# Transformed Pareto distribution

One way to generate new probability distributions from old ones is to raise a distribution to a power. Two previous posts are devoted on this topic – raising exponential distribution to a power and raising a gamma distribution to a power. Many familiar and useful models can be generated in this fashion. For example, Weibull distribution is generated by raising an exponential distribution to a positive power. This post discusses the raising of a Pareto distribution to a power, as a result generating Burr distribution and inverse Burr distribution.

Raising to a Power

Let $X$ be a random variable. Let $\tau$ be a positive constant. The random variables $Y=X^{1/\tau}$, $Y=X^{-1}$ and $Y=X^{-1/\tau}$ are called transformed, inverse and inverse transformed, respectively.

Let $f_X(x)$, $F_X(x)$ and $S_X(x)=1-F_X(x)$ be the probability density function (PDF), the cumulative distribution function (CDF) and the survival function of the random variable $X$ (the base distribution). The goal is to express the CDFs of the “transformed” variables in terms of the base CDF $F_X(x)$. The following table shows how.

Name of Distribution Random Variable CDF
Transformed $Y=X^{1 / \tau}, \ \tau >0$ $F_Y(y)=F_X(y^\tau)$
Inverse $Y=X^{-1}$ $F_Y(y)=1-F_X(y^{-1})$
Inverse Transformed $Y=X^{-1 / \tau}, \ \tau >0$ $F_Y(y)=1-F_X(y^{-\tau})$

If the CDF of the base distribution, as represented by the random variable $X$, is known, then the CDF of the “transformed” distribution can be derived using $F_X(x)$ as shown in this table. Thus the CDF, in many cases, is a good entry point of the transformed distribution.

Pareto Information

Before the transformation, we first list out the information on the Pareto distribution. The Pareto distribution of interest here is the Type II Lomax distribution (discussed here). The following table gives several distributional quantities for a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta$.

Pareto Type II Lomax
Survival Function $S(x)=\displaystyle \biggl( \frac{\theta}{x+\theta} \biggr)^\alpha \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x >0$
Cumulative Distribution Function $F(x)=1-\displaystyle \biggl( \frac{\theta}{x+\theta} \biggr)^\alpha \ \ \ \ \ \ \ \ \ \ \ \ \ x >0$
Probability Density Function $\displaystyle f(x)=\frac{\alpha \ \theta^\alpha}{(x+\theta)^{\alpha+1}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x >0$
Mean $\displaystyle E(X)=\frac{\theta}{\alpha-1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \alpha>1$
Median $\displaystyle \theta \ 2^{\frac{\alpha}{2}}-\theta$
Mode 0
Variance $\displaystyle Var(X)=\frac{\theta^2 \ \alpha}{(\alpha-1)^2 \ (\alpha-2)} \ \ \ \ \ \ \ \alpha>2$
Higher Moments $\displaystyle E(X^k)=\frac{k! \ \theta^k}{(\alpha-1) \cdots (\alpha-k)} \ \ \ \ \ \ \alpha>k \ \ \ k$ is integer
Higher Moments $\displaystyle E(X^k)=\frac{\theta^k \ \Gamma(k+1) \Gamma(\alpha-k)}{\Gamma(\alpha)} \ \ \ \ \alpha>k$

The higher moments in the general case use $\Gamma(\cdot)$, which is the gamma function.

The Distributions Derived from Pareto

Let $X$ be a random variable that has a Pareto distribution (as described in the table in the preceding section). Assume that $X$ has a shape parameter $\alpha$ and scale parameter $\theta$. Let $\tau$ be a positive number. When raising $X$ to the power $1/\tau$, the resulting distribution is a transformed Pareto distribution and is also called a Burr distribution, which then is a distribution with three parameters – $\alpha$, $\theta$ and $\tau$.

When raising $X$ to the power $-1/\tau$, the resulting distribution is an inverse transformed Pareto distribution and it is also called an inverse Burr distribution. When raising $X$ to the power -1, the resulting distribution is an inverse Pareto distribution (it does not have a special name other than inverse Pareto).

