Hermite
Probability mass function The horizontal axis is the indexk, the number of occurrences. The function is only defined at integer values ofk. The connecting lines are only guides for the eye.
Cumulative distribution function The horizontal axis is the indexk, the number of occurrences. The CDF is discontinuous at the integers ofk and flat everywhere else because a variable that is Hermite distributed only takes on integer values.
Notation	${\displaystyle \mathrm{Herm}(a_1,a_2)}$
Parameters	a1 ≥ 0,a2 ≥ 0
Support	x ∈ { 0, 1, 2, ... }
PMF	${\displaystyle x\mapsto e^{-(a_1+a_2)}\sum _{j=0}^{\lfloor x/2\rfloor }\frac{a_1^{x-2j}{a}_2^j}{(x-2j)!j!}}$
CDF	${\displaystyle x\mapsto e^{-a_1+a_2}\sum _{i=0}^{\lfloor x\rfloor }\sum _{j=0}^{\lfloor i/2\rfloor }\frac{a_1^{i-2j}{a}_2^j}{(i-2j)!j!}}$
Mean	${\displaystyle a_1+2a_2}$
Variance	${\displaystyle a_1+4a_2}$
Skewness	${\displaystyle \frac{a_1+8a_2}{(a_1+4a_2{)}^{3/2}}}$
Excess kurtosis	${\displaystyle \frac{a_1+16a_2}{(a_1+4a_2{)}^2}}$
MGF	${\displaystyle \exp (a_1(e^t-1)+a_2(e^{2t}-1))}$
CF	${\displaystyle \exp (a_1(e^{ti}-1)+a_2(e^{2ti}-1))}$
PGF	${\displaystyle \exp (a_1(s-1)+a_2(s^2-1))}$

Inprobability theory andstatistics, theHermite distribution, named afterCharles Hermite, is adiscrete probability distribution used to modelcount data with more than one parameter. This distribution is flexible in terms of its ability to allow a moderateover-dispersion in the data.

The authors C. D. Kemp andA. W. Kemp^[1] have called it "Hermite distribution" from the fact itsprobability function and themoment generating function can be expressed in terms of the coefficients of (modified)Hermite polynomials.

The distribution first appeared in the paperApplications of Mathematics to Medical Problems,^[2] byAnderson 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 thebivariate 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 ofbacteria inleucocytes. Using themethod of moments he fitted the data with the Hermite distribution and found the model more satisfactory than fitting it with aPoisson distribution.

The distribution was formally introduced and published by C. D. Kemp and Adrienne W. Kemp in 1965 in their workSome Properties of ‘Hermite’ Distribution. The work is focused on the properties of this distribution for instance a necessary condition on the parameters and theirmaximum likelihood estimators (MLE), the analysis of theprobability 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 themaximum likelihood method.

Hermite distribution is a special case of discretecompound Poisson distribution with only two parameters.^[3][4]

The same authors published in 1966 the paperAn alternative Derivation of the Hermite Distribution.^[5] In this work established that the Hermite distribution can be obtained formally bycompounding aPoisson distribution with anormal distribution.

In 1971, Y. C. Patel^[6] 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^[7] did a research on a generalized form of Hermite distribution.

LetX₁ andX₂ be two independent Poisson variables with parametersa₁ anda₂. Theprobability distribution of therandom variableY =X₁ + 2X₂ is the Hermite distribution with parametersa₁ anda₂ andprobability mass function is given by^[8]

${\displaystyle p_n=P(Y=n)=e^{-(a_1+a_2)}\sum _{j=0}^{\lfloor n/2\rfloor }\frac{a_1^{n-2j}{a}_2^j}{(n-2j)!j!}}$

where

• n = 0, 1, 2, ...
• a₁,a₂ ≥ 0.
• (n − 2j)! andj! are thefactorials of (n − 2j) andj, respectively.
• $\lfloor n/2\rfloor$ is the integer part of  n/2.

