(a,b,0) class of distributions

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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 pk = 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.

[edit] The a and b parameters

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

\begin{array}{l|c|c|c|c} \hline & P(X = k) & a & b & p_0\\ \hline \mathrm{Poisson}(\lambda) & e^{-\lambda} \, \frac{\lambda^k}{k!} & 0 & \lambda & e^{-\lambda}\\ \mathrm{Bin}(n,p) & {n \choose k} \, p^k \, (1-p)^{n-k} & - \frac{p}{1-p} & (n+1) \, \frac{p}{1-p} & (1-p)^n \\ \mathrm{Neg\,Bin}(r, \beta) & {k+r-1 \choose k} \, \left( \frac{1}{1+\beta} \right)^r \, \left( \frac{\beta}{1+\beta} \right)^k & \frac{\beta}{1+\beta} & (r-1) \, \frac{\beta}{1+\beta} & \left( \frac{1}{1+\beta} \right)^r\\ \hline \end{array}

[edit] 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 pk's need to be done. If nk represents the number of observations having the value k, then \hat{p}_k = \frac{n_k}{n} is an unbiased estimator of the true pk.

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.

[edit] 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