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{{Short description|Family of probability distributions}}
{{Use American English|date = January 2019}}

In [[probability]] and [[statistics]], the '''Tweedie distributions''' are a family of [[probability distribution]]s which include the purely continuous [[normal distribution|normal]], [[gamma distribution|gamma]] and [[inverse gaussian distribution|inverse Gaussian]] distributions, the purely discrete scaled [[Poisson distribution]], and the class of [[compound poisson distribution#Compound Poisson Gamma distribution|compound Poisson–gamma]] distributions which have positive mass at zero, but are otherwise continuous.<ref name="t84">{{cite conference
 |last=Tweedie |first=M.C.K.
 |year=1984
 |title=An index which distinguishes between some important exponential families
 |book-title=Statistics: Applications and New Directions
 |conference=Proceedings of the Indian Statistical Institute Golden Jubilee International Conference
 |editor1-first=J.K. |editor1-last=Ghosh
 |editor2-first=J |editor2-last=Roy
 |pages=579–604
 |publisher=Indian Statistical Institute |location=Calcutta
 |mr=786162
}}</ref>
Tweedie distributions are a special case of [[exponential dispersion model]]s and are often used as distributions for [[generalized linear model]]s.<ref name="Jørgensen-1997">{{cite book
 |last=Jørgensen |first=Bent 
 |year = 1997
 |title = The theory of dispersion models
 |publisher = Chapman & Hall
 |isbn = 978-0412997112
}}</ref>

The Tweedie distributions were named by [[Bent Jørgensen (statistician)|Bent Jørgensen]] in <ref>{{cite journal|last=Jørgensen, B| title=Exponential dispersion models | journal= Journal of the Royal Statistical Society, Series B|year=1987|volume=49|issue=2|pages=127–162 |jstor=2345415}}</ref> after [[Maurice Tweedie]],<ref>{{cite journal
 |last=Smith |first=C.A.B. |author-link=Cedric Smith (statistician)
 |year=1997
 |title=Obituary: Maurice Charles Kenneth Tweedie, 1919–96
 |journal=[[Journal of the Royal Statistical Society, Series A]]
 |volume=160 |issue=1 |pages=151–154 |doi=10.1111/1467-985X.00052
|doi-access=free}}</ref> a statistician and medical physicist at the [[University of Liverpool]], UK, who presented the first thorough study of these distributions in 1982 when the conference <ref name="t84"/> was held.
Around the same time, Bar-Lev and Enis published about the same topic.<ref>{{cite journal
 |last1=Bar-Lev |first1=S.K.
 |last2=Enis |first2=P.
 |year=1985
 |title=Reproducibility in the one-parameter exponential family
 |journal=[[Metrika]]
 |volume=32 |issue=1 |pages=391–394 |doi=10.1007/BF01897827
|doi-access=}}</ref><ref>{{cite journal
 |last1=Bar-Lev |first1=Shall.K.
 |last2=Enis |first2=Peter
 |year=1986
 |title=Reproducibility and Natural Exponential Families with Power Variance Functions
 |journal=[[The Annals of Statistics]]
 |volume=14 |issue=4 |pages=1507-1522 |doi=10.1214/aos/1176350173
|doi-access=free}}</ref>

==Definitions==

The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) [[exponential dispersion model]]s (ED), with a special [[mean]]-[[variance]] relationship. 
A [[random variable]] ''Y'' is Tweedie distributed ''Tw<sub>p</sub>(μ, σ<sup>2</sup>)'', if <math>Y \sim \mathrm{ED}(\mu, \sigma^2)</math> with mean <math>\mu = \operatorname{E}(Y)</math>, positive dispersion parameter <math>\sigma^2</math> and
<math display="block">\operatorname{Var}(Y) = \sigma^2\,\mu^p,</math>
where <math>p \in \mathbf{R}</math> is called the Tweedie power parameter.
The probability distribution ''P''<sub>''θ'',''σ''<sup>2</sup></sub> on the [[Measure (mathematics)|measurable sets]] ''A'', is given by
<math display="block">P_{\theta, \sigma^2}(Y\in A)=\int_A \exp\left(\frac{\theta \cdot z-\kappa_p(\theta)}{\sigma^2}\right)\cdot \nu_\lambda\, (dz),</math>
for some σ-finite measure ''ν<sub>λ</sub>''.
This representation uses the canonical parameter ''θ'' of an exponential dispersion model and [[cumulant|cumulant function]]
<math display="block">\kappa_p(\theta)=
\begin{cases}
  \frac{\alpha-1}{\alpha} \left(\frac{\theta}{\alpha-1}\right)^\alpha,  & \text{for }p\neq 1,2\\
  -\log(-\theta), & \text{for }p=2\\
  e^\theta, & \text{for }p=1
\end{cases}
</math>
where we used <math>\alpha = \frac{p-2}{p-1}</math>, or equivalently <math>p = \frac{\alpha-2}{\alpha-1}</math>.

==Properties==

===Additive exponential dispersion models===

The models just described are in the reproductive form. An [[exponential dispersion model]] has always a [[Duality (mathematics)|dual]]: the additive form. If ''Y'' is reproductive, then <math>Z=\lambda Y</math> with <math>\lambda = \frac{1}{\sigma^2}</math> is in the additive form ED<sup>*</sup>(''θ'',''λ''), for Tweedie ''Tw<sup>*</sup><sub>p</sub>(μ, λ)''. Additive models have the property that the distribution of the sum of independent random variables,
<math display="block">Z_+ = Z_1 +\cdots+ Z_n,</math>
for which ''Z''<sub>''i''</sub>&nbsp;~&nbsp;ED<sup>*</sup>(''θ'',''λ''<sub>''i''</sub>) with fixed ''θ'' and various ''λ'' are members of the family of distributions with the same ''θ'',
<math display="block">Z_+ \sim \operatorname{ED}^*(\theta,\lambda_1+\cdots+\lambda_n).</math>

===Reproductive exponential dispersion models===

A second class of exponential dispersion models exists designated by the random variable
<math display="block">Y=Z/\lambda \sim \operatorname{ED}(\mu,\sigma^2),</math>
where ''σ''<sup>2</sup>&nbsp;=&nbsp;1/''λ'', known as reproductive exponential dispersion models.  They have the property that for ''n'' independent random variables ''Y''<sub>''i''</sub>&nbsp;~&nbsp;ED(''μ'',''σ''<sup>2</sup>/''w''<sub>''i''</sub>), with weighting factors ''w<sub>i</sub>'' and
<math display="block">w= \sum_{i=1}^n w_i,</math>
a weighted average of the variables gives,
<math display="block">w^{-1}\sum_{i=1}^n w_iY_i \sim \operatorname{ED}(\mu,\sigma^2/w).</math>

For reproductive models the weighted average of independent random variables with fixed ''μ'' and ''σ''<sup>2</sup> and various values for ''w<sub>i</sub>'' is a member of the family of distributions with same ''μ'' and ''σ''<sup>2</sup>.

