Goodness-of-Fit Tests for Poisson Distribution

Performs the mean distance goodness-of-fit test and the energy goodness-of-fit test of Poisson distribution with unknown parameter.

poisson.e(x)
poisson.m(x)
poisson.etest(x, R)
poisson.mtest(x, R)
poisson.tests(x, R, test="all")

Arguments

x: vector of nonnegative integers, the sample data
R: number of bootstrap replicates
test: name of test(s)

Details

Two distance-based tests of Poissonity are applied in poisson.tests, "M" and "E". The default is to do all tests and return results in a data frame. Valid choices for test are "M", "E", or "all" with default "all".

If "all" tests, all tests are performed by a single parametric bootstrap computing all test statistics on each sample.

The "M" choice is two tests, one based on a Cramer-von Mises distance and the other an Anderson-Darling distance. The "E" choice is the energy goodness-of-fit test.

R must be a positive integer for a test. If R is missing or 0, a warning is printed but test statistics are computed (without testing).

The mean distance test of Poissonity (M-test) is based on the result that the sequence of expected values E|X-j|, j=0,1,2,... characterizes the distribution of the random variable X. As an application of this characterization one can get an estimator $\hat F(j)$ of the CDF. The test statistic (see poisson.m) is a Cramer-von Mises type of distance, with M-estimates replacing the usual EDF estimates of the CDF: $$M_n = n\sum_{j=0}^\infty (\hat F(j) - F(j\;; \hat \lambda))^2 f(j\;; \hat \lambda).$$

In poisson.tests, an Anderson-Darling type of weight is also applied when test="M" or test="all".

The tests are implemented by parametric bootstrap with R replicates.

An energy goodness-of-fit test (E) is based on the test statistic $$Q_n = n (\frac{2}{n} \sum_{i=1}^n E|x_i - X| - E|X-X'| - \frac{1}{n^2} \sum_{i,j=1}^n |x_i - x_j|, $$ where X and X' are iid with the hypothesized null distribution. For a test of H: X ~ Poisson($\lambda$), we can express E|X-X'| in terms of Bessel functions, and E|x_i - X| in terms of the CDF of Poisson($\lambda$).

If test=="all" or not specified, all tests are run with a single parametric bootstrap. poisson.mtest implements only the Poisson M-test with Cramer-von Mises type distance. poisson.etest implements only the Poisson energy test.

Value

The functions poisson.m and poisson.e return the test statistics. The function poisson.mtest or poisson.etest return an htest object containing

method: Description of test
statistic: observed value of the test statistic
p.value: approximate p-value of the test
data.name: replicates R
estimate: sample mean

poisson.tests returns "M-CvM test", "M-AD test" and "Energy test" results in a data frame with columns

estimate: sample mean
statistic: observed value of the test statistic
p.value: approximate p-value of the test
method: Description of test

which can be coerced to a tibble.

Note

The running time of the M test is much faster than the E-test.

References

Szekely, G. J. and Rizzo, M. L. (2004) Mean Distance Test of Poisson Distribution, Statistics and Probability Letters, 67/3, 241-247. doi:10.1016/j.spl.2004.01.005 .

Szekely, G. J. and Rizzo, M. L. (2005) A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93/1, 58-80, doi:10.1016/j.jmva.2003.12.002 .

Author

Maria L. Rizzo mrizzo@bgsu.edu and Gabor J. Szekely

Examples

 x <- rpois(50, 2)
 poisson.m(x)
#>      M-CvM       M-AD 
#> 0.07368603 0.42332826 
 poisson.e(x)
#>         E 
#> 0.6370008 
 # \donttest{
 poisson.etest(x, R=199)
#> 
#> 	Poisson E-test
#> 
#> data:  replicates: 199
#> E = 0.637, p-value = 0.4623
#> sample estimates:
#> [1] 2.06
#> 
 poisson.mtest(x, R=199)
#> 
#> 	Poisson M-test
#> 
#> data:  x replicates:  199
#> M-CvM = 0.073686, p-value = 0.4422
#> sample estimates:
#> [1] 2.06
#> 
 poisson.tests(x, R=199)
#>       estimate  statistic   p.value      method
#> M-CvM     2.06 0.07368603 0.4773869  M-CvM test
#> M-AD      2.06 0.42332826 0.4673367   M-AD test
#> E         2.06 0.63700084 0.4673367 Energy test
 # }