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 <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

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 <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

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 <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

object of class "bbbvpa", a list consisting of

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

Object of class "bbbvpa3", a list consisting of

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

numeric vector.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

numeric vector.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

a list consisting of

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

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

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

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

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

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 <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

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 <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

return a function.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

A scalar numeric, pseudo log likelihood of the model.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

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

Value

numeric matrix.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Arabin Kumar Dey <arabin@iitg.ac.in>

Examples

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