The Tweedie exponential dispersion models are both additive and reproductive; we thus have the ''duality transformation''
<math display="block">Y \mapsto Z=Y/\sigma^2.</math>

===Scale invariance===

A third property of the Tweedie models is that they are [[Scale invariance|scale invariant]]:  For a reproductive exponential dispersion model ''Tw<sub>p</sub>(μ, σ<sup>2</sup>)'' and any positive constant ''c'' we have the property of closure under scale transformation,
<math display="block">c \operatorname{Tw}_p(\mu,\sigma^2) = \operatorname{Tw}_p(c\mu,c^{2-p}\sigma^2).</math>

===The Tweedie power variance function===

To define the [[variance function]] for exponential dispersion models we make use of the mean value mapping, the relationship between the canonical parameter ''θ'' and the mean ''μ''.  It is defined by the function
<math display="block">\tau(\theta) = \kappa^\prime(\theta) = \mu.</math>
with cumulative function <math>\kappa(\theta)</math>.
The [[Natural exponential family|variance function]] ''V''(''μ'') is constructed from the mean value mapping,
<math display="block">V(\mu) = \tau^\prime[\tau^{-1}(\mu)].</math>

Here the minus exponent in ''τ''<sup>−1</sup>(''μ'') denotes an inverse function rather than a reciprocal.  The mean and variance of an additive random variable is then {{math|1=E(''Z'') = ''λμ''}} and {{math|1=var(''Z'') = ''λV''(''μ'')}}.

Scale invariance implies that the variance function obeys the relationship {{math|1=
''V''(''μ'') = ''μ''<sup> ''p''</sup>}}.<ref 
name="Jørgensen-1997" />

===The Tweedie deviance===
The unit [[Deviance (statistics)|deviance]] of a reproductive Tweedie distribution is given by
<math display="block">d(y,\mu) =
\begin{cases}
  (y-\mu)^2, & \text{for }p=0\\
  2(y \log(y/\mu) + \mu - y),  & \text{for }p=1\\
  2(\log(\mu/y) + y/\mu - 1),  & \text{for }p=2\\
  2\left(\frac{\max(y,0)^{2-p}}{(1-p)(2-p)}-\frac{y\mu^{1-p}}{1-p}+\frac{\mu^{2-p}}{2-p}\right), & \text{else}
\end{cases}
</math>

===The Tweedie cumulant generating functions===
The properties of exponential dispersion models give us two [[differential equation]]s.<ref 
name="Jørgensen-1997" />  The first relates the mean value mapping and the variance function to each other,
<math display="block">\frac{\partial \tau^{-1}(\mu)}{\partial \mu}= \frac{1}{V(\mu)}.</math>

The second shows how the mean value mapping is related to the [[cumulant|cumulant function]],
<math display="block">\frac{\partial \kappa(\theta)}{\partial \theta} = \tau(\theta).</math>

These equations can be solved to obtain the cumulant function for different cases of the Tweedie models.  A cumulant generating function (CGF) may then be obtained from the cumulant function.  The additive CGF is generally specified by the equation
<math display="block">K^*(s) = \log[\operatorname{E}(e^{sZ})] = \lambda[\kappa(\theta+s)-\kappa(\theta)],</math>
and the reproductive CGF by
<math display="block">K(s) = \log[\operatorname{E}(e^{sY})] = \lambda[\kappa(\theta+s/\lambda)-\kappa(\theta)],</math>
where ''s'' is the generating function variable.

For the additive Tweedie models the CGFs take the form,
<math display="block">K^*_p(s;\theta,\lambda) = \begin{cases}
\lambda\kappa_p(\theta)[(1+s/\theta)^\alpha-1]
 & \quad p \ne 1,2, \\
-\lambda \log(1+s/\theta) &  \quad p = 2,  \\
\lambda e^\theta (e^s -1) & \quad p = 1,
\end{cases}</math>
and for the reproductive models,
<math display="block">K_p(s;\theta,\lambda) = \begin{cases}
\lambda\kappa_p(\theta)\left \{ \left[1+s/(\theta \lambda)\right]^\alpha-1 \right \}
& \quad p \ne 1,2, \\[1ex]
-\lambda \log[1+s/(\theta \lambda)]  & \quad p = 2,  \\[1ex]
\lambda e^\theta \left(e^{s/\lambda} -1\right) & \quad p = 1.
\end{cases}
</math>

The additive and reproductive Tweedie models are conventionally denoted by the symbols ''Tw''<sup>*</sup><sub>''p''</sub>(''θ'',''λ'') and ''Tw''<sub>''p''</sub>(''θ'',''σ''<sup>2</sup>), respectively.

The first and second derivatives of the CGFs, with ''s''&nbsp;=&nbsp;0, yields the mean and variance, respectively.  One can thus confirm that for the additive models the variance relates to the mean by the power law,
<math display="block">\mathrm{var} (Z)\propto \mathrm{E}(Z)^p.</math>

===The Tweedie convergence theorem===

The Tweedie exponential dispersion models are fundamental in statistical theory consequent to their roles as foci of [[convergence in distribution|convergence]] for a wide range of statistical processes.  Jørgensen ''et al'' proved a theorem that specifies the asymptotic behaviour of variance functions known as the '''Tweedie convergence theorem'''.<ref>{{cite journal|last=Jørgensen, B |author2=Martinez, JR |author3=Tsao, M|title=Asymptotic behaviour of the variance function|journal=Scandinavian Journal of Statistics|year=1994|volume=21|pages=223–243}}</ref> This theorem, in technical terms, is stated thus:<ref name="Jørgensen-1997" /> The unit variance function is regular of order ''p'' at zero (or infinity) provided that {{math|''V''(''μ'') ~ ''c''<sub>0</sub>''μ''<sup>''p''</sup>}} for ''μ'' as it approaches zero (or infinity) for all real values of ''p'' and ''c''<sub>0</sub>&nbsp;>&nbsp;0.  Then for a unit variance function regular of order ''p'' at either zero or infinity and for
<math display="block">p \notin (0,1),</math>
for any <math>\mu>0</math>, and <math> \sigma^2 > 0</math> we have
<math display="block">c^{-1} \operatorname{ED}(c\mu,\sigma^2c^{2-p}) \rightarrow Tw_p(\mu,c_0 \sigma^2)</math>
as <math>c \downarrow 0</math> or <math>c \rightarrow \infty</math>, respectively, where the convergence is through values of ''c'' such that ''cμ'' is in the domain of ''θ'' and ''c''<sup>''p''&minus;2</sup>/''σ''<sup>2</sup> is in the domain of ''λ''. The model must be infinitely divisible as ''c''<sup>2&minus;''p''</sup> approaches infinity.<ref 
name="Jørgensen-1997" />

