp(200, 1)
h <- 3
Gamma(y = y, k = 90, h = h)

## Not run: 
y <- data(TUNA)
xx <- seq(min(y) + 0.05, max(y), length = 500)
h <- 2
den <- Gamma(x = xx, y = y, k = 200, h = h)

## End(Not run)

## Not run: 
##If user do not mention the number of grid points
y <- rexp(1000, 1)
xx <- seq(0.001, 1000, length = 1000)

#any bandwidth can be used
require(KernSmooth)
h <- dpik(y)
Gamma(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 <- rexp(1000, 1)
band = 3
Gamma(y = y, h = band)

## End(Not run)

#if bandwidth is missing
y <- rexp(100,1)
xx <- seq(0.001, max(y), length = 100)
Gamma(x = xx, y = y, k = 90)

---

Estimate Density Values by Lognormal kernel

Description

The LogN estimate Values of density by using Lognormal Kernel.The Lognomal kernel is developed by Jin and Kawczak (2003). For this too, they claimed that performance of their developed kernel is better near the boundary points in terms of boundary reduction. Lognormal Kernel is

$K_{LN(\ln(x),4\ln(1+h))}=\frac{1}{\sqrt{( 8\pi \ln(1+h))} y)} exp\left[-\frac{(\ln(y)-\ln(x))^2}{(8\ln(1+h))}\right]$

Usage

LogN(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

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 Beta, Erlang, Gamma and BS. To plot its density see plot.LogN and to calculate 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 <- rweibull(350, 1)
xx <- seq(0.001, max(y), length = 500)
h <- 2
den <- LogN(x = xx, y = y, k = 200, h = h)

##If scheme for generating grid points is unknown
n <- 1000
y <- abs(rlogis(n, location = 0, scale = 1))
h <- 3
LogN(y = y, k = 90, h = h)

## Not run: 
##If user do not mention the number of grid points
y <- rweibull(350, 1)
xx <- seq(0.00001, max(y), 500)

#any bandwidth can be used
require(ks)
h <- hscv(y)   #Smooth cross validation bandwidth
LogN(x = xx, y = y, h = h)

## End(Not run)

## Not run: 
#if both scheme and number of grid points are missing then function generate NA
n <- 1000
y <- abs(rlogis(n, location = 0, scale = 1))
band = 3
LogN(y = y, h = band)

## End(Not run)

#if bandwidth is missing
y <- rweibull(350, 1)
xx <- seq(0.001, 100, length = 500)
LogN(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. But distribution of vector should be Exponential, Gamma or Weibull. Any other choice of distribution will result NaN.

Usage

mse(kernel, type)

Arguments

kernel	type of kernel which is to be used
type	mention distribution of vector.If exponential distribution then use "Exp". If use gamma distribution then use "Gamma".If Weibull distribution then use "Weibull".

Value

Mean Squared Error (MSE)

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.

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

• Chen, S. X. 2000. Probability density function estimation using Gamma kernels. Annals of the Institute of Statistical Mathematics 52 (3), 471-480.

• Chen, S. X. 2000. Beta kernel smothers for regression curves. Statistica Sinica 10, 73-91.

Examples

y <- rexp(100, 1)
xx <- seq(min(y) + 0.05, max(y), length = 500)
h <- 2
gr <- Gamma(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 Beta kernel

Description

Plot density by using Beta Kernel.

Usage

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

Arguments

x	an object of class "Beta"
...	Not presently used in this implementation

Value

nothing

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Chen, S. X. 2000. Beta kernel smothers for regression curves. Statistica Sinica 10, 73-91.

See Also

For further kernels see plot.Gamma, plot.Erlang, plot.BS and plot.LogN. To calculate its estimated values see Beta and for MSE see mse.

Examples

y <- runif(100)
h <- 0.5
xx <- sample(0.00001:900, 50, replace = FALSE)/1000
den <- Beta(x = xx, y = y, k = 50, h = h)
plot(den, type = "p")
##other details can also be added
y <- runif(100)
h <- 0.7
xx <- sample(0.00001:900, 50, replace = FALSE)/1000
den <- Beta(x = xx, y = y, k = 50, h = h)
plot(den, type = "l", ylab = "Density Function", lty = 1, xlab = "Time")

---

Density Plot by Birnbaum-Saunders kernel

Description

Plot Kernel density by using Birnbaum-Saunders Kernel.