Theprobability generating function of the probability mass is,^[8]

${\displaystyle G_Y(s)=\sum _{n=0}^{\mathrm{\infty }}{p}_n{s}^n=\exp (a_1(s-1)+a_2(s^2-1))}$

When arandom variableY =X₁ + 2X₂ is distributed by an Hermite distribution, whereX₁ andX₂ are two independent Poisson variables with parametersa₁ anda₂, we write

${\displaystyle Y\sim \mathrm{Herm}(a_1,a_2)}$

Themoment generating function of a random variableX is defined as the expected value ofe^t, as a function of the real parametert. For an Hermite distribution with parametersX₁ andX₂, the moment generating function exists and is equal to

${\displaystyle M(t)=G(e^t)=\exp (a_1(e^t-1)+a_2(e^{2t}-1))}$

Thecumulant generating function is the logarithm of the moment generating function and is equal to^[4]

${\displaystyle K(t)=\log (M(t))=a_1(e^t-1)+a_2(e^{2t}-1)}$

If we consider the coefficient of (it)^rr! in the expansion ofK(t) we obtain ther-cumulant

${\displaystyle k_n=a_1+2^n{a}_2}$

Hence themean and the succeeding threemoments about it are

Order	Moment	Cumulant
1	${\displaystyle {\mu }_1=k_1=a_1+2a_2}$	${\displaystyle \mu }$
2	${\displaystyle {\mu }_2=k_2=a_1+4a_2}$	${\displaystyle {\sigma }^2}$
3	${\displaystyle {\mu }_3=k_3=a_1+8a_2}$	${\displaystyle k_3}$
4	${\displaystyle {\mu }_4=k_4+3k_2^2=a_1+16a_2+3(a_1+4a_2{)}^2}$	${\displaystyle k_4}$

Theskewness is the third moment centered around the mean divided by the 3/2 power of thestandard deviation, and for the hermite distribution is,^[4]

${\displaystyle {\gamma }_1=\frac{{\mu }_3}{{\mu }_2^{3/2}}=\frac{a_1+8a_2}{(a_1+4a_2{)}^{3/2}}}$

• Always${\displaystyle {\gamma }_1>0}$, so the mass of the distribution is concentrated on the left.

Thekurtosis is the fourth moment centered around the mean, divided by the square of thevariance, and for the Hermite distribution is,^[4]

${\displaystyle {\beta }_2=\frac{{\mu }_4}{{\mu }_2^2}=\frac{a_1+16a_2+3(a_1+4a_2{)}^2}{(a_1+4a_2{)}^2}=\frac{a_1+16a_2}{(a_1+4a_2{)}^2}+3}$

Theexcess kurtosis is just a correction to make the kurtosis of the normal distribution equal to zero, and it is the following,

${\displaystyle {\gamma }_2=\frac{{\mu }_4}{{\mu }_2^2}-3=\frac{a_1+16a_2}{(a_1+4a_2{)}^2}}$

• Always${\displaystyle {\beta }_2>3}$, or${\displaystyle {\gamma }_2>0}$ the distribution has a high acute peak around the mean and fatter tails.

In a discrete distribution thecharacteristic function of any real-valued random variable is defined as theexpected value of${\displaystyle e^{itX}}$, wherei is the imaginary unit andt ∈ R

${\displaystyle \varphi (t)=E[e^{itX}]=\sum _{j=0}^{\mathrm{\infty }}{e}^{ijt}P[X=j]}$

This function is related to the moment-generating function via${\displaystyle {\varphi }_x(t)=M_X(it)}$. Hence for this distribution the characteristic function is,^[1]

${\displaystyle {\varphi }_x(t)=\exp (a_1(e^{it}-1)+a_2(e^{2it}-1))}$

Thecumulative distribution function is,^[1]

• This distribution can have any number ofmodes. As an example, the fitted distribution for McKendrick’s^[2] data has an estimated parameters of${\displaystyle {\hat{a}}_1=0.0135}$,${\displaystyle {\hat{a}}_2=0.0932}$. Therefore, the first five estimated probabilities are 0.899, 0.012, 0.084, 0.001, 0.004.