In nontechnical terms this theorem implies that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the [[Attractor|domain of attraction]] of a Tweedie model.  Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form.  Hence many probability distributions have variance functions that express this asymptotic behaviour, and the Tweedie distributions become foci of convergence for a wide range of data types.<ref name=Kendal2011b>{{Cite journal | last1 = Kendal | first1 = W. S. | last2 = Jørgensen | first2 = B. | doi = 10.1103/PhysRevE.84.066120 | title = Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality | journal = Physical Review E | volume = 84 | issue = 6 | year = 2011 | pmid =  22304168|bibcode = 2011PhRvE..84f6120K | page=066120| url = https://portal.findresearcher.sdu.dk/da/publications/7f80e772-1f87-4ff3-8b07-8e38494cc650 }}</ref>

==Related distributions==

The Tweedie distributions include a number of familiar distributions as well as some unusual ones, each being specified by the [[Domain (mathematical analysis)|domain]] of the index parameter.  We have the

*extreme stable distribution, ''p''&nbsp;<&nbsp;0, 
*[[normal distribution]], ''p''&nbsp;=&nbsp;0,
*[[Poisson distribution]], ''p''&nbsp;=&nbsp;1,
*[[Compound Poisson distribution|compound Poisson–gamma distribution]], 1&nbsp;<&nbsp;''p''&nbsp;<&nbsp;2,
*[[gamma distribution]], ''p''&nbsp;=&nbsp;2,
*positive [[stable distribution]]s, 2&nbsp;<&nbsp;''p''&nbsp;<&nbsp;3,
*[[Inverse Gaussian distribution]], ''p''&nbsp;=&nbsp;3,
*positive stable distributions, ''p''&nbsp;>&nbsp;3, and
*extreme stable distributions, ''p''&nbsp;=&nbsp;{{math|&infin;}}.

For 0&nbsp;<&nbsp;''p''&nbsp;<&nbsp;1 no Tweedie model exists. Note that all ''stable'' distributions mean actually ''generated by stable distributions''.

==Occurrence and applications==

===The Tweedie models and Taylor’s power law===

[[Taylor's law]] is an empirical law in [[ecology]] that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a [[power-law]] relationship.<ref name=Taylor1961>{{cite journal | last1 = Taylor | first1 = LR | year = 1961 | title = Aggregation, variance and the mean | journal = Nature | volume = 189 | issue = 4766| pages = 732–735 | doi=10.1038/189732a0| bibcode = 1961Natur.189..732T| s2cid = 4263093 }}</ref>  For the population count ''Y'' with mean ''μ'' and variance var(''Y''), Taylor's law is written,
<math display="block">\operatorname{var}(Y) = a\mu^p,</math>
where ''a'' and ''p'' are both positive constants.   Since L. R. Taylor described this law in 1961 there have been many different explanations offered to explain it, ranging from animal behavior,<ref name="Taylor1961" /> a [[random walk]] model,<ref name=Hanski1980>{{cite journal | last1 = Hanski | first1 = I | year = 1980 | title = Spatial patterns and movements in coprophagous beetles | journal = Oikos | volume = 34 | issue = 3| pages = 293–310 | doi=10.2307/3544289| jstor = 3544289 | bibcode = 1980Oikos..34..293H }}</ref> a [[birth–death process|stochastic birth, death, immigration and emigration model]],<ref name=Anderson1961>{{cite journal | last1 = Anderson | first1 = RD | last2 = Crawley | first2 = GM | last3 = Hassell | first3 = M | year = 1982 | title = Variability in the abundance of animal and plant species | journal = Nature | volume = 296 | issue = 5854| pages = 245–248 | doi=10.1038/296245a0| bibcode = 1982Natur.296..245A | s2cid = 4272853 }}</ref> to a consequence of equilibrium and non-equilibrium [[statistical mechanics]].<ref name=Fronczak2010>{{cite journal | last1 = Fronczak | first1 = A | last2 = Fronczak | first2 = P | year = 2010 | title = Origins of Taylor's power law for fluctuation scaling in complex systems | journal = Phys Rev E | volume = 81 | issue = 6| page = 066112 | doi=10.1103/physreve.81.066112| pmid = 20866483 | arxiv = 0909.1896 | bibcode = 2010PhRvE..81f6112F | s2cid = 17435198 }}</ref>  No consensus exists as to an explanation for this model.

Since Taylor's law is mathematically identical to the variance-to-mean power law that characterizes the Tweedie models, it seemed reasonable to use these models and the Tweedie convergence theorem to explain the observed clustering of animals and plants associated with Taylor's law.<ref name=Kendal2002>{{cite journal | last1 = Kendal | first1 = WS | year = 2002 | title = Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model | journal = Ecological Modelling | volume = 151 | issue = 2–3| pages = 261–269 | doi=10.1016/s0304-3800(01)00494-x}}</ref><ref name=Kendal2004>{{cite journal | last1 = Kendal | first1 = WS | year = 2004 | title = Taylor's ecological power law as a consequence of scale invariant exponential dispersion models | journal = Ecol Complex | volume = 1 | issue = 3| pages = 193–209 | doi=10.1016/j.ecocom.2004.05.001}}</ref>  The majority of the observed values for the power-law exponent ''p'' have fallen in the interval (1,2) and so the Tweedie compound Poisson–gamma distribution would seem applicable.  Comparison of the [[empirical distribution function]] to the theoretical compound Poisson–gamma distribution has provided a means to verify consistency of this hypothesis.<ref name="Kendal2002" />

Whereas conventional models for Taylor's law have tended to involve ''[[ad hoc]]'' animal behavioral or [[population dynamics|population dynamic]] assumptions, the Tweedie convergence theorem would imply that Taylor's law results from a general mathematical convergence effect much as how the [[central limit theorem]] governs the convergence behavior of certain types of random data.  Indeed, any mathematical model, approximation or simulation that is designed to yield Taylor's law (on the basis of this theorem) is required to converge to the form of the Tweedie models.<ref name=Kendal2011b />

===Tweedie convergence and 1/''f'' noise===

[[Pink noise]], or 1/''f'' noise, refers to a pattern of noise characterized by a power-law relationship between its intensities ''S''(''f'') at different frequencies ''f'',
<math display="block">S(f)\propto \frac 1 {f^\gamma},</math>
where the dimensionless exponent ''γ'' ∈ [0,1].  It is found within a diverse number of natural processes.<ref name=Dutta1981>{{cite journal | last1 = Dutta | first1 = P | last2 = Horn | first2 = PM | year = 1981 | title = Low frequency fluctuations in solids: 1/''f'' noise | journal = Rev Mod Phys | volume = 53 | issue = 3| pages = 497–516 | doi=10.1103/revmodphys.53.497 | bibcode=1981RvMP...53..497D}}</ref>  Many different explanations for 1/''f'' noise exist, a widely held hypothesis is based on [[Self-organized criticality]] where dynamical systems close to a [[Critical point (thermodynamics)|critical point]] are thought to manifest [[scale-invariance|scale-invariant]] spatial and/or temporal behavior.

