Package 'bvpa'

Bivariate Pareto Distribution

Description

Implements the EM algorithm with one-step Gradient Descent method to estimate the parameters of the Block-Basu bivariate Pareto distribution with location and scale. We also found parametric bootstrap and asymptotic confidence intervals based on the observed Fisher information scale, shape parameters, and exact confidence intervals for location parameters.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

References

Biplab Paul and Arabin Kumar Dey (2023). An EM algorithm for absolutely continuous Marshall-Olkin bivariate Pareto distribution with location and scale, Preprint.

E L Lehmann and George Casella (1998). Theory of Point Estimation, Springer, New York, doi.org/10.1007/b98854.

Bradley Efron and R J Tibshirani (1994). An Introduction to the Bootstrap, CRC press, New York, doi.org/10.1201/9780429246593.

A P Dempster, N M Laird and D B Rubin (1977). Maximum Likelihood from Incomplete Data via the EM Algorithm, Journal of the royal statistical society: series B (methodological), www.jstor.org/stable/2984875.

---

Observed Fisher information based confidence interval of Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Observed Fisher information based confidence interval of Bivariate BBBVPA distribution.

Usage

conf.intv(
  object,
  conf.lev = 0.95,
  tol = 1e-04,
  intv.m1 = c(0, 2),
  intv.m2 = c(0, 2)
)

Arguments

object	"bbbvpa" class object.
conf.lev	confidence level, $0.95$ (default).
tol	convergence tolerance for confidence intervals, 0.0001 (default).
intv.m1	interval related to confidence interval of $\mu_1$, c(0,2) (default).
intv.m2	interval related to confidence interval of $\mu_1$, c(0,2) (default).

Value

A matrix of lower and upper confidence interval limits (in the first and second column respectively). The matrix rows are labeled by the parameter names (if any) and columns by the corresponding distribution quantiles.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

# see the example of estimation

---

Observed Fisher information based confidence interval of 3-parameter Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Observed Fisher information based confidence interval of 3-parameter BBBVPA distribution.

Usage

conf.intv3(object, conf.lev = 0.95, tol = 1e-04)

Arguments

object	"bbbvpa" class object.
conf.lev	confidence level, $0.95$ (default).
tol	convergence tolerance for confidence intervals, 0.0001 (default).

Value

A matrix of lower and upper confidence interval limits (in the first and second column respectively). The matrix rows are labeled by the parameter names (if any) and columns by the corresponding distribution quantiles.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0, 0, 1.0, 1.0, 2.0, 0.4, 0.5)
conf.intv3(estimates3(dat, 2.4, 0.3, 0.6))

---

Estimation of Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Parameters estimation of BBBVPA distribution.

Usage

estimates(
  I,
  s1.int,
  s2.int,
  a0.int,
  a1.int,
  a2.int,
  tol.est = 1e-05,
  MxIter.no = 2000,
  rate = 1e-04,
  condition = "log.L"
)

Arguments

I	bivariate observations.
s1.int	initial choice of $\sigma_1$.
s2.int	initial choice of $\sigma_2$.
a0.int	initial choice of $\alpha_0$.
a1.int	initial choice of $\alpha_1$.
a2.int	initial choice of $\alpha_2$.
tol.est	convergence tolerance, 0.00001 (default).
MxIter.no	maximum number of iterations, 2000 (default).
rate	step size or learning rate for gradient descent, 0.0001 (default).
condition	convergence criterion, "log.L" (default) and "p.logL".

