Bivariate COM-Poisson Distribution: Formulation , Properties, and Inference

1. Bivariate COM-Poisson Distribution: Formulation, Properties, and Inference Associate Professor of Statistics Department of Mathematics and Statistics Georgetown University Research Mathematical Statistician Center for Statistical Research & Methodology U.S. Census Bureau Kimberly F. Sellers Reference: Sellers KF, Morris DS, Balakrishnan N (2016) Bivariate Conway-Maxwell-Poisson Distribution: Formulation, Properties, and Inference, Journal of Multivariate Analysis 150: 152-168. This presentation describes previously conducted and/or ongoing research and analysis, and is released to inform interested parties and encourage discussion. The views expressed here are those of the presenter and not necessarily those of the U.S. Census Bureau.

2. Background • Marshall and Olkin (1985) used bivariate Bernoulli, exploit well-known relationships between Bernoulli, Poisson, and geometric distributions to generate bivariate analogs

3. Bivariate Bernoulli Distribution(Marshall & Olkin, 1985; Kocherlakota & Kocherlakota, 1992; Johnson et al., 1997)

4. Bivariate Bernoulli: Properties • Covariance and Correlation: • Joint pgf:

5. Bivariate Poisson Joint PGF (Option 1)(Marshall & Olkin, 1985; Kocherlakota & Kocherlakota, 1992; Johnson et al., 1997) • Trivariate reduction: • Let • Let

6. Bivariate Poisson Joint PGF (Option 2)(Marshall & Olkin, 1985; Kocherlakota & Kocherlakota, 1992; Johnson et al., 1997) • Compound bivariate binomial with Poisson: • Let and have joint conditional pgf • Unconditional joint pgf becomes

7. Bivariate Poisson Joint PGF(Marshall & Olkin, 1985; Kocherlakota & Kocherlakota, 1992; Johnson et al., 1997) • Via trivariate reduction method: • Via compounding method:

8. Bivariate Poisson PMF • The pmf is • Form is attained either via trivariate reduction or compounding

9. Bivariate Poisson: Properties • Covariance: • Correlation: • Regression:

10. Problem: Data Dispersion • Bivariate Poisson assumes equi-dispersion in the random variables, but what if the data don’t satisfy this constraint? • If over-dispersed, can consider bivariate negative binomial distribution • What if data are under-dispersed?

11. Bivariate Generalized Poisson Dist.(Famoye and Consul, 1995) • Joint pmf: • when • Obtained via trivariate reduction method • Let • Let • Problem when data significantly under-dispersed

12. Alternative: Consider Bivariate CMP • Conway-Maxwell-Poisson (COM-Poisson or CMP) distribution is flexible, two-parameter count model allowing for data over-, equi-, or under-dispersion • Developing a bivariate CMP distribution should likewise capture data over-/under-dispersion • First, introducing the univariate CMP distribution

13. COM-Poisson Distribution(Conway and Maxwell, 1962; Shmueli et al., 2005) • COM-Poisson pmf for random variable : • Special cases:

14. COM-Poisson Distribution Properties • Has exponential family form • Ratio between probabilities of two consecutive values is

15. COM-Poisson Distribution Properties • Expected value: • Variance: • Approximations hold for n < 1 or l > 10n

16. COM-Poisson Distribution Properties • Moment generating function: • Probability generating function: • Sums of Poissons  Poisson, but sums of COM-Poissons  COM-Poisson!

17. Background: Bivariate CMP • Marshall and Olkin (1985) used bivariate Bernoulli, exploit well-known relationships between Bernoulli, Poisson, and geometric distributions to generate bivariate analogs • This work develops bivariate Conway-Maxwell-Poisson (bivariate COM-Poisson/CMP) distribution: • flexible distribution family capturing over-/under-dispersion • includes bivariate Bernoulli, bivariate geometric, bivariate Poisson distributions as special cases

18. Derivation of BivCMP PGF • Use compounding method: • Let and have conditional pgf • Unconditional joint pgf:

19. Bivariate CMP: Other Generating Functions • Moment generating function: • Factorial moment generating function: • Cumulant generating function:

20. BivCMP PGF : Special Cases • BivCMP captures bivariate special cases • For Biv. Poisson (): • For Biv. Bern. ():

21. BivCMPPGF: Special Cases (cont.) • Bernoulli table of the form,

22. BivCMP PGF : Special Cases (cont.) • For Biv. Geom. : where

23. Bivariate CMP: Marginal PGFs • Marginal pgfs for and , respectively: • Marginal distributions not COM-Poisson • Poisson marginals:

24. Bivariate COM-Poisson PMF pmf has the form:

25. Bivariate COM-Poisson Moments where

26. BivCMP: Other Properties • Denote • Regression of on : • Conditional pgf of given :

27. Parameter Estimation: MLEs • Maximize the log-likelihood, with respect to • Done iteratively via Newton-type algorithm (nlminbin R) • Bivariate Poisson MLEs serve as starting values