In this subsection a mathematical connection between 1/''f'' noise and the Tweedie variance-to-mean power law will be described.  To begin, we first need to introduce [[self-similar process]]es:  For the sequence of numbers
<math display="block">Y = (Y_i :i=0,1,2,\ldots,N)</math>
with mean
<math display="block">\widehat{\mu}=\operatorname{E}(Y_i),</math>
deviations
<math display="block">y_i = Y_i - \widehat{\mu}, </math>
variance
<math display="block">\widehat{\sigma}^2= \operatorname{E}(y_i^2),</math>
and autocorrelation function
<math display="block">r(k) = \frac{ \operatorname{E}(y_i,y_{i+k}) }{ \operatorname{E}(y_i^2)}</math>
with lag ''k'', if the [[autocorrelation]] of this sequence has the long range behavior
<math display="block">r(k)\sim k^{-d} L(k) </math>
as ''k''{{math|&rarr;&infin;}} and where ''L''(''k'') is a slowly varying function at large values of ''k'', this sequence is called a self-similar process.<ref name=Leland1994>{{cite journal | last1 = Leland | first1 = WE | last2 = Taqqu | first2 = MS | last3 = Willinger | first3 = W | last4 = Wilson | first4 = DV | year = 1994 | title =  On the self-similar nature of Ethernet traffic (Extended version)| journal =  IEEE/ACM Transactions on Networking| volume = 2 | pages = 1–15 | doi=10.1109/90.282603| s2cid = 6011907 }}</ref>

The '''method of expanding bins''' can be used to analyze self-similar processes.  Consider a set of equal-sized non-overlapping bins that divides the original sequence of ''N'' elements into groups of ''m'' equal-sized segments (''N/m'' is integer) so that new reproductive sequences, based on the mean values, can be defined:
<math display="block">Y_i^{(m)} = \left(Y_{im-m+1}+\cdots+Y_{im}\right)/m.</math>

The variance determined from this sequence will scale as the bin size changes such that
<math display="block">\operatorname{var}[Y^{(m)}] = \widehat{\sigma}^2 m^{-d}</math>
if and only if the autocorrelation has the limiting form<ref name=Tsybakov1997>{{cite journal | last1 = Tsybakov | first1 = B | last2 = Georganas | first2 = ND | year = 1997 | title = On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution | journal =  IEEE/ACM Transactions on Networking| volume = 5 | issue = 3| pages = 397–409 | doi=10.1109/90.611104| citeseerx = 10.1.1.53.5040 | s2cid = 2205855 }}</ref>
<math display="block">\lim_{k \to \infty} r(k)/k^{-d} = (2-d)(1-d)/2.</math>

One can also construct a set of corresponding additive sequences
<math display="block">Z_i^{(m)} = m Y_i^{(m)},</math>
based on the expanding bins,
<math display="block">Z_i^{(m)}=(Y_{im-m+1}+\cdots+Y_{im}).</math>

Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship
<math display="block"> \operatorname{var}[Z_i^{(m)}] = m^2 \operatorname{var}[Y^{(m)}] = \left(\frac{\widehat{\sigma}^2 }{ \widehat{\mu}^{2-d}} \right) \operatorname{E}[Z_i^{(m)}]^{2-d}</math>

Since <math>\widehat{\mu}</math> and <math>\widehat{\sigma}^2</math> are constants this relationship constitutes a variance-to-mean power law, with ''p''&nbsp;=&nbsp;2&nbsp;-&nbsp;''d''.<ref name="Kendal2011b" /><ref name=Kendal2007>{{cite journal | last1 = Kendal | first1 = WS | year = 2007 | title = Scale invariant correlations between genes and SNPs on Human chromosome 1 reveal potential evolutionary mechanisms | journal = J Theor Biol | volume = 245 | issue = 2| pages = 329–340 | doi=10.1016/j.jtbi.2006.10.010| pmid = 17137602 | bibcode = 2007JThBi.245..329K }}</ref>

The [[Logical biconditional|biconditional]] relationship above between the variance-to-mean power law and power law autocorrelation function, and the [[Wiener–Khinchin theorem]]<ref name=McQuarrie1976>McQuarrie DA (1976) ''Statistical mechanics'' [Harper & Row]</ref> imply that any sequence that exhibits a variance-to-mean power law by the method of expanding bins will also manifest 1/''f'' noise, and vice versa.  Moreover, the Tweedie convergence theorem, by virtue of its central limit-like effect of generating distributions that manifest variance-to-mean power functions, will also generate processes that manifest 1/''f'' noise.<ref name="Kendal2011b" />  The Tweedie convergence theorem thus provides an alternative explanation for the origin of 1/''f'' noise, based its central limit-like effect.

Much as the [[central limit theorem]] requires certain kinds of random processes to have as a focus of their convergence the [[normal distribution|Gaussian distribution]] and thus express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian processes to have as a focus of convergence the Tweedie distributions that express 1/''f'' noise.<ref name="Kendal2011b" />

===The Tweedie models and multifractality===

From the properties of self-similar processes, the power-law exponent ''p''&nbsp;=&nbsp;2&nbsp;-&nbsp;''d'' is related to the [[Hurst exponent]] ''H'' and the [[fractal dimension]] ''D'' by<ref name="Tsybakov1997" />
<math display="block">D = 2-H = 2 - p/2. </math>

A one-dimensional data sequence of self-similar data may demonstrate a variance-to-mean power law with local variations in the value of ''p'' and hence in the value of ''D''.  When fractal structures manifest local variations in fractal dimension, they are said to be  [[multifractal system|multifractals]].  Examples of data sequences that exhibit local variations in ''p'' like this include the eigenvalue deviations of the [[Random matrix|Gaussian Orthogonal and Unitary Ensembles]].<ref name="Kendal2011b" />  The Tweedie compound Poisson–gamma distribution has served to model multifractality based on local variations in the Tweedie exponent ''α''.  Consequently, in conjunction with the variation of ''α'', the Tweedie convergence theorem can be viewed as having a role in the genesis of such multifractals.