The paralogistic family of distributions is created from the Burr distribution by collapsing two of the parameters into one. Let $\alpha$, $\theta$ and $\tau$ be the parameters of a Burr distribution. By equating $\tau=\alpha$, the resulting distribution is a paralogistic distribution. By equating $\tau=\alpha$ in the corresponding inverse Burr distribution, the resulting distribution is an inverse paralogistic distribution.

Transformed Pareto = Burr

There are two ways to create the transformed Pareto distribution. One is to start with a base Pareto with shape parameter $\alpha$ and scale parameter 1 and then raise it to $1/\tau$. The scale parameter $\theta$ is added at the end. Another way is to start with a base Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^\tau$ and then raise it to the power $1/\tau$. Both ways would generate the same CDF. We take the latter approach since it generates both the CDF and moments quite conveniently.

Let $X$ be a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^\tau$. The following table gives the distribution information on $Y^{1/\tau}$.

Burr Distribution
CDF $F_Y(y)=\displaystyle 1-\biggl( \frac{1}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$
Survival Function $S_Y(x)=\displaystyle \biggl( \frac{1}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$
Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha \ \tau \ (y/\theta)^\tau}{y \ [(y/\theta)^\tau+1 ]^{\alpha+1}}$ $y >0$
Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1/\tau+1) \Gamma(\alpha-1/\tau)}{\Gamma(\alpha)}$ $1 <\alpha \ \tau$
Median $\displaystyle \theta \ (2^{1/\alpha}-1)^{1/\tau}$
Mode $\displaystyle \theta \ \biggl(\frac{\tau-1}{\alpha \tau+1} \biggr)^{1/\tau}$ $\tau >1$, else 0
Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(k/\tau+1) \Gamma(\alpha-k/\tau)}{\Gamma(\alpha)}$ $-\tau

The distribution displayed in the above table is a three-parameter distribution. It is called the Burr distribution with parameters $\alpha$ (shape), $\theta$ (scale) and $\tau$ (power).

To obtain the moments, note that $E(Y^k)=E(X^{k/\tau})$, which is derived using the Pareto moments. The Burr CDF has a closed form that is relatively easy to compute. Thus percentiles are very accessible. The moments rely on the gamma function and are usually calculated by software.

Inverse Transformed Pareto = Inverse Burr

One way to generate inverse transformed Pareto distribution is to raise a Pareto distribution with shape parameter $\alpha$ and scale parameter 1 to the power of -1 and then add the scale parameter. Another way is to raise a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^{-\tau}$. Both ways derive the same CDF. As in the preceding case, we take the latter approach.

Let $X$ be a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^{-\tau}$. The following table gives the distribution information on $Y^{-1/\tau}$.

Inverse Burr Distribution
CDF $F_Y(y)=\displaystyle \biggl( \frac{(y/\theta)^\tau}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$
Survival Function $S_Y(x)=\displaystyle 1-\biggl( \frac{(y/\theta)^\tau}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$
Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha \ \tau \ (y/\theta)^{\tau \alpha}}{y \ [1+(y/\theta)^\tau]^{\alpha+1}}$ $y >0$
Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1-1/\tau) \Gamma(\alpha+1/\tau)}{\Gamma(\alpha)}$ $1 <\tau$
Median $\displaystyle \theta \ \biggl[\frac{1}{ 2^{1/\alpha}-1} \biggr]^{1/\tau}$
Mode $\displaystyle \theta \ \biggl(\frac{\alpha \tau-1}{\tau+1} \biggr)^{1/\tau}$ $\alpha \tau >1$, else 0
Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(1-k/\tau) \Gamma(\alpha+k/\tau)}{\Gamma(\alpha)}$ $-\alpha \tau

The distribution displayed in the above table is a three-parameter distribution. It is called the Inverse Burr distribution with parameters $\alpha$ (shape), $\theta$ (scale) and $\tau$ (power).

Note that both the moments for Burr and inverse Burr distributions are limited, the Burr limited by the product of the parameters $\alpha$ and $\tau$ and the inverse Burr limited by the parameter $\tau$. This is not surprising since the base Pareto distribution has limited moments. This is one indication that all of these distributions have a heavy right tail.

