Draft: Notes on the Pareto_distribution.tex

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\title{\textbf{Draft: Notes on the Pareto distribution}}
\author{John Creighton\\
        Nobody else}
\date{}
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\begin{document}

\maketitle

\section{Introduction}

The Pareto distribution is a well knon model for wealth and income distribution and sometimes (e.g. Jordan Peterson) used as a model for varations in productivity.However, for this to be an acurate model of productivity in the tail of the distribution (AKA survival function) One must assume that one's pay is proportional to their productivity. This is a highly contentious claim and for instance would contradict Marx's labour theory of value. Under Marx's theory the capitalist as relativly productive as indicated by wealth and income but instead apears so due to taking a disproporationate amount of the surplus value that is produced by the collective enterprise.

\section{Cross Dicipline Interest}

The most important cross discipline relevance of the Pareto distribution is the tail behaviour. Near the tail the shape of the curve is obscured by an integrating constant which approaches one at infitinity. This asymptotic constant dispears in the survival function \cite{wikipediaSurvivalFunction}.


\begin{equation}
S(t)=P(\{T>t\})=\int _{t}^{\infty }f(u)\,du=1-F(t).
\end{equation}

which is more suitable for curve fitting both in a numerical or statsitcial sense if one is interested primarly in the tail behavior. The survival function is often written with a bar on top of the cumulative distribution function. This bar denotes logical negation.

Formally the Pereto distrubtion is usually defined in terms of the survival function:

\begin{equation}
{\displaystyle {\overline {F}}(x)=\Pr(X>x)={\begin{cases}\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }&x\geq x_{\mathrm {m} },\\1&x<x_{\mathrm {m} },\end{cases}}}
\end{equation}
although the transformation into the cumulative distribution funciton is trival:

\begin{equation}
F_X(x) = \begin{cases}
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
0 & x < x_\mathrm{m}.\label{eq:CDF_pareto}
\end{cases}
\end{equation}
but must be transformed to something like the survival function to produce a nice linear fit in log space.

\includegraphics[width=0.9\textwidth]{Probability_density_function_of_Pareto_distribution.png}

Infact the orginal law was orginally developed by linear regression of the wealth distirubiton in log space. \cite{JulietteFournierSep2015} \cite{Persky1992}  (verify)

\begin{equation}
log \, N = A - a \, log \, x
\end{equation}
where $N$ is the number of households which income is greater than $x$. If we are using the natural logrithm than for notational covinence:

\begin{equation}
C=e^A
\end{equation}

$C$ is related to the scale paramater which is for practicle puroses obtained by first tasking the natural exponential of each side and then deviding by the number of hosholds.In which case

\begin{equation}
{A \over N(X_m)}=\left(X_m\right)^{\alpha}
\end{equation}

where $Xm$ is the minimum value of $X$ and is typically called "The Scale Paramater" \cite{wikipediaSurvivalFunction}.

In the case where the Pareto distribution is used to model income then Xm might be thought of as the minimum income required to sustain a worker, and should include indirect income such as gifits, charity, subsidies.

Such indirect income could take monitary form, of some physical good or service (e.g. parental labour). For wealth one would need to convert the above effective income into a present value based on the time value of money. $Xm$ is known as the scale paramater. This is somewhat problematic because we don't know what a worker will earn in the future and the time value of money is variable based on context (e.g. credit score and earning potential.)

\section{Is the Pareto Distribution "Tail Heavy"?}

\subsection{The Moments of the Parato Distribution}

The Parato Disribution is defined for $alpha>2$. This can be scene from it's probability density function:

\begin{equation}
f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases}
\end{equation}

which is a power law distribution . If we try to integrate this pdf (probability density function) at infinity convergence requires $\alpha>0$ and the origin it requires $\alpha<-1$ These two conditions can not be simutaneously true and this is why the Pareto distribution is defined in terms of the lower limit $Xm$.

