Exponential-Type Distribution

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What is an exponential-type distribution?

The exponential-type distribution is a broad class of probability distributions that includes many common probability distributions such as the exponential distribution, the gamma distribution, the Gumbel distribution, the log-normal distribution, the normal distribution and the Weibull distribution.

Although sometimes used interchangeably, the exponential distribution and exponential-type distribution are not the same thing; the exponential distribution is a sub-type characterized by a constant hazard rate. Although the exponential-type distribution can be used to model a wide variety of phenomena, the exponential distribution is simpler and often easier to work with.

The exponential-type distribution can model a wide variety of phenomena. It is often used in reliability engineering, where it can be used to model a system’s time to failure. In finance, it can be used to model the distribution of asset prices.

Exponential-type distribution properties

Exponential-type distributions are defined by probability densities (pdfs) given by [1]:

f(y; δ) = exp [yδ + q(δ)] , a < δ < b

 

with respect to a σ-ίinite measure μ over a Euclidean sample space

Exponential-type distributions have finite moments of all orders [3].

Exponential-type distributions can be classified as convex, simple, or concave, depending on the behavior of the following extremal intensity function (extremal failure rate). Suppose we want to find extreme observations in a large sample n. We can obtain limits I_α and U_α such that [4]

where γ_n  , the extremal intensity function, determines the class:

1. Convex exponential type: γ_n → ∞ as n → ∞ (e.g. normal distributions).
2. Simple exponential type:  γ_n → c >0 as n → ∞ (e.g. exponential distributions).
3. Concave exponential type:  γ_n → 0 as n → ∞ (e.g. logistic distributions).

Exponential-type functions

The exponential-type distribution and the exponential-type function are closely related concepts in probability theory.

While the exponential-type distribution is a family of probability distributions that share a common asymptotic behavior, the exponential-type function represents the PDF of the exponential-type distribution. It can be used to calculate the probability of a random variable from the exponential-type distribution taking on a specific value.

References

1. Bolger, E. & Harkness, W. Some Characterizations of exponential-type distributions. Pacific Journal of Mathematics. Vol 16, No. 1, 1966.
2. Sobczyk, K. & Spencer, B. Random Fatigue: From Data to theory. (1992) Academic Press.
3. Csorgo, M. and Krishnaiah, P. (2010). From Finite Sample to Asymptotic Methods. Cambridge University Press.
4. Por Pranab K. Sen, Julio M. Singer, Antonio C. Pedroso de Lima. From Finite Sample to Asymptotic Methods in Statistics.