Value

object of class "bbbvpa", a list consisting of

mu1, mu2, sigma1, sigma2, alpha0, alpha1, alpha2, iter.no	estimates of parameters and number of iteration.
data	the supplied data I.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

# Read data
data(precipitation)
data <- as.vector(precipitation[,2])
data[is.na(data)]<-0
n <- length(data)
# Construct the three-dimensional data set
data3d <- function(data){
 u <- 12
 Y <- c()
 indx <- indx1 <- indx2 <- indx3 <- 0
 r <- 5
 i <- 2
 while(i < n){
   i <- i + 1
   if(data[i] > u || sum(data[(i-1):i]) > u || sum(data[(i-2):i]) > u){
     if(data[i] > u){imax <- i}
     if(sum(data[(i-1):i]) > u){imax <- i - 3 + which(data[(i-1):i] == max(data[(i-1):i]))[1]}
     if(sum(data[(i-2):i]) > u){imax <- i - 3 + which(data[(i-2):i] == max(data[(i-2):i]))[1]}
     if(max(indx) > (imax-r)){
       cluster <- data[(max(indx)+3):(imax+r)]
     } else{
       cluster <- data[(imax-r):(imax+r)]
     }
     cluster2 <- sapply(c(1:(length(cluster)-1)), function(j) sum(cluster[j:(j+1)]))
     cluster3 <- sapply(c(1:(length(cluster)-2)), function(j) sum(cluster[j:(j+2)]))
     indx1 <- append(indx1,imax-r-1+which(cluster==max(cluster))[1])
     indx2 <- append(indx2,imax-r-1+which(cluster2==max(cluster2)))
     indx3 <- append(indx3,imax-r-1+which(cluster3==max(cluster3)))
     Y <- rbind(Y, c(max(cluster),max(cluster2),max(cluster3)))
     indx <- append(indx,imax)
     i <- i + r
   }
 }
 return(Y)
}
I <- data3d(data)[,c(1,3)]
iniz <- intliz(I)
iniz
est <- estimates(I, iniz[1], iniz[2], iniz[3], iniz[4], iniz[5])
est[-9]
param.boot(I, iniz[1], iniz[2], iniz[3], iniz[4], iniz[5])
conf.intv(est)

---

Estimation of 3-parameter Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Parameters estimation of 3-parameter BBBVPA distribution.

Usage

estimates3(
  I,
  a0.int,
  a1.int,
  a2.int,
  tol.est = 1e-05,
  MxIter.no = 2000,
  condition = "log.L"
)

Arguments

I	bivariate observations.
a0.int	initial choice of $\alpha_0$.
a1.int	initial choice of $\alpha_1$.
a2.int	initial choice of $\alpha_2$.
tol.est	convergence tolerance, 0.0001 (default).
MxIter.no	maximum number of iterations, 2000 (default).
condition	convergence criterion, "log.L" (default) and "p.logL".

Value

Object of class "bbbvpa3", a list consisting of

alpha0, alpha1, alpha2, iter.no	estimates of parameters and number of iteration.
data	the supplied data I.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0, 0, 1.0, 1.0, 2.0, 0.4, 0.5)
estimates3(dat, 2.4, 0.3, 0.6)[-5]

---

Initialization of Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Return initial choice parameters of BBBVPA distribution.

Usage

intliz(
  data,
  ini.run = 100,
  tol.ini = 0.001,
  proc = "ML",
  intv.s1 = c(0, 5),
  intv.s2 = c(0, 5),
  intv.a0 = c(0, 5),
  intv.a1 = c(0, 5),
  intv.a2 = c(0, 5),
  ...
)

Arguments

data	bivariate observations.
ini.run	number of random initializations.
tol.ini	convergence tolerance, 0.001 (default)..
proc	different procedures, "ML" (default) and "S.EM".
intv.s1	interval for random initialization of $\sigma_1$.
intv.s2	interval for random initialization of $\sigma_2$.
intv.a0	interval for random initialization of $\alpha_0$.
intv.a1	interval for random initialization of $\alpha_1$.
intv.a2	interval for random initialization of $\alpha_2$.
...	further arguments to pass to estimates.

Value

numeric vector.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

# see the example of estimation

---

Initialization of 3-parameter Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Return initial choice parameters of 3-parameter BBBVPA distribution.