Moreover, for the moments to be well defined $m < \alpha$, \cite{WikipediaPowerLaw} which means that that for the mean to be well defined $\alpha>1$

In general the moments of the Pareto distribution are expressed as:

\begin{equation}
\langle x^{m} \rangle = \int_{xm}^{\infty} x^{m} p(x) dx = { \alpha \over \alpha - m } \left(x_m\right)^m, \; where \;\; m>\alpha
\end{equation}

From which one can derive:

\begin{equation}
{\displaystyle \operatorname {E} (X)={\begin{cases}\infty &\alpha \leq 1,\\{\frac {\alpha x_{\mathrm {m} }}{\alpha -1}}&\alpha >1.\end{cases}}}
\end{equation}

and
\begin{equation}
{\displaystyle \operatorname {Var} (X)={\begin{cases}\infty &\alpha \in (1,2],\\\left({\frac {x_{\mathrm {m} }}{\alpha -1}}\right)^{2}{\frac {\alpha }{\alpha -2}}&\alpha >2.\end{cases}}}
\end{equation}

\subsection{Typical values of alpha and the 80-20 rule}

Recall from the previous section for the Parato distribution to even converge $\alpha>1$ and for the mean to be well defined $\alpha>2$. Without some further limiting factor (e.g. exponential limiting), the Pareto will always be a bit tail heavy becasue there will be some limit on how high an order of central moments is defined.

Furtermore, not only are the higher order moments not gaurnteed to be defined but even the first order moment is heavilyt tail dependent. Sometimes this is known as (Breaking the curve) where a few exceptionaly high values can play a large role in the mean.

The rule of thumb for a Pareto distirubtion is that 20\% of all people receive 80\% of all income. As given on wikipedia with this rule we have,
\begin{equation}
{\displaystyle \alpha_{(80-20)} =\log _{4}5={\cfrac {\log _{10}5}{\log _{10}4}}\approx 1.161}
\end{equation}

as we will show later this number is close to what one might infer from Oxfam data for wealth.


\subsection{Kurtosis}

Kurtosis plays an important role in determinging how quickly various estimates of central moments converage and is also a measure of how tail heavy a function is. The Kurtosis for the Pareto distribution is (\href{https://en.wikipedia.org/w/index.php?title=Pareto_distribution&oldid=965348211#Relation_to_the_\%22Pareto_principle\%22}{from wikipedia} \cite{WikipediaParetoDistribution} ):

\begin{equation}
\text{Excess kurtosis}=\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4
\end{equation}

The Pareto distribution has an Excess Kurtosis value which is greater than one. This type of distirubtion is refered to as \href{https://en.wikipedia.org/w/index.php?title=Kurtosis&oldid=965968832#Leptokurtic}{Leptokurtic} \cite{WikipediaKurtosis} and is characterized by a fatter tail. Other examples of such distibutions are  Student's t-distribution, Rayleigh distribution, Laplace distribution, Poisson distribution and the logistic distribution.

\section{Lorenz curve} \label{LorenzCurve}

The Lorenz Curve provides a good way to visualize inequality (\href{https://www.facebook.com/groups/280538506628903/permalink/285154029500684/}{faceook})(\href{https://en.wikipedia.org/w/index.php?title=Pareto_distribution&oldid=967068451#Lorenz_curve_and_Gini_coefficient}{wikipedia} \cite{WikipediaParetoDistribution} \cite{WikipediaLorenzCurve}).

\includegraphics[width=0.9\textwidth]{ParetoLorenzSVG.png}

The formal definition is:

\begin{equation}
L(F)=\frac{\int_{x_\mathrm{m}}^{x(F)}xf(x)\,dx}{\int_{x_\mathrm{m}}^\infty xf(x)\,dx} =\frac{\int_0^F x(F')\,dF'}{\int_0^1 x(F')\,dF'}
\end{equation}
where x(F) is the inverse of the CDF.

