Goodness of fit test for the bivariate Poisson distribution {bivpois}

Goodness of fit test for the bivariate Poisson distribution

Description

Goodness of fit test for the bivariate Poisson distribution.

Usage

bp.gof(x1, x2 = NULL, R = 999)
bp.gof2(x1, x2 = NULL, R = 999)

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.

R

The number of Monte Carlo replicates to use.

Details

Kocherlakota and Kocherlakota (1992) mention the following a goodness of fit test for the bivariate Poisson distribution, the index of dispersion test. They mention that Loukas and Kemp (1986) developed this test as an extension of the univariate dispersion test. They test for departures from the bivariate Poisson againsta alternatives which involve an increase in the generalised variance, the determinant of the covariance matrix of the two variables.

Rayner, Thas and Best (2009) mentions a revised version of this test whose test statistic is now given by

I_{B^*}=\frac{n}{1-r^2}\left(\frac{S_1^2}{\bar{x}_1}-2r^2\sqrt{\frac{S_1^2}{\bar{x}_1}\frac{S_2^2}{\bar{x}_2}}+\frac{S_2^2}{\bar{x}_2}\right),

where n is the sample size, r is the sample Pearson correlation coefficient, S_1^2 and S_2^2 are the two sample variances and \bar{x}_1 and \bar{x}_2 are the two sample means. Under the null hypothesis the I_{B^*} follows asymptotically a \chi^2 with 2n-3 degrees of freedom. However, I did some simulations and I saw that it does not perform very well in terms of the type I error. If you see the simulations in their book (page 132) you will see this. For this reason, the function calculates the p-value of the I_{B^*} using Monte Carlo (or parametric bootstrap).

The second function, bp.gof2(), is a vectorised version of the first, and much faster. I put both of them here to show how one can vectorize a function and make it faster.

Value

A list including:

runtime

The duration of the algorithm.

tab

The contingency table of the two variables.

pvalue

The Monte-Carlo based estimated p-value.

Author(s)

Michail Tsagris.

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

References

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

Loukas S. and Kemp C. (1986). The index of dispersion test for the bivariate Poisson distribution. Biometrics, 42(4): 941–948.

Rayner J. C., Thas O. and Best D. J. (2009). Smooth Tests of Goodness of Fit: Using R. John Wiley & Sons.

See Also

bp.mle

Examples

x <- rbp( 300, c(3, 5, 2) )
bp.gof(x)
bp.gof2(x)