Maximum likelihood estimation of the bivariate Poisson distribution {bivpois}

Maximum likelihood estimation of the bivariate Poisson distribution

Description

Maximum likelihood estimation of the bivariate Poisson distribution.

Usage

bp.mle(x1, x2 = NULL)
bp.mle2(x1, x2 = NULL)

Arguments

x1

Either a numerical vector with the values of the first variable or a matrix with 2 columns containing both variables. In the latter case, x2 must be NULL.

x2

A numerical vector with the values of the second. If x1 is a matrix with 2 columns containing both variables, x2 must be NULL.

Details

Using the addition method (see function rbp) to simulate random values from the bivariate Poisson, its representation is given by

P\left(X=x,Y=y \right)e^{-\left(\lambda_1+\lambda_2+\lambda_3\right)}\frac{\lambda_1^x}{x!}\frac{\lambda_2^y}{y!} \sum_{k=0}^{\min{\left(x,y\right)}} \binom {x} {k} \binom {y} {k} k! \left(\frac{\lambda_3}{\lambda_1\lambda_2}\right)^k.

The above form is found in Karlis and Ntzoufras (2003). This bivariate distribution allows for dependence between the two random variables. Marginally each random variable follows a Poisson distribution with E\left(X\right)=\lambda_1+\lambda_3 and E\left(Y\right)=\lambda_2+\lambda_3. In addition, Cov\left(X,Y\right)=\lambda_3. If \lambda_3=0, the above expression becomes a product of two Poisson distributions. Hence, \lambda_3 is a measure of dependence between the two random variables.

The function bp.mle() returns a lot of information and is slower than bp.mle2(), which returns fewer information, but is faster.

Value

For the function bp.mle() a list including:

lambda

A vector with the estimated values of (\lambda_1, \lambda_2) and \lambda_3. Note that \hat{\lambda}_1=\bar{x}_1 - \lambda_3 and \hat{\lambda}_1=\bar{x}_1 - \lambda_3, where \bar{x}_1 and \bar{x}_2 are the two sample means.

rho

The estimated correlation coefficient, that is: \dfrac{\hat{\lambda}_3}{\sqrt{\left(\hat{\lambda}_1 + \hat{\lambda_3}\right)\left(\hat{\lambda}_2 + \hat{\lambda_3}\right)}}.

ci

The 95% Confidence intervals using the observed and the asymptotic information matrix.

loglik

The log-likelihood values assuming independence (\lambda_3=0) and assuming the bivariate Poisson distribution.

pvalue

Three p-values for testing \lambda_3=0. These are based on the log-likelihood ratio and two Wald tests using the observed and the asymptotic information matrix.

For the function bp.mle2() a list including:

lambda

A vector with the estimated values of (\lambda_1, \lambda_2) and \lambda_3. Note that \hat{\lambda}_1=\bar{x}_1 - \lambda_3 and \hat{\lambda}_1=\bar{x}_1 - \lambda_3, where \bar{x}_1 and \bar{x}_2 are the two sample means.

loglik

The log-likelihood values assuming independence (\lambda_3=0) and assuming the bivariate Poisson distribution.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Kawamura K. (1984). Direct calculation of maximum likelihood estimator for the bivariate Poisson distribution. Kodai Mathematical Journal, 7(2): 211–221.

Kocherlakota S. and Kocherlakota K. (1992). Bivariate discrete distributions. CRC Press.

Karlis D. and Ntzoufras I. (2003). Analysis of sports data by using bivariate poisson models. Journal of the Royal Statistical Society: Series D (The Statistician), 52(3): 381–393.

See Also

rbp

Examples

x <- rbp( 300, c(3, 5, 2) )
bp.mle(x)