The CDF is given in euation \eqref{eq:CDF_pareto} and has the following inverse.

\begin{equation}
x(F)=\frac{x_\mathrm{m}}{(1-F)^{\frac{1}{\alpha}}} \label{eq:inv_cdf_pareto}
\end{equation}
where, $F(X)=P(x<X)$ and the Lorenz curve is threfore:

\begin{equation}
L(F) = 1-(1-F)^{1-\frac{1}{\alpha}}, \label{eq:Lorenz_Curve}
\end{equation}

reagranging:

\begin{equation}
F=1-(1-L(F))^{\alpha \over (1-\alpha)}
\end{equation}

differentiating:
\begin{equation}
f(L)=dF/dL=(1-L(F))^{2 \alpha - 1 \over (1-\alpha)}  \label{eq:Lorenz_Curveasdf}
\end{equation}

\section{The log-logistic Function and the Type II Pareto Distribution}

\subsection{Pareto Distibutions Type I to Type IV}
The Pareto distirubtion is a statment of the survival function. In this respect various functions are asymptotically equivalent to the Pareto function but differ for low values of the random variable.

\subsubsection{Type II Parato Distributions and the Lowmax Distirbution}

Pareto Considered Several of these. The first of these is the Type II pareto distirbution (\href{https://en.wikipedia.org/w/index.php?title=Pareto_distribution&oldid=967068451#Pareto_types_I\%E2\%80\%93IV}{wikipedia}  \cite{WikipediaParetoDistribution} )(\href{https://www.facebook.com/groups/280538506628903/permalink/281084956574258/?comment_id=281090623240358}{facebook}) wich reduces to the lowmax distirution (\href{https://en.wikipedia.org/w/index.php?title=Lomax_distribution&oldid=958008294#Characterization}{wikipedia}  \cite{WikipediaLomax} )(\href{https://www.facebook.com/groups/280538506628903/permalink/281084956574258/}{facebook})  when the paramater $\mu=0$ is zero.

\begin{equation}
{\displaystyle {\overline {F}}(x)=\Pr(X>x)=1-F(x) = \left[1+{\frac {x-\mu }{\sigma }}\right]^{-\alpha }}
\end{equation}

the lomax distribution was refered to by Johnson \& Kotz (1970) \cite{Johnson1970} as a Pareto distribution of the second kind ( \href{https://www.facebook.com/groups/280538506628903/permalink/281081976574556/}{facebook} \cite{Clark1999}), which has the following probability mass function.

\begin{equation}
{\displaystyle {\displaystyle p(x)={{\alpha \lambda ^{\alpha }} \over {(x+\lambda )^{\alpha +1}}}}={\alpha  \over \lambda }\left[{1+{x \over \lambda }}\right]^{-(\alpha +1)},\qquad x\geq 0,}
\end{equation}

the mean for the Type II pareto distirbution is given by:

\begin{equation}
 E[X]=\frac{ \sigma }{\alpha-1}
\end{equation}

and in general the central moments are:

\begin{equation}
 E[X^\delta]= \frac{ \sigma^\delta \Gamma(\alpha-\delta)\Gamma(1+\delta)}{\Gamma(\alpha)}
\end{equation}


where for positive integers (\href{https://en.wikipedia.org/w/index.php?title=Gamma_function&oldid=962235242}{wikipedia} \cite{WikipediaGammaFn})
\begin{equation}
{\displaystyle \Gamma (n)=(n-1)!\ .}
\end{equation}

\subsubsection{Type III \& IV Parato Distributions}

A type IV Parato Distriubtion can genearlizes Types I through to III as follows:

\begin{equation}
P(IV)(\sigma, \sigma, 1, \alpha) = P(I)(\sigma, \alpha),
\end{equation}
\begin{equation}
P(IV)(\mu, \sigma, 1, \alpha) = P(II)(\mu, \sigma, \alpha),
\end{equation}
\begin{equation}
P(IV)(\mu, \sigma, \gamma, 1) = P(III)(\mu, \sigma, \gamma).
\end{equation}

The survival function for the Type IV pareto distibution is:

\begin{equation}
{\displaystyle {\overline {F}}(x)=P(X>x)=1-F(x) = \left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }}
\end{equation}
where, ${\displaystyle x\geq \mu }$ and $\mu \in R \;\;$ $\sigma, \gamma > 0, \alpha$ \newline

and has the following central moments

\begin{equation}
E[X^\delta]= \frac{\sigma^\delta\Gamma(\alpha-\gamma \delta)\Gamma(1+\gamma \delta)}{\Gamma(\alpha)}
\end{equation}

where $\alpha$ is the tail index, $\mu$ is location, $\sigma$ is scale, $\gamma$ is an inequality parameter.

