In probability theory and statistics, the Hermite distribution, named after Charles Hermite, is a discrete probability distribution used to model count data with more than one parameter. This distribution is flexible in terms of its ability to allow a moderate over-dispersion in the data. The authors C. D. Kemp and A. W. Kemp have called it "Hermite distribution" from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of (modified) Hermite polynomials.
History The distribution first appeared in the paper Applications of Mathematics to Medical Problems, by Anderson Gray McKendrick in 1926. In this work the author explains several mathematical methods that can be applied to medical research. In one of this methods he considered the bivariate Poisson distribution and showed that the distribution of the sum of two correlated Poisson variables follow a distribution that later would be known as Hermite distribution. As a practical application, McKendrick considered the distribution of counts of bacteria in leucocytes. Using the method of moments he fitted the data with the Hermite distribution and found the model more satisfactory than fitting it with a Poisson distribution. The distribution was formally introduced and published by C. D. Kemp and Adrienne W. Kemp in 1965 in their work Some Properties of ‘Hermite’ Distribution. The work is focused on the properties of this distribution for instance a necessary condition on the parameters and their maximum likelihood estimators (MLE), the analysis of the probability generating function (PGF) and how it can be expressed in terms of the coefficients of (modified) Hermite polynomials. An example they have used in this publication is the distribution of counts of bacteria in leucocytes that used McKendrick but Kemp and Kemp estimate the model using the maximum likelihood method. Hermite distribution is a special case of discrete compound Poisson distribution with only two parameters. The same authors published in 1966 the paper An alternative Derivation of the Hermite Distribution. In this work established that the Hermite distribution can be obtained formally by compounding a Poisson distribution with a normal distribution. In 1971, Y. C. Patel did a comparative study of various estimation procedures for the Hermite distribution in his doctoral thesis. It included maximum likelihood, moment estimators, mean and zero frequency estimators and the method of even points. In 1974, Gupta and Jain did a research on a generalized form of Hermite distribution.
Definition
Probability mass function Let X1 and X2 be two independent Poisson variables with parameters a1 and a2. The probability distribution of the random variable Y = X1 + 2X2 is the Hermite distribution with parameters a1 and a2 and probability mass function is given by
p n = P ( Y = n ) = e − ( a 1 + a 2 ) ∑ j = 0 ⌊ n / 2 ⌋ a 1 n − 2 j a 2 j ( n − 2 j ) ! j ! {\displaystyle p_{n}=P(Y=n)=e^{-(a_{1}+a_{2})}\sum _{j=0}^{\lfloor n/2\rfloor }{\frac {a_{1}^{n-2j}a_{2}^{j}}{(n-2j)!j!}}}
where
n = 0, 1, 2, ... a1, a2 ≥ 0. (n − 2j)! and j! are the factorials of (n − 2j) and j, respectively.
⌊ n / 2 ⌋ {\textstyle \lfloor n/2\rfloor } is the integer part of n/2. The probability generating function of the probability mass is,
G Y ( s ) = ∑ n = 0 ∞ p n s n = exp ( a 1 ( s − 1 ) + a 2 ( s 2 − 1 ) ) {\displaystyle G_{Y}(s)=\sum _{n=0}^{\infty }p_{n}s^{n}=\exp(a_{1}(s-1)+a_{2}(s^{2}-1))}
Notation When a random variable Y = X1 + 2X2 is distributed by an Hermite distribution, where X1 and X2 are two independent Poisson variables with parameters a1 and a2, we write
Y ∼ Herm ( a 1 , a 2 ) {\displaystyle Y\ \sim \operatorname {Herm} (a_{1},a_{2})\,}
Properties
Moment and cumulant generating functions The moment generating function of a random variable X is defined as the expected value of et, as a function of the real parameter t. For an Hermite distribution with parameters X1 and X2, the moment generating function exists and is equal to
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