(a,b,0) class of distributions

In probability, a discrete probability density function of a random variable $X$ is said to be a member of the ("a", "b", 0) class of distributions if

: $frac\{p\_k\}\{p\_\{k-1\; =\; a\; +\; frac\{b\}\{k\},\; qquad\; k\; =\; 1,\; 2,\; 3,\; dots$

where $p\_k\; =\; P(X\; =\; k)$ (provided $a$ and $b$ exist and are real).

There are only three density functions that satisfy this relationship: the Poisson, binomial and negative binomial distributions.

The "a" and "b" parameters

For each density function, the values of $a$ and $b$ can be found using the parameters of the distribution.

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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

* Klugman, Stuart; Panjer, Harry; Gordon, Willmot (2004). "Loss Models: From Data to Decisions", 2nd edition, New Jersey: Wiley Series in Probability and Statistics. ISBN 0-471-21577-5