Loss Models: From Data to Decisions, 4th Edition

B.3 The (a, b, 1) class

To distinguish this class from the (a, b, 0) class, the probabilities are denoted Pr(N = k) = p^M_k or Pr (N = k) = p^T_k depending on which subclass is being represented. For this class, p^M₀ is arbitrary (i.e., it is a parameter), and then p^M₁ or p^T₁ is a specified function of the parameters a and b. Subsequent probabilities are obtained recursively as in the (a, b, 0) class: p^M_k = (a + b/k)p^M_k-1, k = 2, 3,…, with the same recursion for p^T_k. There are two subclasses of this class. When discussing their members, we often refer to the “corresponding” member of the (a, b, 0) class. This refers to the member of that class with the same values for a and b. The notation p_k will continue to be used for probabilities for the corresponding (a, b, 0) distribution.

B.3.1 The zero-truncated subclass

The members of this class have p^T₀ = 0, and therefore it need not be estimated. These distributions should only be used when a value of zero is impossible. The first factorial moment is μ₍₁₎ = (a + b)/[(1 − a)(1 − p₀)], where p₀ is the value for the corresponding member of the (a, b, 0) class. For the logarithmic distribution (which has no corresponding member), μ₍₁₎ = β/ln(1 + β). Higher factorial moments are obtained recursively with the same formula as with the (a, b, 0) class. The variance is (a + b) [1 − (a + b + 1)p₀]/[(1 − a)(1 − p₀)]². For those members of the subclass that have corresponding (a, b, 0) distributions, p^T_k = P_k/(1 − p₀).

B.3.1.1 Zero-truncated Poisson ...