\subsection{The Log-Logistic Distirbution} \label{LogLogisticDist}

The cumulative distribution for the Type IV pareto distribution can be written as:

\begin{equation}
{\displaystyle F(x)=1-{\overline F}(x)=P(X<x)= 1-\left[1+\left({\frac {x-\mu }{\sigma }}\right)^{1/\gamma }\right]^{-\alpha }} \label{eq:GenLogLogDist}
\end{equation}
when $\mu=0$ then
\begin{equation}
{\displaystyle F(x)= \frac{\left[ \sigma^{1/\gamma} + \left(x-\mu\right)^{1/\gamma } \right]^{\alpha}}{\left[ \sigma^{1/\gamma} + \left(x-\mu\right)^{1/\gamma } \right]^{\alpha}}- {\sigma^{\alpha / \gamma} \over \left[ \sigma^{1/\gamma} + \left(x-\mu\right)^{1/\gamma } \right]^{\alpha} } }
\end{equation}

and when $\alpha=1$ is close to one this is approximatly

\begin{equation}
{\displaystyle F(x)= \frac{\left(x-\mu\right)^{\beta }}{ \sigma^{\beta} + \left(x-\mu\right)^{\beta }}}
\end{equation}
where, $\beta=\alpha/\gamma$ \newline

which when $\mu=0$ is the logistic distribution.

\subsubsection{Alpha ($\alpha$) aproximatly equal to one is exact for Type III Pareto}

The above aproximation $\alpha=1$ is exact for a "Type III" pareto distribution. Also for large values of X the "\emph{Type IV Parato distribution}" reduces to a "\emph{Type I Pareto distribtion}". With $alpha=1$ then $\gamma$ is a free parmater to fit the tail behavior which is the same for Pareto Distribtions of Types I to IV.

\subsubsection{The Moments of the Log-Logistic Distribution}

The log-logistic distibution has the following central moments:

\begin{equation}
\operatorname {E}(X)=\alpha b/\sin b,\quad \beta >1, \; b=\pi /\beta
\end{equation}
\begin{equation}
\operatorname {Var}(X)=\alpha ^{2}\left(2b/\sin 2b-b^{2}/\sin ^{2}b\right),\quad \beta >2,  \; b=\pi /\beta
\end{equation}

\section{Derivation of Log-Type Distributions}

The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. In general we can consider a random variable of the form:
\begin{equation}
X=e^{\mu +\sigma Z}
\end{equation}

Where Z is a random variable of a given type (e.g. logistic or normal) and X is a variable who is a distribution of that type. Stated formally:

\begin{equation}
{\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}
\end{equation}

and it follows:

\begin{equation}
{\displaystyle {\begin{aligned}f_{X}(x)&={\frac {\rm {d}}{{\rm {d}}x}}\Pr(X\leq x)={\frac {\rm {d}}{{\rm {d}}x}}\Pr(\ln X\leq \ln x)={\frac {\rm {d}}{{\rm {d}}x}}\Phi \left({\frac {\ln x-\mu }{\sigma }}\right)\\[6pt]&=\varphi \left({\frac {\ln x-\mu }{\sigma }}\right){\frac {\rm {d}}{{\rm {d}}x}}\left({\frac {\ln x-\mu }{\sigma }}\right)=\varphi \left({\frac {\ln x-\mu }{\sigma }}\right){\frac {1}{\sigma x}}\\[6pt].\end{aligned}}}
\end{equation}

and if $\phi(x)$ is a normal distrubution then
\begin{equation}
f_{X}(x)={\frac {1}{x}}\cdot {\frac {1}{\sigma {\sqrt {2\pi \,}}}}\exp \left(-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right)
\end{equation}

alternatively if $\phi(x)$ is a logistic distrubution then

\begin{equation}
f(x;\alpha ,\beta )={\frac  {(\beta /\alpha )(x/\alpha )^{{\beta -1}}}{\left(1+(x/\alpha )^{{\beta }}\right)^{2}}}
\end{equation}

