Geometric
Brownian motion and Gaussian distributions have been used to describe
financial returns, until empirical investigations presented the case
of heavy tailed and asymmetric behavior. This set off a burst of interest
among economists to develop new approaches to the theory of random walk
in financial (asset) prices. Innovative models have been developed using
the Pareto law to characterize heavy tailed behavior in the family of
Lévy distributions. Subsequently, distributions with innovations
to model asymmetry have matured. It is in this spirit of inherent thinking
we propose a four parameter doubly-Pareto uniform (DPU) distribution
over the entire real line with capability to model heavy tailed and
asymmetric behavior.
The
DPU density and cumulative distribution functions are constructed by
seamlessly concatenating left and right Pareto tails via a uniform central
stage. We present a detailed derivation of the properties of the DPU
distribution. A maximum likelihood estimation (MLE) procedure is developed
in analytic form. Applications and advantages of our developed methodology
are illustrated in the analyses of data. Two primary motivating examples
are drawn from financial markets. The DPU model is extended to a third
data set describing biometric characteristics of athletes, thereby presenting
applicability of the model beyond the financial realm.
An
additional application of the DPU model is the utilization of the uniform
distribution as an input distribution to uncertainty analysis over bounds
, where the parameters of the uniform distribution are extracted via
quantile specification by experts. To take this a step further, a modal
range can be elicited from experts as an alternate to point estimates
of central tendency where the range lies between the upper and lower
quantiles. These quantile ranges contribute to the parameter estimation
of the input distribution in uncertainty analysis.
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