• This distribution is closed under addition or closed under convolutions.^[9] Like thePoisson distribution, the Hermite distribution has this property. Given two Hermite-distributed random variables${\displaystyle X_1\sim \mathrm{Herm}(a_1,a_2)}$ and${\displaystyle X_2\sim \mathrm{Herm}(b_1,b_2)}$, thenY =X₁ +X₂ follows an Hermite distribution,${\displaystyle Y\sim \mathrm{Herm}(a_1+b_1,a_2+b_2)}$.
• This distribution allows a moderateoverdispersion, so it can be used when data has this property.^[9] A random variable has overdispersion, or it is overdispersed with respect the Poisson distribution, when its variance is greater than its expected value. The Hermite distribution allows a moderate overdispersion because the coefficient of dispersion is always between 1 and 2,

${\displaystyle d=\frac{\mathrm{Var}(Y)}{\mathrm{E}(Y)}=\frac{a_1+4a_2}{a_1+2a_2}=1+\frac{2a_2}{a_1+2a_2}}$

Themean and thevariance of the Hermite distribution are${\displaystyle \mu =a_1+2a_2}$ and${\displaystyle {\sigma }^2=a_1+4a_2}$, respectively. So we have these two equation,

${\displaystyle \{\begin{array}{l}\overline{x}=a_1+2a_2\\ {\sigma }^2=a_1+4a_2\end{array}}$

Solving these two equation we get the moment estimators${\displaystyle \widehat{a_1}}$ and${\displaystyle \widehat{a_2}}$ ofa₁ anda₂.^[6]

${\displaystyle \widehat{a_1}=2\overline{x}-{\sigma }^2}$

${\displaystyle \widehat{a_2}=\frac{{\sigma }^2-\hat{x}}{2}}$

Sincea₁ anda₂ both are positive, the estimator${\displaystyle \widehat{a_1}}$ and${\displaystyle \widehat{a_2}}$ are admissible (≥ 0) only if,.

Given a sampleX₁, ...,X_m areindependent random variables each having an Hermite distribution we wish to estimate the value of the parameters${\displaystyle \widehat{a_1}}$ and${\displaystyle \widehat{a_2}}$. We know that the mean and the variance of the distribution are${\displaystyle \mu =a_1+2a_2}$ and${\displaystyle {\sigma }^2=a_1+4a_2}$, respectively. Using these two equation,

${\displaystyle \{\begin{array}{l}{a}_1=\mu (2-d)\\ a_2={\displaystyle \frac{\mu (d-1)}{2}}\end{array}}$

We can parameterize the probability function by μ andd

${\displaystyle P(X=x)=\exp \left(-\left(\mu (2-d)+\frac{\mu (d-1)}{2}\right)\right)\sum _{j=0}^{[x/2]}\frac{(\mu (2-d){)}^{x-2j}{\left(\frac{\mu (d-1)}{2}\right)}^j}{(x-2j)!j!}}$

Hence thelog-likelihood function is,^[9]

where

• ${\displaystyle q_i(\theta )=\sum _{j=0}^{[x_i/2]}\frac{{\theta }^j}{(x_i-2j)!j!}}$
• ${\displaystyle \theta =\frac{d-1}{2\mu (2-d{)}^2}}$

From the log-likelihood function, thelikelihood equations are,^[9]

${\displaystyle \frac{\mathrm{\partial }l}{\mathrm{\partial }\mu }=m\left(-1+\frac{d-1}{2}\right)+\frac{1}{\mu }\sum _{i=1}^m{x}_i-\frac{d-1}{2{\mu }^2(2-d{)}^2}\sum _{i=1}^m\frac{q_i^{{}^{\prime }}(\theta )}{q_i(\theta )}}$

${\displaystyle \frac{\mathrm{\partial }l}{\mathrm{\partial }d}=m\frac{\mu }{2}-\frac{\sum _{i=1}^m{x}_i}{2-d}-\frac{d}{2\mu (2-d{)}^3}\sum _{i=1}^m\sum _{i=1}^m\frac{q_i^{{}^{\prime }}(\theta )}{q_i(\theta )}}$

Straightforward calculations show that,^[9]

• ${\displaystyle \mu =\overline{x}}$
• Andd can be found by solving,

${\displaystyle \sum _{i=1}^m\frac{q_i^{{}^{\prime }}(\tilde{\theta })}{q_i(\tilde{\theta })}=m(\overline{x}(2-d){)}^2}$

where${\displaystyle \tilde{\theta }=\frac{d-1}{2\overline{x}(2-d{)}^2}}$

• It can be shown that thelog-likelihood function is strictly concave in the domain of the parameters. Consequently, the MLE is unique.