\section{The Quantile Function}

In section \ref{LogLogisticDist} we used the aproximation $\alpha=1$ to argue that the Pareto distribution reduces to the log-logistic distirubtion when $\mu=0$. As stated, this aproximation is exact for a "\emph{Type III pareto distribution}". The "\emph{Type III Pareto Distribugion}", generalizes the \emph{"Type I Pareto Distirubution"} and all types (i.e.Types I to IV) converge to the "\emph{Type I Pareto Distribution}" for large values of $X$.

For the following analysis with the cumulative distirbution function given in equation \eqref{eq:GenLogLogDist} since it is more general than the log-logistic distirubtion. We can invert this equation by solving for $X$ in terms of the value of the cumulative distribution function.

\begin{equation}
x=\mu + \left[ {\sigma^\beta F(x) \over 1 - F(x) } \right]^{\left( 1/\beta\right) }=\mu + \sigma \left[ {F(x) \over 1 - F(x) } \right]^{\left( 1/\beta\right) }
\end{equation} \label{eq:inv_cdf_paretoIII}

The result is the Quantile Function for the Type III Pareto distibution and if we set $\mu=0$ this is the Quantile FUnction for the log-logistic distirbution (\href{https://en.wikipedia.org/w/index.php?title=Logistic_distribution&oldid=955853904#Quantile_function}{wikipedia} \cite{WikipediaQuantileFunction} )(\href{https://www.facebook.com/groups/280538506628903/permalink/280655966617157/}{facebook}).

\subsection{The Asymptoic Quantile Function For Types I \& III Parato Distributions}

The form of the Type III Pareto Distirugiton Quantile Function \eqref{eq:inv_cdf_paretoIII} is notacibly different than the Type I Parto Distirguion Quantile Function (i.e. equation  \eqref{eq:inv_cdf_pareto}). The main distinquishing factor is the $F(x)$ in the numberator is not present in the Type I version of the Quantile Function.

The asymptoic simmilarity can be shown by using a new variable $\epsilon = 1-F(x)$. With this substitution \eqref{eq:inv_cdf_paretoIII} becomes:

\begin{equation}
x=\mu + \sigma \left[ \frac{1}{\epsilon} -1 \right]^{\left( 1/\beta\right) }\cong \mu + \sigma\left[ \frac{1}{\epsilon} \right]^{\left( 1/\beta\right) }
\end{equation} \label{eq:inv_cdf_paretoIII}

when both $\mu=0$ and $\sigma=(x_m)^{1/\beta}$ we get the same asymptotic result for both the types I and III pareto distributions.

\section{The Isograph}

The Isograph can be obtained by first subtracing $\mu$ from each side of equation \eqref{eq:inv_cdf_paretoIII} and then dividing by the median to yield a log-logit transformation of the Type III pareto disribution.

This should be s stright line with slope $(1/\beta)$ and intercept $\frac{\sigma}{\sigma_{median}}=\sigma^\prime$ \newline

As can be scene this log-logit transformation produces a faily straight line \cite{isographslide} \newline

\includegraphics[width=1.2\textwidth]{LogitWealth.png}

Dividing the Y variable \cite{Mishra2017} \cite{Chauvel2018} we get the isograph, which is a constant when the curve is a Type III pareto distribution. The deviation from this constant represents unexpected inequality within a given group. Typically the logrithm used for this graph is the natural logarithm. The isograph essentially shows where the Type III pareto is not a good fit.

\section{The Quantile Function}


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\end{document}