(a,b,0) class of distributions

In probability theory, a member of the (a, b, 0) class of distributions is any distribution of a discrete random variable N whose values are nonnegative integers whose probability mass function satisfies the recurrence formula

{\frac {p_{k}}{p_{k-1}}}=a+{\frac {b}{k}},\qquad k=1,2,3,\dots

for some real numbers a and b, where $p_{k}=P(N=k)$ .

The (a,b,0) class of distributions is also known as the Panjer,^[1]^[2] the Poisson-type or the Katz family of distributions,^[3]^[4] and may be retrieved through the Conway–Maxwell–Poisson distribution.

Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF).

More general distributions can be defined by fixing some initial values of p_j and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of k:^[5] for example the logarithmic distribution and the discrete uniform distribution.

The (a, b, 0) class of distributions has important applications in actuarial science in the context of loss models.^[6]

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Captions are on. To turn off, click the CC button at bottom right. Follow us on Twitter (@amoebasisters) and Facebook! There’s a lot of phenotypes that are easy to tell. Your eye color, your hair texture, your height, whether you have a straight thumb or hitchhiker thumb. But one phenotype that you can’t just tell by looking is your blood type. Your blood is really made of many things----platelets, plasma, and red blood cells. But you have probably heard before that when blood is donated, it’s important that it is matched correctly? And that’s true, because blood type phenotypes vary. It really boils down to the fact that red blood cells are not naked. They have proteins on their surface. And it turns out that your immune system is very protective and if it gets blood donated that have different proteins that it’s not used to, it will attack them! With blood type, you can have several different phenotypes: A, B, AB, or O. These letters stand for antigens that are found on red blood cells. So type A blood, for example, has A antigens on the surface of red blood cells. Type B blood, for example, has B antigens on the surface of red blood cells. Type AB blood has both A and B antigens on the surface of red blood cells. Type O---I like to think of it is looking like a zero---it doesn’t have A or B antigens. It’s naked! Well, ok, it does have other proteins on its surface. But not A or B. So think O is zero! So if you are type B blood, you have B antigens on the surface of red blood cells. That means, a person with type B can accept another type B person’s blood because it recognizes the B antigen. But if you try to give that person a type A blood type, that’s an antigen that the immune system does not recognize. It will attack! It would also attack AB blood, because that includes the A that it doesn’t recognize. Type O would be safe though. Remember O looks like a zero---it doesn’t have A or B antigens. So type O can donate to everyone! Now while O individuals can donate to everyone, they can only receive blood from another type O. Since type O blood does not have A or B antigens, their immune system will attack any other blood type that does. Neither of us have type AB blood, but this is a cool blood type to have in the sense that you could receive blood from anyone. If an AB person receives blood from a type A person, they’ve got that so it’s all good. If an AB person receives blood from a type B person, they’ve got that, so it’s all good. They can receive blood from type O of course as well---there’s not any antigens to even worry about. One thing we want to add that makes all of this a bit more complicated---blood types also have a plus or minus sign listed by the blood type. This also makes a big difference with blood donations. If you have a plus, it means that you have another little protein called Rh factor on the surface of your blood cells. If you have a negative, it means that you do not have this little protein called Rh factor on the surface of your red blood cells. We are not going to be able to go into that in this short clip so---to the google for that if you would like to learn more about that---it is also an inherited trait just like blood type. Blood type is genetically inherited and a great example of multiple alleles. Remember that alleles are a form of a gene---like a flavor. Just an analogy. So let’s put this into practice. Let’s say a couple gives birth to a baby boy. Both parents have type A blood. But then, there was a mixup at the hospital! And now----haha, sorry I get real into my drama----there are these two baby boys and the hospital doesn’t know which one belongs to the couple! Probably not a hospital where you want to have kids but let’s try and use our blood type genetic problem solving skills to help out here. The babies are named baby Phil and baby Sylvester. Two names that have never been in my classroom before. Baby Phil is type B blood. Baby Sylvester is type O blood. Could either of these babies belong to the couple, who both have type A blood? Let’s talk phenotypes and genotypes. The phenotype of type A blood is A. But the genotype is written like this IAIA or IAi. This format of writing helps with multiple alleles like blood type problems, and we can show you why when we work out the Punnett square. The I, in case you are wondering, stands for immunoglobin. Now you may notice that I said that blood type A could be IAIA or IAi. Without testing, you don’t really know. You can consider IAIA to be homozygous and IAi to be heterozygous. The phenotype of type B blood is B. But the genotype is written is IBIB or IBi. The phenotype of type AB blood is AB. But the genotype is written IAIB. There is no other way to write that one. The phenotype of type O blood is O. And the genotype is written ii. Remember how I said that type O looks like a zero and you can think of that as having zero blood type antigens? Well, that’s what it is. ii. No coefficient. None. Back to the babies---we have baby Phil with type B blood and baby Sylvester with type O blood. We are told the parents both have type A, but remember that we don’t know whether that means they are IAIA or IAi. And the mom could be one of those genotypes and the father could be the other genotype---or they could both be the same genotype---we don’t know. So let’s just try it out with all possibilities. Remember that we write the parent genotypes on the top and side of the Punnett squares, like this. We fill in the squares and make sure to have capital I’s first just for formatting. Ok so is it possible to get baby Phil---who has type B in any of these offspring? No. He must be someone else’s baby. What about baby Sylvester with his type O? YES! But both parents would have to be heterozygous A---that means both were IAi----then, yes, you have a 25% chance here of having a baby with type O blood. Of course, if I was advising the couple, I would also insist on a DNA test as this blood type problem only shows that it’s possible that baby Sylvester is theirs. Well that’s it for the amoeba sisters---and we remind you to stay curious! This video has a handout at www.amoebasisters.com. Find us on Facebook and Twitter (@amoebasisters)!