The likelihood equation does not always have a solution like as it shows the following proposition,

Proposition:^[9] LetX₁, ...,X_m come from a generalized Hermite distribution with fixedn. Then the MLEs of the parameters are${\displaystyle \hat{\mu }}$ and${\displaystyle \tilde{d}}$ if only if${\displaystyle m^{(2)}/{\overline{x}}^2>1}$, where${\displaystyle m^{(2)}=\sum _{i=1}^n{x}_i(x_i-1)/n}$ indicates the empirical factorial momement of order 2.

• Remark 1: The condition${\displaystyle m^{(2)}/{\overline{x}}^2>1}$ is equivalent to${\displaystyle \tilde{d}>1}$ where${\displaystyle \tilde{d}={\sigma }^2/\overline{x}}$ is the empirical dispersion index
• Remark 2: If the condition is not satisfied, then the MLEs of the parameters are${\displaystyle \hat{\mu }=\overline{x}}$ and${\displaystyle \tilde{d}=1}$, that is, the data are fitted using the Poisson distribution.

A usual choice for discrete distributions is the zero relative frequency of the data set which is equated to the probability of zero under the assumed distribution. Observing that${\displaystyle f_0=\exp (-(a_1+a_2))}$ and${\displaystyle \mu =a_1+2a_2}$. Following the example of Y. C. Patel (1976) the resulting system of equations,

${\displaystyle \{\begin{array}{l}\overline{x}=a_1+2a_2\\ f_0=\exp (-(a_1+a_2))\end{array}}$

We obtain thezero frequency and themean estimatora₁ of${\displaystyle \widehat{a_1}}$ anda₂ of${\displaystyle \widehat{a_2}}$,^[6]

${\displaystyle \widehat{a_1}=-(\overline{x}+2\log (f_0))}$

${\displaystyle \widehat{a_2}=\overline{x}+\log (f_0)}$

where${\displaystyle f_0=\frac{n_0}{n}}$, is the zero relative frequency, n > 0

It can be seen that for distributions with a high probability at 0, the efficiency is high.

• For admissible values of${\displaystyle \widehat{a_1}}$ and${\displaystyle \widehat{a_2}}$, we must have

When Hermite distribution is used to model a data sample is important to check if thePoisson distribution is enough to fit the data. Following the parametrizedprobability mass function used to calculate the maximum likelihood estimator, is important to corroborate the following hypothesis,

${\displaystyle \{\begin{array}{l}{H}_0:d=1\\ H_1:d>1\end{array}}$

Thelikelihood-ratio test statistic^[9] for hermite distribution is,

${\displaystyle W=2(\mathcal{L}(X;\hat{\mu },\hat{d})-\mathcal{L}(X;\hat{\mu },1))}$

Where${\displaystyle \mathcal{L}()}$ is the log-likelihood function. Asd = 1 belongs to the boundary of the domain of parameters, under the null hypothesis,W does not have an asymptotic${\displaystyle {\chi }_1^2}$ distribution as expected. It can be established that the asymptotic distribution ofW is a 50:50 mixture of the constant 0 and the${\displaystyle {\chi }_1^2}$. The α upper-tail percentage points for this mixture are the same as the 2α upper-tail percentage points for a${\displaystyle {\chi }_1^2}$; for instance, for α = 0.01, 0.05, and 0.10 they are 5.41189, 2.70554 and 1.64237.

The score statistic is,^[9]

${\displaystyle S_2=2m{\left[\frac{m^{(2)}-{\overline{x}}^2}{2\overline{x}}\right]}^2=\frac{m(\tilde{d}-1{)}^2}{2}}$

wherem is the number of observations.

The asymptotic distribution of the score test statistic under the null hypothesis is a${\displaystyle {\chi }_1^2}$ distribution. It may be convenient to use a signed version of the score test, that is,${\displaystyle \mathrm{sgn}(m^{(2)}-{\overline{x}}^2)\sqrt{S}}$, following asymptotically a standard normal.

• Discrete distributions

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