(a,b,0) class of distributions

In probability theory, the distribution of a discrete random variable N whose values are nonnegative integers is said to be a member of the (a, b, 0) class of distributions if its probability mass function obeys

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

where $p_{k}=P(N=k)$ (provided $a$ and $b$ exist and are real).

There are only three discrete distributions that satisfy the full form of this relationship: the Poisson, binomial and negative binomial distributions. 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:^[1] 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.^[2]

Properties

Sundt^[3] 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. 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	$P[N=k]\,$	$a\,$	$b\,$	$p_{0}\,$	$W_{N}(x)\,$	$E[N]\,$	$Var(N)\,$
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	${\frac {\Gamma (r+k)}{k!\,\Gamma (r)}}\,p^{r}\,(1-p)^{k}\,$	$1-p\,$	$(1-p)(r-1)\,$	$p^{r}\,$	$\left({\frac {p}{1-x(1-p)}}\right)^{r}\,$	${\frac {r(1-p)}{p}}\,$	${\frac {r(1-p)}{p^{2}}}\,$

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

↑ 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 2009-06-20. 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 0-471-21577-5.
↑ Sundt, Bjørn; Jewell, William S. (1981). "Further results on recursive evaluation of compound distributions" (PDF). ASTIN Bulletin. International Actuarial Association. 12 (1): 27–39.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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(a,b,0) class of distributions

Properties

Plotting

See also

References