Benesi–Hildebrand method

2013-10-17T07:46:35Z

130.238.59.66: /* Limitations */

In [[probability theory]], the distribution of a [[discrete random variable]] <math>N</math> is said to be a member of the '''(''a'', ''b'', 0) class of distributions''' if its [[probability mass function]] obeys

: <math>\frac{p_k}{p_{k-1}} = a + \frac{b}{k}, \qquad k = 1, 2, 3, \dots</math>

where <math>p_k = P(N = k)</math> (provided <math>a</math> and <math>b</math> exist and are real).

There are only three discrete distributions that satisfy the full form of this relationship: the [[Poisson distribution|Poisson]], [[Binomial distribution|binomial]] and [[Negative binomial distribution|negative binomial]] distributions. These are also the three discrete distributions among the six members of the [[natural exponential family#Quadratic variance functions|natural exponential family with quadratic variance functions]] (NEF–QVF).

More general distributions can be defined by fixing some initial values of ''p<sub>j</sub>'' 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'':<ref name="Schmidt"/> for example the [[logarithmic distribution]] and the discrete [[Uniform distribution (discrete)|uniform distribution]].

The (''a'', ''b'', 0) class of distributions has important applications in [[actuarial science]] in the context of loss models.<ref name=Klugman/>

== Properties ==
Sundt<ref name=Sundt/> 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 <math>W_N(x)\,</math> denotes the [[probability generating function]].

{| class="wikitable"
! class="hintergrundfarbe6" |Distribution
! class="hintergrundfarbe6" |<math> P[N=k]\, </math>
! class="hintergrundfarbe6" |<math> a\, </math>
! class="hintergrundfarbe6" |<math> b \,</math>
! class="hintergrundfarbe6" |<math> p_0\, </math>
! class="hintergrundfarbe6" |<math> W_N(x)\, </math>
! class="hintergrundfarbe6" |<math> E[N]\, </math>
! class="hintergrundfarbe6" |<math> Var(N)\, </math>
|-
|[[Binomial distribution|Binomial]]
|<math>\binom{n}{k} p^k (1-p)^{n-k} </math>
|<math> \frac{-p}{1-p} </math>
|<math> \frac{p(n+1)}{1-p} </math>
|<math> (1-p)^n\, </math>
|<math> (px+(1-p))^{n} \,</math>
|<math> np\, </math>
|<math> np(1-p) \,</math>
|-
|[[Poisson distribution|Poisson]]
|<math> e^{-\lambda}\frac{ \lambda^k}{k!}\, </math>
|<math> 0\, </math>
|<math> \lambda \,</math>
|<math> e^{- \lambda}\, </math>
|<math> e^{\lambda(x-1)} \,</math>
|<math> \lambda\, </math>
|<math> \lambda \,</math>
|-
|[[Negative binomial distribution|Negative binomial]]
|<math> \frac{\Gamma(r+k)}{k!\,\Gamma(r)}\,p^r\,(1-p)^k \,</math>
|<math> 1-p\, </math>
|<math> (1-p)(r-1)\, </math>
|<math> p^r \,</math>
|<math> \left( \frac{p}{1 - x(1-p)}\right) ^r \,</math>
|<math> \frac{r(1-p)}{p} \, </math>
|<math> \frac{r(1-p)}{p^2} \,</math>
|-
|}

== 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 <math>k</math>, you get

: <math>k \, \frac{p_k}{p_{k-1}} = ak + b,</math>

which shows that the left side is obviously a linear function of <math>k</math>. When using a sample of <math>n</math> data, an approximation of the <math>p_k</math>'s need to be done. If <math>n_k</math> represents the number of observations having the value <math>k</math>, then <math>\hat{p}_k = \frac{n_k}{n}</math> is an [[Estimator bias|unbiased]] estimator of the true <math>p_k</math>.

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 <math>a</math>, while the ordinate at the origin would be <math>b</math>.

== References ==
{{Reflist|refs=
<ref name="Schmidt">{{cite journal
| first1 = Klaus Th.
| last1 = Hess
| first2 = Anett | last2 = Liewald | first3 = Klaus D. | last3 = Schmidt
| year = 2002
| title = An extension of Panjer's recursion
| journal = ASTIN Bulletin
| volume = 32
| issue = 2
| pages = 283–297
| url = http://www.casact.org/library/astin/vol32no2/283.pdf
| format=PDF
| doi = 10.2143/AST.32.2.1030
| archiveurl=http://www.webcitation.org/5hg38Pbjx
| archivedate=2009-06-20
| deadurl=no
| accessdate=2009-06-18
}}</ref>
<ref name=Klugman>{{cite book
| last1 = Klugman | first1 = Stuart | last2 = Panjer | first2 = Harry | last3 = Gordon | first3 = Willmot
| year = 2004
| title = Loss Models: From Data to Decisions
| edition = 2nd
| location = New Jersey
| publisher = Wiley
| series = Series in Probability and Statistics
| isbn = 0-471-21577-5
}}</ref>
<ref name=Sundt>{{cite journal
| first1 = Bjørn | last1 = Sundt | first2 = William S. | last2 = Jewell
| title = Further results on recursive evaluation of compound distributions
| journal = ASTIN Bulletin
| volume = 12
| issue = 1
| year = 1981
| pages = 27–39
| publisher = [[International Actuarial Association]]
| url = http://www.casact.org/library/astin/vol12no1/27.pdf
| format = PDF
}}</ref>}}

{{DEFAULTSORT:A,b,0 class of distributions}}
{{ProbDistributions|families}}
[[Category:Discrete distributions]]
[[Category:Systems of probability distributions]]
[[Category:Actuarial science]]
[[Category:Probability distributions]]

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Benesi–Hildebrand method