Usage

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

Arguments

x	An object of class "BS"
...	Not presently used in this implementation

Value

Nothing

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 plot.Beta, plot.Erlang, plot.Gamma and plot.LogN. For estimated values BS and for MSE mse.

Examples

alpha = 10
theta = 15 / 60
y <- rgamma(n = 10000, shape = alpha, scale = theta)
h <- 1.5
xx <- seq(min(y) + 0.05, max(y), length = 200)
den <- BS(x = xx, y = y, k = 200, h = h)
plot(den, type = "l")

##other details can also be added
y <- rgamma(n = 10000, shape = alpha, scale = theta)
h <- 0.79 * IQR(y) * length(y) ^ (-1/5)  #Normal Scale Rule Bandwidth
gr <- BS(x = xx, 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 Birnbaum-Saunders Kernel"),
col=c("red", "black"), lty = c(1,2))

---

Density Plot by Erlang kernel

Description

Plot Kernel density by using Erlang Kernel.

Usage

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

Arguments

x	An object of class "Erlang"
...	Not presently used in this implementation

Value

Nothing

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 plot.Beta, plot.BS, plot.Gamma and plot.LogN. For estimated values Erlang and for calculating MSE see mse.

Examples

y <- rlnorm(100, meanlog = 0, sdlog = 1)
h <- 1.5
xx <- seq(min(y) + 0.05, max(y), length = 200)
den <- Erlang(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 <- Erlang(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 Erlang Kernel"),
col=c("red", "black"), lty=c(1,2))

---

Density Plot by Gamma kernel

Description

Plot density by using Gamma Kernel.

Usage

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

Arguments

x	an object of class "Gamma"
...	Not presently used in this implementation

Value

nothing

Author(s)

Javaria Ahmad Khan, Atif Akbar.

References

Chen, S. X. 2000. Probability density function estimation using Gamma kernels. Annals of the Institute of Statistical Mathematics 52 (3), 471-480.

See Also

For further kernels see plot.Beta, plot.Erlang, plot.BS and plot.LogN. To calculate its estimated values see Gamma and for MSE mse.

Examples

y <- rexp(100, 1)
h <- 1.5
xx <- seq(min(y) + 0.05, max(y), length =200)
den <- Gamma(x=xx, y=y, k=200, h=h)
plot(den, type = "l")

##other details can also be added
y <- rexp(100, 2)
h <- 0.79 * IQR(y) * length(y) ^ (-1/5)
gr <- Gamma(x=xx, 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 Gamma Kernel"),
col=c("red", "black"), lty=c(1,2))

---

Density Plot by Lognormal kernel

Description

Plot Kernel density by using Lognormal Kernel.

Usage

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

Arguments

x	An object of class "LogN"
...	Not presently used in this implementation

Value

Nothing

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 plot.Beta, plot.Erlang, plot.Gamma and plot.BS. To calculate MSE use mse and for estimated values for density estimation see LogN.

Examples

n <- 1000
y <- abs(rlogis(n, location = 0, scale = 1))
xx <- seq(min(y) + 0.05, max(y), length =90)
h <- 0.00003
den <- LogN(x = xx, y = y, k = 90, h = h)
plot(den, type = "l")

##other details can also be added
y <- abs(rlogis(n, location = 0, scale = 1))
h <- 3
gr <- LogN(x = xx, y = y, k = 90, 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("topleft", c("Real Density", "Density by Lognormal Kernel"),
col = c("green", "black"), lty = c(1,2))

---

Data of Tuna fish

Description

Data is about Tuna, which is saltwater fish. Its seasonal migration is between waters off the coast of Australia and the Indian Ocean. The data represents a line transect aerial survey of Southern Bluefin Tuna in the Great Australian Bight in summer when the tuna tend to stay on the surface. The abundance D is measured by

$D = \frac{N}{A}$

, where N is the total number of surface schools in the Bight and A is the survey area. To estimate D, an aircraft with two spotters on board is used to fly randomly allocated transect lines to detect tuna schools. Each school sighted from transect is counted and its perpendicular distance to transect is measured.

Usage

TUNA

Format

A vector with 64 observations

References

Buckland, S. T.; Burnham, K. P.; Anderson, D. R.; Laake, J. L. 1993. Density Estimation using Distance Sampling. Chapman & Hall, London.

---