Usage

intliz3(
  data,
  ini.run = 100,
  tol.ini = 0.001,
  proc = "ML",
  intv.a0 = c(0, 5),
  intv.a1 = c(0, 5),
  intv.a2 = c(0, 5),
  ...
)

Arguments

data	bivariate observations.
ini.run	number of random initializations.
tol.ini	convergence tolerance, 0.001 (default)..
proc	different procedures, "ML" (default) and "S.EM".
intv.a0	interval for random initialization of $\alpha_0$.
intv.a1	interval for random initialization of $\alpha_1$.
intv.a2	interval for random initialization of $\alpha_2$.
...	further arguments to pass to estimates3.

Value

numeric vector.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0, 0, 1.0, 1.0, 2.0, 0.4, 0.5)
intliz3(dat)

---

Log-likelihood function of Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Return the log likelihood value.

Usage

logL(I, mu1, mu2, s1, s2, a0, a1, a2)

Arguments

I	baivariate observations.
mu1	value of $\mu_1$.
mu2	value of $\mu_2$.
s1	value of $\sigma_1$.
s2	value of $\sigma_2$.
a0	value of $\alpha_0$.
a1	value of $\alpha_1$.
a2	value of $\alpha_2$.

Value

a list consisting of

logLik	A scalar numeric, log likelihood of the model.
n1, n2	$n_1$ and $n_2$.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5)
logL(dat, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5)

---

Marginal log-liklihood function of variable X1

Description

Return the marginal log-liklihood value of variable $X_1$.

Usage

mLf1(I, mu1, s1, a0, a1, a2)

Arguments

I	baivariate observations.
mu1	value of $\mu_1$.
s1	value of $\sigma_1$.
a0	value of $\alpha_0$.
a1	value of $\alpha_1$.
a2	value of $\alpha_2$.

Value

A scalar numeric, the marginal log-liklihood value of variable $X_1$.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5)
mLf1(dat, 0.1, 0.8, 2.0, 0.4, 0.5)

---

Marginal log-liklihood function of variable X2

Description

Return the marginal log-liklihood value of variable $X_2$.

Usage

mLf2(I, mu2, s2, a0, a1, a2)

Arguments

I	baivariate observations.
mu2	value of $\mu_2$.
s2	value of $\sigma_2$.
a0	value of $\alpha_0$.
a1	value of $\alpha_1$.
a2	value of $\alpha_2$.

Value

A scalar numeric, the marginal log-liklihood value of variable $X_2$.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5)
mLf2(dat, 0.1, 0.8, 2.0, 0.4, 0.5)

---

Parametric bootstrap confidence intervals of parameters of Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Parametric bootstrap confidence interval of parameters of BBBVPA distribution.

Usage

param.boot(
  data,
  s1.int,
  s2.int,
  a0.int,
  a1.int,
  a2.int,
  conf.lev = 0.95,
  intv.m1 = c(0, 2),
  intv.m2 = c(0, 2),
  no.paboot = 100,
  tol = 1e-04,
  ...
)

Arguments

data	bivariate observations.
s1.int	initial choice of $\sigma_1$.
s2.int	initial choice of $\sigma_2$.
a0.int	initial choice of $\alpha_0$.
a1.int	initial choice of $\alpha_1$.
a2.int	initial choice of $\alpha_2$.
conf.lev	confidence level, defult $0.95$.
intv.m1	interval related to confidence interval of $\mu_1$, c(0,2) (default).
intv.m2	interval related to confidence interval of $\mu_1$, c(0,2) (default).
no.paboot	number of bootstrap samples, 100 (default).
tol	convergence tolerance for confidence interval of $\mu_1$. and $\mu_2$, 0.0001 (default).
...	further arguments to pass to estimates.