The Paralogistic Family

With the facts of the Burr distribution and the inverse Burr distribution established, paralogistic and inverse paralogistic distributions can now be obtained. A paralogistic distribution is simply a Burr distribution with $\tau=\alpha$. An inverse paralogistic distribution is simply an inverse Burr distribution with $\tau=\alpha$. In the above tables for Burr and inverse Burr, replacing $\tau$ by $\alpha$ gives the following table.

Paralogistic Distribution
CDF $F_Y(y)=\displaystyle 1-\biggl( \frac{1}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$
Survival Function $S_Y(x)=\displaystyle \biggl( \frac{1}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$
Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha^2 \ \ (y/\theta)^\alpha}{y \ [(y/\theta)^\alpha+1 ]^{\alpha+1}}$ $y >0$
Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1/\alpha+1) \Gamma(\alpha-1/\alpha)}{\Gamma(\alpha)}$ $1 <\alpha^2$
Median $\displaystyle \theta \ (2^{1/\alpha}-1)^{1/\alpha}$
Mode $\displaystyle \theta \ \biggl(\frac{\alpha-1}{\alpha^2+1} \biggr)^{1/\alpha}$ $\alpha >1$, else 0
Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(k/\alpha+1) \Gamma(\alpha-k/\alpha)}{\Gamma(\alpha)}$ $-\alpha
Inverse Paralogistic Distribution
CDF $F_Y(y)=\displaystyle \biggl( \frac{(y/\theta)^\alpha}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$
Survival Function $S_Y(x)=\displaystyle 1-\biggl( \frac{(y/\theta)^\alpha}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$
Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha^2 \ (y/\theta)^{\alpha^2}}{y \ [1+(y/\theta)^\alpha]^{\alpha+1}}$ $y >0$
Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1-1/\alpha) \Gamma(\alpha+1/\alpha)}{\Gamma(\alpha)}$ $1 <\alpha$
Median $\displaystyle \theta \ \biggl[\frac{1}{ 2^{1/\alpha}-1} \biggr]^{1/\alpha}$
Mode $\displaystyle \theta \ (\alpha-1)^{1/\alpha}$ $\alpha^2 >1$, else 0
Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(1-k/\alpha) \Gamma(\alpha+k/\alpha)}{\Gamma(\alpha)}$ $-\alpha^2

Inverse Pareto Distribution

The distribution that has not been discussed is the inverse Pareto. Again, we have the option of deriving it by raising to a base Pareto with just the shape parameter to -1 and then add the scale parameter. We take the approach of raising a base Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^{-1}$. Both approaches lead to the same CDF.

Inverse Pareto Distribution
CDF $F_Y(y)=\displaystyle \biggl( \frac{y}{\theta+y} \biggr)^\alpha$ $y >0$
Survival Function $S_Y(x)=\displaystyle 1-\biggl( \frac{y}{\theta+y} \biggr)^\alpha$ $y >0$
Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha \ \theta \ y^{\alpha-1}}{[\theta+y ]^{\alpha+1}}$ $y >0$
Median $\displaystyle \frac{\theta}{2^{1/\alpha}-1}$
Mode $\displaystyle \theta \ \frac{\alpha-1}{2}$ $\alpha >1$, else 0
Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(1-k) \Gamma(\alpha+k)}{\Gamma(\alpha)}$ $-\alpha

The distribution described in the above table is an inverse Pareto distribution with parameters $\alpha$ (shape) and $\theta$ (scale). Note that the moments are even more limited than the Burr and inverse Burr distributions. For inverse Pareto, even the mean $E(Y)$ is nonexistent.

Remarks

The Burr and paralogistic families of distributions are derived from the Pareto family (Pareto Type II Lomax). The Pareto connection helps put Burr and paralogistic distributions in perspective. The Pareto distribution itself can be generated as a mixture of exponential distributions with gamma mixing weight (see here). Thus from basic building blocks (exponential and gamma), vast families of distributions can be created, thus expanding the toolkit for modeling. The distributions discussed here are found in the appendix that is found in this link.

$\copyright$ 2017 – Dan Ma