The variation of ''α'' has been found to obey the asymmetric [[Laplace distribution]] in certain cases.<ref name=Kendal2014>{{cite journal | last1 = Kendal | first1 = WS | year = 2014 | title = Multifractality attributed to dual central limit-lie convergence effects | journal = Physica A | volume = 401 | pages = 22–33 | doi=10.1016/j.physa.2014.01.022| bibcode = 2014PhyA..401...22K}}</ref>  This distribution has been shown to be a member of the family of geometric Tweedie models,<ref name="Jørgensen2011">{{cite journal | last1 = Jørgensen | first1 = B | last2 = Kokonendji | first2 = CC | year = 2011 | title = Dispersion models for geometric sums | journal = Braz J Probab Stat | volume = 25 | issue = 3| pages = 263–293 | doi=10.1214/10-bjps136| doi-access = free }}</ref> that manifest as limiting distributions in a convergence theorem for geometric dispersion models.

===Regional organ blood flow===

Regional organ blood flow has been traditionally assessed by the injection of [[Isotopic labeling|radiolabelled]] [[polyethylene microspheres]] into the arterial circulation of animals, of a size that they become entrapped within the [[microcirculation]] of organs.  The organ to be assessed is then divided into equal-sized cubes and the amount of radiolabel within each cube is evaluated by [[liquid scintillation counting]] and recorded. The amount of radioactivity within each cube is taken to reflect the blood flow through that sample at the time of injection.  It is possible to evaluate adjacent cubes from an organ in order to additively determine the blood flow through larger regions.  Through the work of '''J B Bassingthwaighte''' and others an empirical power law has been derived between the relative dispersion of blood flow of tissue samples (''RD''&nbsp;=&nbsp;standard&nbsp;deviation/mean) of mass ''m'' relative to reference-sized samples:<ref name=Bassingthwaighte1989>{{cite journal | last1 = Bassingthwaighte | first1 = JB | year = 1989 | title = Fractal nature of regional myocardial blood flow heterogeneity | journal = Circ Res | volume = 65 | issue = 3| pages = 578–590 | doi=10.1161/01.res.65.3.578| pmc = 3361973 | pmid=2766485}}</ref>
<math display="block">RD(m)=RD(m_\text{ref})\left (\frac{m}{m_\text{ref}}\right )^{1-D_s}</math>

This power law exponent ''D<sub>s</sub>'' has been called a fractal dimension.  '''Bassingthwaighte's power law''' can be shown to directly relate to the variance-to-mean power law.  Regional organ blood flow can thus be modelled by the Tweedie compound Poisson–gamma distribution.,<ref name=Kendal2001>{{cite journal | last1 = Kendal | first1 = WS | year = 2001 | title = A stochastic model for the self-similar heterogeneity of regional organ blood flow | journal = Proc Natl Acad Sci U S A | volume = 98 | issue = 3| pages = 837–841 | doi=10.1073/pnas.98.3.837| pmid = 11158557 | pmc = 14670| bibcode = 2001PNAS...98..837K| doi-access = free }}</ref>  In this model tissue sample could be considered to contain a random (Poisson) distributed number of entrapment sites, each with [[gamma distribution|gamma distributed]] blood flow.  Blood flow at this microcirculatory level has been observed to obey a gamma distribution,<ref>{{cite journal | last1 = Honig | first1 = CR | last2 = Feldstein | first2 = ML | last3 = Frierson | first3 = JL | year = 1977 | title = Capillary lengths, anastomoses, and estimated capillary transit times in skeletal muscle | journal = Am J Physiol Heart Circ Physiol | volume = 233 | issue = 1| pages = H122–H129 | doi=10.1152/ajpheart.1977.233.1.h122| pmid = 879328 }}</ref> thus providing support for this hypothesis.

===Cancer metastasis===

The "experimental cancer [[metastasis]] assay"<ref name=Fidler1977>{{cite journal|last=Fidler|first=IJ|author2=Kripke, M |title=Metastasis results from preexisting variant cells within a malignant tumor|journal=Science|year=1977|volume=197|issue=4306|pages=893–895|bibcode = 1977Sci...197..893F |doi = 10.1126/science.887927 |pmid=887927}}</ref>  has some resemblance to the above method to measure regional blood flow.  Groups of [[syngeneic]] and age matched mice are given intravenous injections of equal-sized aliquots of suspensions of cloned cancer cells and then after a set period of time their lungs are removed and the number of cancer metastases enumerated within each pair of lungs.  If other groups of mice are injected with different cancer cell [[Clone (cell biology)|clones]] then the number of metastases per group will differ in accordance with the metastatic potentials of the clones.  It has been long recognized that there can be considerable intraclonal variation in the numbers of metastases per mouse despite the best attempts to keep the experimental conditions within each clonal group uniform.<ref name=Fidler1977 /> This variation is larger than would be expected on the basis of a [[Poisson distribution]] of numbers of metastases per mouse in each clone and when the variance of the number of metastases per mouse was plotted against the corresponding mean a power law was found.<ref name=Kendal1987>{{cite journal | last1 = Kendal | first1 = WS | last2 = Frost | first2 = P | year = 1987 | title = Experimental metastasis: a novel application of the variance-to-mean power function | doi = 10.1093/jnci/79.5.1113 | journal = J Natl Cancer Inst | volume = 79 | issue = 5 | pages = 1113–1115 | pmid = 3479636 }}</ref>

