2. The Katz Family of Distributions
3. The Poisson, Negative Binomial, and Katz Regression Models
3.1. The Poisson Regression Model
3.2. The Classical Negative Binomial Regression Model
3.3. The Katz Regression Model
4. Testing for Over-dispersion in Poisson Regression model
4.1. The Katz Likelihood
4.2. Point Optimal Tests
4.3. Score Test
4.4. Simulation Experiments
4.4.1. The Unconditional Model
4.4.2. The Katz Regression
5. Hellinger Distance
5.1. The Poisson Distribution
5.2. The Katz Distribution
Conflicts of Interest
Appendix A. On Left-Truncated Katz Distributions
Appendix A.1. Support and Parameter Spaces
- From (A1) we see that if for any then for all . In particular, if then for all . But this leads to violation of (4), that is, probabilities do not sum to unity, and so we exclude from further consideration. Equally, if , so that the pmf of Y is degenerate at L, which is a case that we have already excluded from further analysis. Hereafter, we assume that .
- If and then we have a pmf degenerate at L unless , which will be assumed hereafter. In this case there is no implied restriction on the upper bound of , that is, . Of course, the concern when generating an infinite sequence of probabilities is to ensure that the associated series, , converges. This can be examined by considering the quantityFrom the limit version of d’Alembert’s ratio test we see that the series converges because .
- Similar in effect to the previous case, if then . Given , as assumed above, if and only if which will be assumed, hereafter, for all cases where .
- Because , if and we see that for all and so here the support of the pmf of Y is unbounded from above and independent of the values taken by and . Again, we can establish convergence of the corresponding series. HereAppealing again to the limit version of d’Alembert’s ratio test we see that the series converges if , diverges if , but the test is inconclusive if . Expanding the denominator of in power series yieldsApplying Gauss’s test,23 we see that the series will converge absolutely if and only if but will otherwise diverge. Here we have assumed that and so . Hence, the series is divergent for .
- In this case there is no value of y that satisfies and so cannot belong to . Moreover, this statement remains true even if . Consequently, in this case, the pmf of Y is degenerate at L, a situation that we have chosen to exclude from further consideration.
- of different signIn this case we see that can change sign as y increases, unlike the situation of the previous two cases. Let n denote the smallest value of y such that . Then n is the largest value in . There are only two cases to consider here (having treated that of above): (i) , and (ii) .
- If then the pmf of Y will be degenerate at L which, as explained above, is statistically uninteresting and a situation that we will assume away. That is, if then we will assume that . In particular, if then this requirement reduces to . As y increases, will approach zero from above. That value of y for which is first less than or equal to zero is the largest value of y in and shall be denoted by n, so that is well-defined but is not.24 That is, n is the smallest integer greater than or equal to . This is the definition of the so-called ceiling function, written . In summary, if then we see that the upper bound on the support of the pmf of Y is a function of the parameters and , with the space of subject to the constraint .
- Here is an increasing function of y but the pmf of Y is non-degenerate at L if and only if . As we have already excluded from further consideration pmfs degenerate at L we here assume this to be the case. In particular, when we have a contradiction as we are assuming both , which is required when (see ), and ; we conclude that and can only arise when . As for all , will be unbounded from above provided that the series of probabilities so formed is convergent. Using the analysis outlined in , applying the ratio test we find convergence for all provided that . Moreover, if , Gauss’s test gives convergence provided that is strictly negative, that is, .
Appendix A.2. Probability Mass Functions and Their Properties
Appendix A.2.1. L = 0
Appendix A.2.2. L > 0
- if then
- if then, on noting that ,
- if then
- If thenIn the special case , the quantity in the square brackets reduces to unity and
- If thenA comparison of this expression with that at (A8) reveals a remarkable similarity to the case where and . As in the earlier case we see that (i) the ratio is negative, (ii) there is a scale factor reflecting left-truncation, with the only substantial difference being that whereas here we have a series reducing to the term , in the earlier case we had a sum that only offers a similar simplification when is integer.