Properties

Sundt^[7] proved that only the binomial distribution, the Poisson distribution and the negative binomial distribution belong to this class of distributions, with each distribution being represented by a different sign of a. Furthermore, it was shown by Fackler^[2] that there is a universal formula for all three distributions, called the (united) Panjer distribution.

The more usual parameters of these distributions are determined by both a and b. The properties of these distributions in relation to the present class of distributions are summarised in the following table. Note that $W_{N}(x)\,$ denotes the probability generating function.

Distribution	$P[N=k]\,$	$a\,$	$b\,$	$p_{0}\,$	$W_{N}(x)\,$	$\operatorname {E} [N]\,$	$\operatorname {Var} (N)\,$
Binomial	${\binom {n}{k}}p^{k}(1-p)^{n-k}$	${\frac {-p}{1-p}}$	${\frac {p(n+1)}{1-p}}$	$(1-p)^{n}\,$	$(px+(1-p))^{n}\,$	$np\,$	$np(1-p)\,$
Poisson	$e^{-\lambda }{\frac {\lambda ^{k}}{k!}}\,$	$0\,$	$\lambda \,$	$e^{-\lambda }\,$	$e^{\lambda (x-1)}\,$	$\lambda \,$	$\lambda \,$
Negative binomial	${\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}\,$	$1-p\,$	$(1-p)(r-1)\,$	$p^{r}\,$	$\left({\frac {p}{1-x(1-p)}}\right)^{r}\,$	${\frac {r(1-p)}{p}}\,$	${\frac {r(1-p)}{p^{2}}}\,$
Panjer distribution	$\left(1+{\frac {\lambda }{\alpha }}\right)^{-\alpha }{\frac {\lambda ^{k}}{k!}}\prod _{i=0}^{k-1}{\frac {\alpha +i}{\alpha +\lambda }}\,$	${\frac {\lambda }{\alpha +\lambda }}\,$	${\frac {(\alpha -1)\lambda }{\alpha +\lambda }}\,$	$\left(1+{\frac {\lambda }{\alpha }}\right)^{-\alpha }\,$	$\left(1-{\frac {\lambda }{\alpha }}(x-1)\right)^{-\alpha }\,$	$\lambda \,$	$\lambda \left(1+{\frac {\lambda }{\alpha }}\right)\,$

Note that the Panjer distribution reduces to the Poisson distribution in the limit case $\alpha \rightarrow \pm \infty$ ; it coincides with the negative binomial distribution for positive, finite real numbers $\alpha \in \mathbb {R} _{>0}$ , and it equals the binomial distribution for negative integers $-\alpha \in \mathbb {Z}$ .

Plotting

An easy way to quickly determine whether a given sample was taken from a distribution from the (a,b,0) class is by graphing the ratio of two consecutive observed data (multiplied by a constant) against the x-axis.

By multiplying both sides of the recursive formula by $k$ , you get

k\,{\frac {p_{k}}{p_{k-1}}}=ak+b,

which shows that the left side is obviously a linear function of $k$ . When using a sample of $n$ data, an approximation of the $p_{k}$ 's need to be done. If $n_{k}$ represents the number of observations having the value $k$ , then ${\hat {p}}_{k}={\frac {n_{k}}{n}}$ is an unbiased estimator of the true $p_{k}$ .

Therefore, if a linear trend is seen, then it can be assumed that the data is taken from an (a,b,0) distribution. Moreover, the slope of the function would be the parameter $a$ , while the ordinate at the origin would be $b$ .

References

^ Panjer, Harry H. (1981). "Recursive Evaluation of a Family of Compound Distributions" (PDF). ASTIN Bulletin. 12 (1): 22–26.
^ ^a ^b Fackler, Michael (2009). "Panjer class united - one formula for the Poisson, Binomial and Negative Binomial distribution" (PDF). ASTIN Colloquium. International Actuarial Association.
^ Katz, Leo (1965). Ganapati Patil (ed.). Unified treatment of a broad class of discrete probability distributions. Classical and Contagious Discrete Distributions. Pergamon Press, Oxford. pp. 175–182.
^ Gathy, Maude; Lefèvre, Claude (2010). "On the Lagrangian Katz family of distributions as a claim frequency modelDistributions". Insurance: Mathematics and Economics. 47 (1): 78–83. doi:10.1016/j.insmatheco.2010.03.010.
^ Hess, Klaus Th.; Liewald, Anett; Schmidt, Klaus D. (2002). "An extension of Panjer's recursion" (PDF). ASTIN Bulletin. 32 (2): 283–297. doi:10.2143/AST.32.2.1030. Archived (PDF) from the original on 2009-07-11. Retrieved 2009-06-18.
^ Klugman, Stuart; Panjer, Harry; Gordon, Willmot (2004). Loss Models: From Data to Decisions. Series in Probability and Statistics (2nd ed.). New Jersey: Wiley. ISBN 978-0-471-21577-6.
^ Sundt, Bjørn; Jewell, William S. (1981). "Further results on recursive evaluation of compound distributions" (PDF). ASTIN Bulletin. 12 (1): 27–39. doi:10.1017/S0515036100006802.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda