Transformed exponential distributions

The processes of creating distributions from existing ones are an important topic in the study of probability models. Such processes expand the tool kit in the modeling process. Two examples: new distributions can be generated by taking independent sum of old ones or by mixing distributions (the result would be called a mixture). Another way to generate distributions is through raising a distribution to a power, which is the subject of this post. Start with a random variable X (the base distribution). Then raising it to a constant generates a new distribution. In this post, the base distribution is the exponential distribution. The next post discusses transforming the gamma distribution.

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Raising to a Power

Let X be a random variable. Let \tau be a nonzero constant. The new distribution is generated when X is raised to the power of 1 / \tau. Thus the random variable Y=X^{1 / \tau} is the subject of the discussion in this post.

When \tau >0, the distribution for Y=X^{1 / \tau} is called transformed. When \tau=-1, the distribution for Y=X^{1 / \tau} is called inverse. When \tau <0 and \tau \ne -1, the distribution for Y=X^{1 / \tau} is called inverse transformed.

If the base distribution is exponential, then raising it to 1 / \tau would produce a transformed exponential distribution for the case of \tau >0, an inverse exponential distribution for the case of \tau=-1 and an inverse transformed exponential distribution for the case \tau <0 with \tau \ne -1. If the base distribution is a gamma distribution, the three new distributions would be transformed gamma distribution, inverse gamma distribution and inverse transformed gamma distribution.

For the case of inverse transformed, we make the random variable Y=X^{-1 / \tau} by letting \tau >0. The following summarizes the definition.

Name of Distribution Parameter \tau Random Variable
Transformed \tau >0 Y=X^{1 / \tau}
Inverse \tau=-1 Y=X^{1 / \tau}
Inverse Transformed \tau >0 Y=X^{-1 / \tau}

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Transforming Exponential

The “transformed” distributions discussed here have two parameters, \tau and \theta (\tau=1 for inverse exponential). The parameter \tau is the shape parameter, which comes from the exponent 1 / \tau. The scale parameter \theta is added after raising the base distribution to a power.

Let X be the random variable for the base exponential distribution. The following shows the information on the base exponential distribution.

Base Exponential
Density Function f_X(x)=e^{-x} \ \ \ \ \ \ \ \ \ \ x>0
CDF F_X(x)=1-e^{-x} \ \ \ \ x>0
Survival Function S_X(x)=e^{-x} \ \ \ \ \ \ \ \ \ \ x>0

Note that the above density function and CDF do not have the scale parameter. Once the base distribution is raised to a power, the scale parameter will be added to the newly created distribution.

The following gives the CDF and the density function of the transformed exponential distribution. The density function is obtained by taking the derivative of the CDF.

    \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{1 / \tau} \le y) \\&=P(X \le y^\tau)\\&=F_X(y^\tau) \\&=1-e^{- y^\tau} \ \ \ \ \ \ \ \ \ \ \ \ \ y>0 \end{aligned}

    \displaystyle f_Y(y)=\tau \ y^{\tau-1} \ e^{- y^\tau} \ \ \ \ \ \ \ \ \ y>0

The following gives the CDF and the density function of the inverse exponential distribution.

    \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{-1} \le y) \\&=P(X \ge 1/y)\\&=S_X(1/y) \\&=e^{- 1/y} \ \ \ \ \ \ \ \ \ \ \ \ \ y>0 \end{aligned}

    \displaystyle f_Y(y)=\frac{1}{y^2} \ e^{- 1/y} \ \ \ \ \ \ \ \ \ y>0

The following gives the CDF and the density function of the inverse transformed exponential distribution.

    \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{- 1 / \tau} \le y) \\&=P(X \ge y^{- \tau})\\&=S_X(y^{- \tau}) \\&=e^{- y^{- \tau}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y>0 \end{aligned}

    \displaystyle f_Y(y)= \frac{\tau}{y^{\tau+1}} \ e^{- 1/y^\tau} \ \ \ \ \ \ \ \ \ y>0

The above derivation does not involve the scale parameter. Now it is added to the results.

Transformed Distribution
Transformed Exponential
CDF F_Y(y)=1-e^{- (y/\theta)^\tau} y>0
Survival Function S_Y(y)=e^{- (y/\theta)^\tau} y>0
Density Function f_Y(y)=(\tau / \theta) \ (y/\theta)^{\tau-1} \ e^{- (y/\theta)^\tau} y>0
Inverse Exponential
CDF F_Y(y)=e^{- \theta/y} y>0
Survival Function S_Y(y)=1-e^{- \theta/y} y>0
Density Function f_Y(y)=\frac{\theta}{y^2} \ e^{- \theta/y} y>0
Inverse Transformed Exponential
CDF F_Y(y)=e^{- (\theta/y)^{\tau}} y>0
Survival Function S_Y(y)=1-e^{- (\theta/y)^{\tau}} y>0
Density Function f_Y(y)=\tau ( \theta / y )^\tau \ (1/y) \ e^{- (\theta/y)^{\tau}} y>0

The transformed exponential distribution and the inverse transformed distribution have two parameters \tau and \theta. The inverse exponential distribution has only one parameter \theta. The parameter \theta is the scale parameter. The parameter \tau, when there is one, is the shape parameter and it comes from the exponent when the exponential is raised to a power.

The above transformation starts with the exponential distribution with mean 1 (without the scale parameter) and the scale parameter \theta is added back in at the end. We can also accomplish the same result by starting with an exponential variable X with mean (scale parameter) \theta^\tau. Then raising X to 1/\tau, -1, and -1/\tau would generate the three distributions described in the above table. In this process, the scale parameter \theta is baked into the base distribution. This makes it easier to obtain the moments of the “transformed” exponential distributions since the moments would be derived from exponential moments.
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Connection with Weibull Distribution

Compare the density function for the transformed exponential distribution with the density of the Weibull distribution discussed here. Note that the two are identical. Thus raising an exponential distribution to 1 / \tau where \tau >0 produces a Weibull distribution.

On the other hand, raising a Weillbull distribution to -1 produces an inverse Weillbull distribution (by definition). Let F_X(x)=1-e^{- x^\tau} be the CDF of the base Weibull distribution where \tau >0. Let’s find the CDF of Y=X^{-1}. Then add the scale parameter.

    \displaystyle F_Y(y)=P(Y \le y)=P(X \ge 1 / y)=e^{- (1/y)^\tau}

    \displaystyle F_Y(y)=e^{- (\theta/y)^\tau} (scale parameter added)

Note the the CDF of inverse Weibull distribution is identical to the one for inverse transformed exponential distribution. Thus transformed exponential distribution is identical to a Weibull distribution and inverse transformed exponential distribution is identical to an inverse Weibull distribution.

Since Weibull distribution is the same as transformed exponential distribution, the previous post on Weibull distribution can inform us on transformed exponential distribution. For example, assuming that the Weibull distribution (or transformed exponential) is a model for the time until death of a life, varying the shape parameter \tau yields different mortality patterns. The following are two graphics from the proevious post.

Figure 1

weibull-pdfs-2

Figure 2

weibull-failure-rates

Figure 1 shows the Weibull density functions for different values of the shape parameter (the scale parameter \theta is fixed at 1). The curve for \tau=1 is the exponential density curve. It is clear that the green density curve (\tau=2) approaches the x-axis at a faster rate then the other two curves and thus has a lighter tail than the other two density curves. In general, the Weibull (transformed exponential) distribution with shape parameter \tau >1 has a lighter tail than the Weibull with shape parameter 0<\tau <1.

Figure 2 shows the failure rates for the Weibull (transformed exponential) distributions with the same three values of \tau. Note that the failure rate for \tau=0.5 (blue) decreases over time and the failure rate for \tau=2 increases over time. The failure rate for \tau=1 is constant since it is the exponential distribution.

What is being displayed in Figure 2 describes a general pattern. When the shape parameter is 0<\tau<0.5, the failure rate decreases as time increases and the Weibull (transformed exponential) distribution is a model for infant mortality, or early-life failures. Hence these Weibull distributions have a thicker tail as shown in Figure 1.

When the shape parameter is \tau >1, the failure rate increases as time increases and the Weibull (transformed exponential) distribution is a model for wear-out failures. As times go by, the lives are fatigued and “die off.” Hence these Weibull distributions have a lighter tail as shown in Figure 1.

When \tau=1, the resulting Weibull (transformed exponential) distribution is exponential. The failure rate is constant and it is a model for random failures (failures that are independent of age).

Thus the transformed exponential family has a great deal of flexibility for modeling the failures of objects (machines, devices).

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Moments and Other Distributional Quantities

The moments for the three “transformed” exponential distributions are based on the gamma function. The two inverse distributions have limited moments. Since the transformed exponential distribution is identical to Weibull, its moments are identical to that of the Weibull distribution. The moments of the “transformed” exponential distributions are E(Y)=E(X^{1 / \tau}) where X has an exponential distribution with mean (scale parameter) \theta^\tau. See here for the information on exponential moments. The following shows the moments of the “transformed” exponential distributions.

Name of Distribution Moment
Transformed Exponential E(Y^k)=\theta^k \Gamma(1+k/\tau) k >- \tau
Inverse Exponential E(Y^k)=\theta^k \Gamma(1-k) k <1
Inverse Transformed Exponential E(Y^k)=\theta^k \Gamma(1-k/\tau) k <\tau

The function \Gamma(\cdot) is the Gamma function. The transformed exponential moment E(Y^k) exists for all k >- \tau. The moments are limited for the other two distributions. The first moment E(Y) does not exist for the inverse exponential distribution. The inverse transformed exponential moment E(Y^k) exist only for k<\tau. Thus the inverse transformed exponential mean and variance exist only if the shape parameter \tau is larger than 2.

The distributional quantities that are based on moments can be calculated (e.g. variance, skewness and kurtosis) when the moments are available. For all three "transformed" exponential distributions, percentiles are easily computed since the CDFs contain only one instance of the unknown y. The following gives the mode of the three distributions.

Name of Distribution Mode
Transformed Exponential \displaystyle \theta \biggl(\frac{\tau-1}{\tau} \biggr)^{1/\tau} for \tau >1, else 0
Inverse Exponential \theta / 2
Inverse Transformed Exponential \displaystyle \theta \biggl(\frac{\tau}{\tau+1} \biggr)^{1/\tau}

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\copyright 2017 – Dan Ma

4 thoughts on “Transformed exponential distributions

  1. Pingback: Transformed gamma distribution | Topics in Actuarial Modeling

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