The variance-to-mean power law for metastases was found to also hold for '''spontaneous murine metastases'''<ref>{{cite journal | last1 = Kendal | first1 = WS | year = 1999 | title = Clustering of murine lung metastases reflects fractal nonuniformity in regional lung blood flow | journal = Invasion and Metastasis | volume = 18 | issue = 5–6| pages = 285–296 | doi=10.1159/000024521| pmid = 10729773 | s2cid = 46835513 }}</ref>  and for cases series of human metastases.<ref>{{cite journal | last1 = Kendal | first1 = WS | last2 = Lagerwaard | first2 = FJ | last3 = Agboola | first3 = O | year = 2000 | title = Characterization of the frequency distribution for human hematogenous metastases: evidence for clustering and a power variance function | journal = Clin Exp Metastasis | volume = 18 | issue = 3| pages = 219–229 | doi = 10.1023/A:1006737100797 | pmid = 11315095 | s2cid = 25261069 }}</ref> 
Since hematogenous metastasis occurs in direct relationship to regional blood flow<ref>{{cite journal | last1 = Weiss | first1 = L | last2 = Bronk | first2 = J | last3 = Pickren | first3 = JW | last4 = Lane | first4 = WW | year = 1981 | title = Metastatic patterns and targe organ arterial blood flow | journal = Invasion and Metastasis | volume = 1 | issue = 2 | pages = 126–135 | pmid = 7188382 }}</ref> and videomicroscopic studies indicate that the passage and entrapment of cancer cells within the circulation appears analogous to the microsphere experiments<ref>{{cite journal | last1 = Chambers | first1 = AF | last2 = Groom | first2 = AC | last3 = MacDonald | first3 = IC | year = 2002 | title = Dissemination and growth of cancer cells in metastatic sites | journal = Nature Reviews Cancer | volume = 2 | issue = 8| pages = 563–572 | doi=10.1038/nrc865| pmid = 12154349 | s2cid = 135169 }}</ref> it seemed plausible to propose that the variation in numbers of hematogenous metastases could reflect heterogeneity in regional organ blood flow.<ref>{{cite journal | last1 = Kendal | first1 = WS | year = 2002 | title = A frequency distribution for the number of hematogenous organ metastases | journal = Invasion and Metastasis | volume = 1 | issue = 2 | pages = 126–135 | doi = 10.1006/jtbi.2002.3021 | pmid = 12202114 | bibcode = 2002JThBi.217..203K }}</ref>  
The blood flow model was based on the Tweedie compound Poisson–gamma distribution, a distribution governing a continuous random variable.  For that reason in the metastasis model it was assumed that blood flow was governed by that distribution and that the number of regional metastases occurred as a [[Poisson process]] for which the intensity was directly proportional to blood flow.  This led to the description of the Poisson negative binomial (PNB) distribution as a [[discrete probability distribution|discrete equivalent]] to the Tweedie compound Poisson–gamma distribution.  The [[probability-generating function|probability generating function]] for the PNB distribution is
<math display="block">G(s) = \exp \left [\lambda \frac {\alpha-1}{\alpha} \left( \frac{\theta} {\alpha-1} \right)^\alpha \left\{ \left(1- \frac{1} {\theta}+ \frac {s} {\theta}\right)^\alpha-1 \right\}\right]</math>

The relationship between the mean and variance of the PNB distribution is then
<math display="block">\operatorname{var}(Y) = a\operatorname{E}(Y)^b + \operatorname{E}(Y),</math>
which, in the range of many experimental metastasis assays, would be indistinguishable from the variance-to-mean power law.  For sparse data, however, this discrete variance-to-mean relationship would behave more like that of a Poisson distribution where the variance equaled the mean.

===Genomic structure and evolution===

The local density of [[Single-nucleotide polymorphism|Single Nucleotide Polymorphisms]] (SNPs) within the [[human genome]], as well as that of [[gene]]s, appears to cluster in accord with the variance-to-mean power law and the Tweedie compound Poisson–gamma distribution.<ref name=Kendal2003>{{cite journal | last1 = Kendal | first1 = WS | year = 2003 | title = An exponential dispersion model for the distribution of human single nucleotide polymorphisms | journal = Mol Biol Evol | volume = 20 | issue = 4| pages = 579–590 | doi=10.1093/molbev/msg057| pmid = 12679541 | doi-access = free }}</ref><ref name=KendalGenes>{{cite journal|last=Kendal|first=WS|title=A scale invariant clustering of genes on human chromosome 7|journal=BMC Evol Biol|year=2004|volume=4|pages=3|doi=10.1186/1471-2148-4-3|pmid=15040817|pmc=373443 |doi-access=free }}</ref>  In the case of SNPs their observed density reflects the assessment techniques, the availability of genomic sequences for analysis, and the [[Nucleotide diversity|nucleotide heterozygosity]].<ref>{{cite journal | year = 2001 | title = A map of human genome variation containing 1.42 million single nucleotide polymorphisms | doi = 10.1038/35057149 | journal = Nature | volume = 409 | issue = 6822| pages = 928–933 | pmid=11237013 | last1 = Sachidanandam | first1 = R | last2 = Weissman | first2 = D | last3 = Schmidt | first3 = SC |display-authors=etal| doi-access = free | bibcode = 2001Natur.409..928S }}</ref>  The first two factors reflect ascertainment errors inherent to the collection methods, the latter factor reflects an intrinsic property of the genome.

In the [[Coalescent theory|coalescent model]] of population genetics each genetic locus has its own unique history.  Within the evolution of a population from some species some genetic loci could presumably be traced back to a relatively [[Most recent common ancestor|recent common ancestor]] whereas other loci might have more ancient [[Genetic genealogy|genealogies]].  More ancient genomic segments would have had more time to accumulate SNPs and to experience [[Genetic recombination|recombination]].  '''R R Hudson''' has proposed a model where recombination could cause variation in the time to [[Most recent common ancestor|most common recent ancestor]] for different genomic segments.<ref>{{cite journal | last1 = Hudson | first1 = RR | year = 1991 | title = Gene genealogies and the coalescent process | journal = Oxford Surveys in Evolutionary Biology | volume = 7 | pages = 1–44 }}</ref>  A high recombination rate could cause a chromosome to contain a large number of small segments with less correlated genealogies.

Assuming a constant background rate of mutation the number of SNPs per genomic segment would accumulate proportionately to the time to the most recent common ancestor.  Current [[population genetics|population genetic theory]] would indicate that these times would be [[gamma distribution|gamma distributed]], on average.<ref>{{cite journal | last1 = Tavare | first1 = S | last2 = Balding | first2 = DJ | last3 = Griffiths | first3 = RC | last4 = Donnelly | first4 = P | year = 1997 | title = Inferring coalescent times from DNA sequence data | journal = Genetics | volume = 145 | issue = 2 | pages = 505–518 | doi = 10.1093/genetics/145.2.505 | pmid = 9071603 | pmc = 1207814 }}</ref>  The Tweedie compound Poisson–gamma distribution would suggest a model whereby the SNP map would consist of multiple small genomic segments with the mean number of SNPs per segment would be gamma distributed as per Hudson's model.

