The Bivariate Binomial Conditionals Distribution (BBCD)"

In BCD: Bivariate Distributions via Conditional Specification

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

Introduction

This vignette introduces the Bivariate Binomial Conditionals Distribution (BBCD), defined via conditional specifications, as proposed by Ghosh, Marques, and Chakraborty (2025). The BCD package provides functions to evaluate the joint and cumulative distributions, perform random sampling, and estimate parameters via maximum likelihood.

Joint Probability: dbinomBCD()

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

[ P(X = x, Y = y) = K \cdot \binom{n_1}{x} \binom{n_2}{y} p_1^x (1 - p_1)^{n_1 - x} p_2^y (1 - p_2)^{n_2 - y} \lambda^{xy}, ]

where ( K ) is a normalizing constant ensuring the probabilities sum to 1.

Example

dbinomBCD(x = 2, y = 1, n1 = 5, n2 = 5, p1 = 0.5, p2 = 0.4, lambda = 0.5)
# independence case
dbinomBCD(x = 2, y = 1, n1 = 5, n2 = 5, p1 = 0.5, p2 = 0.4, lambda = 1.0) 

Cumulative Distribution: pbinomBCD()

The function pbinomBCD() computes the cumulative distribution:

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

Example

pbinomBCD(x = 2, y = 5, n1 = 5, n2 = 5, p1 = 0.5, p2 = 0.4, lambda = 0.5)
pbinomBCD(x = 1, y = 1, n1 = 10, n2 = 10, p1 = 0.3, p2 = 0.6, lambda = 1)

Random Sampling: rbinomBCD()

Generate samples from the BBCD using:

rbinomBCD(n, n1, n2, p1, p2, lambda)

Example

set.seed(123)
samples <- rbinomBCD(n = 100, n1 = 10, n2 = 10, p1 = 0.5, p2 = 0.4, lambda = 1.2)
head(samples)

Maximum Likelihood Estimation: MLEbinomBCD()

Estimate the parameters of the distribution from data.

Example

data <- rbinomBCD(n = 100, n1 = 6, n2 = 4, p1 = 0.6, p2 = 0.3, lambda = 1.5)
fit <- MLEbinomBCD(data)
fit

You may also fix known values for n1 and n2:

MLEbinomBCD(data, fixed_n1 = 6, fixed_n2 = 4)

Real Data Example

The dataset shacc is related to accident records for 122 experienced railway shunters across two historical periods.

data(shacc)
head(shacc)
plot(shacc$X, shacc$Y, xlab = "Accidents 1937–42", ylab = "Accidents 1943–47")

fit <- MLEbinomBCD(shacc, fixed_n1 = 33, fixed_n2 = 27)
FTtest(shacc, "BBCD", params = fit, num_params = 3)

The dataset seedplant records the number of seeds sown and the number of resulting plants grown over plots of fixed area (5 square feet).

data(seedplant)
head(seedplant)
plot(seedplant$X, seedplant$Y, xlab = "Seeds Sown", ylab = "Plants Grown")

EstParams <- MLEbinomBCD(shacc, fixed_n1 = 13, fixed_n2 = 11)
FTtest(shacc, "BBCD", params = EstParams, num_params = 3)

Reference: Ghosh, I., Marques, F., & Chakraborty, S. (2025). A form of bivariate binomial conditionals distributions. Communications in Statistics - Theory and Methods, 54(2), 534–553.

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