The Bivariate Poisson Conditionals Distribution (BPCD)"

In BCD: Bivariate Distributions via Conditional Specification

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(BCD)

Introduction

This vignette introduces the Bivariate Poisson Conditionals Distribution (BPCD), defined via conditional specifications, as proposed by Ghosh, Marques, and Chakraborty (2021). The BCD package provides functions to evaluate the joint and cumulative distributions, perform random sampling, and estimate parameters via maximum likelihood.

Joint Probability: dpoisBCD()

The joint probability mass function (p.m.f.) of the BBCD is given by:

[ P(X = x, Y = y) = K \frac{\lambda_1^x \lambda_2^y \lambda_3^{xy}}{x! y!}, ]

where ( K ) is a normalizing constant ensuring the probabilities sum to 1 and \eqn{ x, y = 0, 1, 2, \ldots }.

Example

dpoisBCD(x = 1, y = 2, lambda1 = 0.5, lambda2 =  0.5, lambda3 =  0.5)
dpoisBCD(x = 0, y = 1, lambda1 = 0.5, lambda2 =  0.5, lambda3 =  1)

Cumulative Distribution: ppoisBCD()

The function ppoisBCD() computes the cumulative distribution:

[ P(X \leq x, Y \leq y) ]

Example

ppoisBCD(x = 1, y = 1, lambda1 = 0.5, lambda2 = 0.5, lambda3 = 0.5)
ppoisBCD(x = 2, y = 2, lambda1 = 1.0, lambda2 = 1.0, lambda3 = 0.8)

Random Sampling: rpoisBCD()

Generate samples from the BPCD using:

rpoisBCD(n, lambda1, lambda2, lambda3)

Example

set.seed(123)
samples <- rpoisBCD(n = 100, lambda1 = 0.5, lambda2 = 0.5, lambda3 = 0.5)
head(samples)
cor(samples$X, samples$Y) # Should be negative

Maximum Likelihood Estimation: MLEpoisBCD()

Estimate the parameters of the distribution from data.

Example

data <- rpoisBCD(n = 100, lambda1 = 0.5, lambda2 = 0.5, lambda3 = 0.5)
fit <- MLEpoisBCD(data)
fit

Real Data Example

The dataset lensfaults records counts of surface faults $X$ and interior faults $Y$ observed in 100 optical lenses.

data(lensfaults)
head(lensfaults)
plot(lensfaults$X, lensfaults$Y, xlab = "X", ylab = "Y")

fit <- MLEpoisBCD(lensfaults)
FTtest(lensfaults, "BPCD", params = fit, num_params = 3)

The dataset eplSeasonGoals is a list of data frames for five consecutive seasons (2014/15 to 2018/19) from the English Premier League. Each data frame contains the number of full-time home ($X$) and away ($Y$) goals scored in each match of the season.

data(eplSeasonGoals)
plot(eplSeasonGoals[["1415"]]$X, eplSeasonGoals[["1415"]]$Y, xlab = "X", ylab = "Y")
plot(eplSeasonGoals[["2425"]]$X, eplSeasonGoals[["2425"]]$Y, xlab = "X", ylab = "Y")

fit <- MLEpoisBCD(eplSeasonGoals[["1415"]])
FTtest(eplSeasonGoals[["1415"]], "BPCD", params = fit, num_params = 3)

fit <- MLEpoisBCD(eplSeasonGoals[["2425"]])
FTtest(eplSeasonGoals[["2425"]], "BPCD", params = fit, num_params = 3)

Reference: Ghosh, I., Marques, F., and Chakraborty, S. (2021). A new bivariate Poisson distribution via conditional specification: properties and applications. \emph{Journal of Applied Statistics}, 48(16), 3025-3047.

BCD documentation built on June 25, 2025, 5:09 p.m.