The distribution of genes within the human genome also demonstrated a variance-to-mean power law, when the method of expanding bins was used to determine the corresponding variances and means.<ref name=KendalGenes /> Similarly the number of genes per enumerative bin was found to obey a Tweedie compound Poisson–gamma distribution.  This probability distribution was deemed compatible with two different biological models: the '''microarrangement model''' where the number of genes per unit genomic length was determined by the sum of a random number of smaller genomic segments derived by random breakage and reconstruction of protochormosomes.  These smaller segments would be assumed to carry on average a gamma distributed number of genes.

In the alternative '''gene cluster model''', genes would be distributed randomly within the protochromosomes.  Over large evolutionary timescales there would occur [[gene duplication|tandem duplication]], [[mutation|mutations, insertions, deletions]] and [[chromosomal rearrangement|rearrangements]] that could affect the genes through a stochastic [[Birth–death process|birth, death and immigration process]] to yield the Tweedie compound Poisson–gamma distribution.

Both these mechanisms would implicate [[Neutral theory of molecular evolution|neutral evolutionary processes]] that would result in regional clustering of genes.

===Random matrix theory===

The '''[[Random matrix|Gaussian unitary ensemble]]''' (GUE) consists of complex [[Hermitian matrix|Hermitian matrices]] that are invariant under [[unitary transformation]]s whereas the '''[[Random matrix|Gaussian orthogonal ensemble]]''' (GOE) consists of real symmetric matrices invariant under [[orthogonal transformation]]s.  The ranked [[Eigenvalues and eigenvectors|eigenvalues]] ''E<sub>n</SUB>'' from these random matrices obey '''[[Wigner semicircle distribution|Wigner's semicircular distribution]]''': For a ''N''&times;''N'' matrix the average density for eigenvalues of size ''E'' will be
<math display="block">\bar{\rho}(E) = \begin{cases}
\sqrt{2N-E^2}/\pi
& \quad \left\vert E \right\vert < \sqrt{2N} \\ 0 & \quad \left\vert E \right\vert > \sqrt{2N}
\end{cases} </math>
as ''E''{{math|&rarr; &infin;}}.  Integration of the semicircular rule provides the number of eigenvalues on average less than ''E'',
<math display="block">\bar{\eta}(E) = \frac{1}{2\pi}\left [E\sqrt{2N-E^2}+2N \arcsin \left( \frac{E}{\sqrt{2N}} \right )+ \pi N \right ]. </math>

The ranked eigenvalues can be '''unfolded''', or renormalized, with the equation
<math display="block">e_n = \bar{\eta}(E)= \int_{-\infty}^{E_n} \, dE' \bar{\rho}(E'). </math>

This removes the trend of the sequence from the fluctuating portion.  If we look at the absolute value of the difference between the actual and expected cumulative number of eigenvalues
<math display="block">\left | \bar{D}_n \right | = \left |  n- \bar{\eta}(E_n) \right | </math>
we obtain a sequence of '''eigenvalue fluctuations''' which, using the method of expanding bins, reveals a variance-to-mean power law.<ref name="Kendal2011b" />
The eigenvalue fluctuations of both the GUE and the GOE manifest this power law with the power law exponents ranging between 1 and 2, and they similarly manifest 1/''f'' noise spectra.  These eigenvalue fluctuations also correspond to the Tweedie compound Poisson–gamma distribution and they exhibit multifractality.<ref name="Kendal2011b" />

===The distribution of [[prime number]]s===

The '''second [[Chebyshev function]]''' ''&psi;''(''x'') is given by,
<math display="block"> \psi(x) = \sum_{\widehat{p\,}^k\le x}\log \widehat{p\,}=\sum_{n \leq x} \Lambda(n) </math>
where the summation extends over all prime powers <math>\widehat{p\,}^k </math> not exceeding&nbsp;''x'', ''x'' runs over the positive real numbers, and <math>\Lambda(n)</math> is the [[von Mangoldt function]]. The function ''&psi;''(''x'') is related to the [[prime-counting function]] ''&pi;''(''x''), and as such provides information with regards to the distribution of prime numbers amongst the real numbers.  It is asymptotic to&nbsp;''x'', a statement equivalent to the [[prime number theorem]] and it can also be shown to be related to the zeros of the [[Riemann zeta function]] located on the critical strip ''ρ'', where the real part of the zeta zero ''ρ'' is between 0 and&nbsp;1.  Then ''ψ'' expressed for ''x'' greater than one can be written:
<math display="block">\psi_0(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac12 \ln(1-x^{-2})</math>
where
<math display="block">\psi_0(x) = \lim_{\varepsilon \rightarrow 0}\frac{\psi(x-\varepsilon)+\psi(x+\varepsilon)}2.</math>

The [[Riemann hypothesis]] states that the [[root of a function|nontrivial zeros]] of the [[Riemann zeta function]] all have [[real part]] {{1/2}}.  These zeta function zeros are related to the [[prime number theorem|distribution of prime numbers]].  [[Lowell Schoenfeld|'''Schoenfeld''']]<ref>{{cite journal|last=Schoenfeld|first=J|title=Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II|journal=Mathematics of Computation|year=1976|volume=30|issue=134|pages=337–360|doi=10.1090/s0025-5718-1976-0457374-x|doi-access=free}}</ref>  has shown that if the Riemann hypothesis is true then
<math display="block"> \Delta(x)=\left\vert \psi(x)-x \right\vert < \sqrt{x} \log^{2}(x)/(8 \pi)</math>
for all <math>x>73.2</math>. If we analyze the Chebyshev deviations Δ(''n'') on the integers ''n'' using the method of expanding bins and plot the variance versus the mean a variance to mean power law can be demonstrated.{{citation needed|date=September 2019}} Moreover, these deviations correspond to the Tweedie compound Poisson-gamma distribution and they exhibit 1/''f'' noise.