28. Hypothesis Testing • Does there exist statistically significant data dispersion? • LRT statistic: • Analogous tests for and Biv Poisson MLE likelihood Biv COM-Poisson MLE likelihood

29. R Computing • Numerical equivalences can be shown between bivCMP and special case distribution functions (pbivpois (in bivpois) and dbinom2.or (VGAM)) • bivpois: Bivariate Poisson models using the EM algorithm (Karlis and Ntzoufras, 2005) • Contains functions for fitting bivariate Poisson models using the EM algorithm • pbivpois computes bivariate Poisson probability at (x,y)

30. R Computing (cont.) • VGAM: Vector Generalized Linear and Additive Models (Yee, 2014) • Six major classes of regression models implemented, where models and distributions are estimated via maximum likelihood estimation or penalized MLE • dbinom2.or/rbinom.or computes density or random number generation for bivariate binary regression model using an odds ratio as dependency measure

31. dbivCMP vs. pbivpois • Returns the probability of bivariate distribution at () • pbivpois: bivariate Poisson • Inputs: • dbivCMP: bivariate COM-Poisson • Inputs: • Equivalence achieved when, for any and , set

32. dbivCMP vs. dbinom2.or • Returns the probability of bivariate distribution at () • dbinom2.or: bivariate Bernoulli • Inputs: • dbivCMP: bivariate COM-Poisson • Inputs: • Equivalence achieved when, for any and , dbinom2.or probabilities, dbivCMP probabilities

33. What about comparing with the bivariate geometric distribution? • There is no pre-packaged function available to compare our dbivCMPfunction to the bivariate geometric distribution.

34. Simulated Example 1: Bivariate Poisson • Generated 500 data pairs via rpois (stats) • , , • Use trivariate reduction to produce bivariate data: • and

35. Simulated Example 1: Bivariate Poisson (cont.) • Close log-likelihoods • Estimated dispersion close to 1, yet potentially indicating slight underdispersion in data simulation ()

36. Simulated Example 1: Bivariate Poisson (cont.) • Three approaches considered for estimating via the COM-Poisson MLEs: • Assume ,, • Replace in the computations described in Option 1 with • Replace in the computations described in Option 1 with

37. Simulated Example 1: Bivariate Poisson (cont.)

38. Simulated Example 1: Bivariate Poisson (cont.) • Is the simulated bivariate Poisson dataset actually significantly (under-) dispersed? • Data dispersion not statistically significant (; )

39. Simulated Example 2: Bivariate Bernoulli • Generated 500 data pairs using rbinom2.or (VGAM) with , ,noting exchangeable distribution • Equates to generating data from a bivariate Bernoulli distribution of the form (rounded to three significant digits) :

40. Simulated Example 2: Bivariate Bernoulli (cont.) • Bivariate COM-Poisson outperforms bivariate Poisson • , indicating dataset is (approximately) Bernoulli distributed • Statistically significant data dispersion • ;

41. Simulated Example 2: Bivariate Bernoulli (cont.) • Assuming the Bernoulli model vs. Truth Estimate

42. Simulated Example 3: Bivariate Geometric • Li and Dhar (2013) provide a simulated sample dataset of 20 data pairs generated from the bivariate geometric model of Basu and Dhar (1995)

43. Simulated Example 3: Bivariate Geometric (cont.) • Bivariate CMP outperforms bivariate Poisson • Statistically significant data over-dispersion detected • bivariate geometric distribution? • Biv. geometric reasonable

44. Example: Accident Proneness Data set regards number of accidents incurred by 122 shunters in 1937-1942 and 1943-1947

45. Example: Accident Proneness Several authors consider various bivariate models to describe this dataset: • Bivariate Poisson: Holgate(1964) • Bivariate negative binomial: Adelstein (1951), and Arbous and Kerrick (1951) • Bivariate generalized Poisson: Famoye and Consul (1995) How does the bivariate COM-Poisson compare?

46. Example: Accident Proneness (cont.) • data overdispersion • Stat. signif. data over-dispersion detected • Potentially modeled via bivariate geometric

47. Example: Accident Proneness (cont.)

48. Further Generalized Bivariate Distribution • Bivariate COM-Poisson captures bivariate Poisson, bivariate Bernoulli, and bivariate geometric distribution • Summing each of the special case distributions leads to bivariate forms of other classical distributions • Consider bivariate sum-of-COM-Poissons(sCOM-Poisson or sCMP) distribution

49. (univariate) sCMP Distribution • Let where iid • has pmf • Special cases: • : Poisson • : NegBin • : Bin • : COM-Poisson

50. sCOM-Poisson: Properties • Invariance under addition • and