Value

A matrix of lower and upper confidence interval limits (in the first and second column respectively). The matrix rows are labeled by the parameter names (if any) and columns by the corresponding distribution quantiles.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

# see the example of estimation

---

Parametric bootstrap confidence intervals of parameters of 3-parameter Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Parametric bootstrap confidence interval of parameters of 3-parameter BBBVPA distribution.

Usage

param.boot3(
  data,
  a0.int,
  a1.int,
  a2.int,
  conf.lev = 0.95,
  no.paboot = 100,
  ...
)

Arguments

data	bivariate observations.
a0.int	initial choice of $\alpha_0$.
a1.int	initial choice of $\alpha_1$.
a2.int	initial choice of $\alpha_2$.
conf.lev	confidence level, defult 0.95.
no.paboot	number of bootstrap samples, 100 (default).
...	further arguments to pass to estimates3.

Value

A matrix of lower and upper confidence interval limits (in the first and second column respectively). The matrix rows are labeled by the parameter names (if any) and columns by the corresponding distribution quantiles.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0, 0, 1.0, 1.0, 2.0, 0.4, 0.5)
param.boot3(dat, 2.4, 0.3, 0.6)

---

Survival functions of pivots of estimators of locations.

Description

Survival functions of pivots of estimators of locations $\mu_1$ and $\mu_2$. These are required to calculate the critical value of confidence intervals for $\mu_1$ and $\mu_2$.

Usage

pctl.fun(z, n, a0, a1, a2, pct, select = 1)

Arguments

z	quantiles.
n	number of observations.
a0	value of $\alpha_0$.
a1	value of $\alpha_1$.
a2	value of $\alpha_2$.
pct	probabilities.
select	Allows to select the function for different location parameters. a single model term to be selected for printing. e.g. if you just want the function for $\mu_1$ set $select=1$ (default).

Value

return a function.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

uniroot(pctl.fun, interval=c(0,2), n = 500, a0 = 2.0, a1 = 0.4, a2 = 0.5,
 pct = 0.025, tol = 0.0001)[[1]]

---

Precipitation data

Description

The dataset contains daily accumulated precipitation data (in mm) from Abisko Scientific Research Station in northern Sweden for 100 years, from 1st January 1913 to 31st December 2012.

Usage

data(precipitation)

Format

A data frame with 36524 rows and 2 columns and the following variables:

Time

1st column represents Day.

Precipitation..mm.

2nd column represents daily accumulated precipitation (in mm) of the day.

Source

<https://www.polar.se/stoed-till-polarforskning/abisko-naturvetenskapliga-station/>

Examples

data(precipitation)

---

Pseudo log-likelihood function of Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Return the pseudo log likelihood value.

Usage

pseu.logL(I, mu1, mu2, s1, s2, a0, a1, a2)

Arguments

I	baivariate observations.
mu1	value of $\mu_1$.
mu2	value of $\mu_2$.
s1	value of $\sigma_1$.
s2	value of $\sigma_2$.
a0	value of $\alpha_0$.
a1	value of $\alpha_1$.
a2	value of $\alpha_2$.

Value

A scalar numeric, pseudo log likelihood of the model.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

dat <- rbb.bvpa(500, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5)
pseu.logL(dat, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5)

---

Simulate from a Block-Basu Bivariate Pareto (BBBVPA) distribution

Description

Produces one or more samples from the specified BBBVPA distribution.

Usage

rbb.bvpa(n, mu1, mu2, sig1, sig2, alp0, alp1, alp2)

Arguments

n	number of observations.
mu1	value of $\mu_1$
mu2	value of $\mu_2$
sig1	value of $\sigma_1$
sig2	value of $\sigma_2$
alp0	value of $\alpha_0$
alp1	value of $\alpha_1$
alp2	value of $\alpha_2$

Value

numeric matrix.

Author(s)

Biplab Paul <[email protected]> and Arabin Kumar Dey <[email protected]>

Examples

cor(rbb.bvpa(500, 0.1, 0.1, 0.8, 0.8, 2.0, 0.4, 0.5))

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