===Other applications===

Applications of Tweedie distributions include:
* actuarial studies<ref>{{cite journal | last1 = Haberman | first1 = S. | last2 = Renshaw | first2 = A. E. | year = 1996 | title = Generalized linear models and actuarial science | journal = The Statistician | volume = 45 | issue = 4| pages = 407–436 | doi=10.2307/2988543| jstor = 2988543 }}</ref><ref>Renshaw, A. E. 1994.
Modelling the claims process in the presence of covariates. ASTIN Bulletin 24: 265–286.</ref><ref>{{cite journal | last1 = Jørgensen | first1 = B. | last2 = Paes | last3 = Souza | first3 = M. C. | year = 1994 | title = Fitting Tweedie's compound Poisson model to insurance claims data | journal = Scand. Actuar. J. | volume = 1 | pages = 69–93 | doi = 10.1080/03461238.1994.10413930 | citeseerx = 10.1.1.329.9259 }}</ref><ref>Haberman, S., 
and Renshaw, A. E. 1998.
Actuarial applications of generalized linear models. In Statistics in Finance, D. J. Hand and S. D. Jacka (eds), Arnold, London.</ref><ref>Mildenhall, S. J. 1999. A systematic relationship between minimum bias and generalized linear models. 1999 
Proceedings of the Casualty Actuarial Society 86: 393–487.</ref><ref>Murphy, K. P., Brockman, M. J., and Lee, P. K. W. (2000). Using generalized linear models to build dynamic 
pricing systems. Casualty Actuarial Forum, Winter 2000.</ref><ref>{{cite journal
 |last1=Smyth |first1=G.K.
 |last2=Jørgensen |first2=B.
 |year=2002
 |title=Fitting Tweedie's compound Poisson model to insurance claims data: dispersion modelling
 |journal=ASTIN Bulletin 
 |volume=32 |pages=143–157
 |url=http://www.casact.org/library/astin/vol32no1/143.pdf
 |doi=10.2143/ast.32.1.1020
|doi-access=free
 }}</ref>
* assay analysis <ref>{{cite journal | last1 = Davidian | first1 = M | author1-link = Marie Davidian | year = 1990 | title = Estimation of variance functions in assays with possible unequal replication and nonnormal data | journal = Biometrika | volume = 77 | pages = 43–54 | doi=10.1093/biomet/77.1.43}}</ref><ref>{{cite journal | last1 = Davidian | author1-link = Marie Davidian | first1 = M. | last2 = Carroll | first2 = R. J. | last3 = Smith | first3 = W. | year = 1988 | title = Variance functions and the minimum detectable concentration in assays | url = http://www.lib.ncsu.edu/resolver/1840.4/3795| journal = Biometrika | volume = 75 | issue = 3| pages = 549–556 | doi=10.1093/biomet/75.3.549}}</ref>
* survival analysis<ref>{{cite journal | last1 = Aalen | first1 = O. O. | year = 1992 | title = Modelling heterogeneity in survival analysis by the compound Poisson distribution | journal = Ann. Appl. Probab. | volume = 2 | issue = 4| pages = 951–972 | doi=10.1214/aoap/1177005583| doi-access = free }}</ref><ref>{{cite journal | last1 = Hougaard | first1 = P. | last2 = Harvald | first2 = B. | last3 = Holm | first3 = N. V. | year = 1992 | title = Measuring the similarities between the lifetimes of adult Danish twins born between 1881–1930 | journal = Journal of the American Statistical Association | volume = 87 | issue = 417| pages = 17–24 | doi=10.1080/01621459.1992.10475170}}</ref><ref>{{cite journal | last1 = Hougaard | first1 = P | year = 1986 | title = Survival models for heterogeneous populations derived from stable distributions | journal = Biometrika | volume = 73 | issue = 2| pages = 387–396 | doi=10.1093/biomet/73.2.387}}</ref>
* ecology <ref name=Kendal2002 />
* analysis of alcohol consumption in British teenagers <ref>Gilchrist, R. and Drinkwater, D. 1999.
Fitting Tweedie models to data with probability of zero responses. 
Proceedings of the 14th International
Workshop on Statistical Modelling, Graz, pp. 207–214.</ref>
* medical applications <ref name="smyth1996">Smyth, G. K. 1996. 
Regression analysis of quantity data with exact zeros.
Proceedings of the Second Australia—Japan Workshop on Stochastic Models in Engineering, Technology and Management. Technology Management Centre, University of Queensland, 572–580.</ref>
* health economics <ref>{{cite journal | last1 = Kurz | first1 = Christoph F. | year = 2017 | title = Tweedie distributions for fitting semicontinuous health care utilization cost data | journal = BMC Medical Research Methodology | volume = 17 | issue = 171 | pages = 171 | doi = 10.1186/s12874-017-0445-y | pmid = 29258428 | pmc = 5735804 | doi-access = free }}</ref>
* meteorology and climatology <ref name="smyth1996"/><ref>{{cite journal | last1 = Hasan | first1 = M.M. | last2 = Dunn | first2 = P.K. | year = 2010 | title = Two Tweedie distributions that are near-optimal for modelling monthly rainfall in Australia | journal = International Journal of Climatology | volume =  31| issue = 9 | pages =  1389–1397| doi = 10.1002/joc.2162 | s2cid = 140135793 }}</ref>
* fisheries <ref>{{cite journal | last1 = Candy | first1 = S. G. | year = 2004 | title = Modelling catch and effort data using generalized linear models, the Tweedie distribution, random vessel effects and random stratum-by-year effects | journal = CCAMLR Science | volume = 11 | pages = 59–80 }}</ref>
* [[Mertens function]]<ref name=Kendal2011a>{{cite journal | last1 = Kendal | first1 = WS | last2 = Jørgensen | first2 = B | year = 2011 | title = Taylor's power law and fluctuation scaling explained by a central-limit-like convergence | journal = Phys. Rev. E | volume = 83 | issue = 6| page = 066115 | doi=10.1103/physreve.83.066115| pmid = 21797449 | bibcode = 2011PhRvE..83f6115K }}</ref>
*[[self-organized criticality]]<ref name=Kendal2015>
{{cite journal
 | author = Kendal, WS
 | year = 2015
 | title = Self-organized criticality attributed to a central limit-like convergence effect
 | journal =Physica A 
 | volume = 421
 | pages =141&ndash;150
 | doi = 10.1016/j.physa.2014.11.035
|bibcode =2015PhyA..421..141K | author-link = Wayne Kendal
 }}</ref>

==References==

{{reflist|40em}}

== Further reading ==

* {{cite book  | last = Dunn| first = P.K. |author2=Smyth, G.K. |  title = Generalized Linear Models With Examples in R | publisher = New York: Springer | year = 2018 | isbn=978-1-4419-0118-7 | doi=10.1007/978-1-4419-0118-7}} Chapter 12 is about Tweedie distributions and models.
* Kaas, R. (2005). [http://ucs.kuleuven.be/seminars_events/other/files/3afmd/Kaas.PDF "Compound Poisson distribution and GLM’s – Tweedie’s distribution"]. In ''Proceedings of the Contact Forum "3rd Actuarial and Financial Mathematics Day"'', pages 3–12. Brussels: Royal Flemish Academy of Belgium for Science and the Arts. 
* {{cite journal | last1 = Tweedie | first1 = M.C.K. | year = 1956 | title = Some statistical properties of Inverse Gaussian distributions | journal = Virginia J. Sci. |series=New Series | volume = 7 | pages = 160–165 }}

{{ProbDistributions|families}}

[[Category:Continuous distributions]]
[[Category:Systems of probability distributions]]