- The final case to consider is that where and . Here
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This family of distributions, and extensions to it, have proved important in the actuarial modelling of claims; see, for example, Hess et al. (2002); Panjer (1981); Sundt and Jewell (1981); Willmot (1988), and Pestana and Velosa (2004). Johnson et al. (1993, chp. 2) provides an extensive discussion of both the Katz family and various other, often related, families of discrete distributions. Although, in respect of the Katz family of distributions alone, the treatment in Johnson and Kotz (1969, chp. 2.4) is more complete; see also Gurland (2006) for a more recent treatment.
The one caveat to this observation is that the use of higher order moments may provide some power against models which share low order moments, thereby creating a class of implicit null hypotheses (Davidson and MacKinnon 1987).
Numerous extensions soon followed; see, for example, Bardwell and Crow (1964); Crow and Bardwell (1965); Ord (1967a, 1967b); Staff (1964, 1967) and Kemp (1968). Here we only briefly sketch some key ideas. For a more complete treatment of such families of distributions see, for example, any of Johnson et al. (1993, chp. 2.3), Ord (1972, chp. 5), or Dacey (1972).
Observe that the Pochhammer symbol , where y is a non-negative integer. Note that r can be negative. If r is a negative integer then for all . If r is a positive integer then .
When is integer the resulting pmfs are sometimes referred to as those of Pascal distributions, with the term negative binomial reserved for the more general case of not necessarily integer.
Similarly, the Poisson approximation to the Binomial reduces to for fixed , which is also a more intuitive statement of how parameters must evolve for the approximation to work than is typically encountered.
We shall persist with the abuse of notation inherent in expressions like rather than, say, a more complete notation along the lines of , for the sake of the notational economy it affords.
Common variants of this argument include: (i) Lee (1986), who specifies the gamma distribution in terms of the shape and scale (or inverse rate) () parameters, that is, , and (ii) Cameron and Trivedi (1986), who use the so-called index form of the gamma distribution, which is specified in terms of the shape and mean () parameters, that is, . Cameron and Trivedi (1986) call the shape parameter () the index or precision parameter.
Moments for the gamma distribution specifications given in Footnote 10 follow immediately on making the appropriate substitution for .
Other values of k yield the Negbin P, or NBP, model (Greene 2008).
Strictly, it is not a generalized linear model as it stands but, conditioning on one of the parameters allows it to be treated so. This parameter can then be estimated conditional on the remaining parameters, which yields a two-step iterative estimation procedure. See, for example, either Hilbe (2011) or Hilbe (2014) for a discussion of the steps involved.
Alternatively, using similar averaging arguments to those seen previously for the NBRM, if we average with respect to , where and , then we obtain a more common form of the negative binomial pmf.
Note that the mean and variance of this distribution are given by (Appendix A.2.2). In contrast with the developments of (11), there is nothing in this model that requires that both the parameters of the mixing gamma distribution vary with the index i. Nor need they be linked in any restrictive way. Specifically, if we were to follow the developments of Greene (2008) who equates the parameters of the mixing distribution, we find that
Strictly, Katz (1965) adopted an approach more in keeping with a method of moments test. Specifically, he looked at the difference between estimators for the mean and variance, which should be equal under the null and then scaled this difference appropriately to obtain a distribution under the null. In any event, the statistic so obtained is the same as the one proposed by Lee (1986) that we consider here.
Lee (1986) proposed other tests than the one considered here, although he did not compare them numerically The results recorded in Miller (1998) suggests that those involving third order moments may have better power properties. For now we are primarily concerned with proof of concept and do not explore these other tests in light of the simplicity of (17).
We also considered an alternative test based on the t-statistic in the regression of on a constant but there was little difference in performance. These tests correspond to the Negbin I and II cases above.
See, for example, https://en.wikipedia.org/wiki/Hellinger_distance.
In their extensions to this class of distributions, Panjer (1981); Sundt and Jewell (1981) and Willmot (1988) adopt a slightly different parameterization, specifically , . Equivalence with (A1) is seemingly established on setting and , although there are differences in the support of the resulting variables. In particular, is specifically excluded from this definition and hence many of the probability distributions claimed to satisfy the recursion in this form are not completely defined by it.
In essence, this is the same as adopting the convention that any negative probabilities are set to zero. It might be argued that this is at odds with Katz’s original assumptions and should be excluded. Our justification for the inclusion in our analysis of these distributions where is non-integer, is that Katz himself included them.
The condition integer obviously requires .
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