Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication
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CellPress
Abstract
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The use of quantile functions of probability distributions whose cumulative distribution is intractable is often
limited in Monte Carlo simulation, modeling, and random number generation. Gamma distribution is one of such
distributions, and that has placed limitations on the use of gamma distribution in modeling fading channels and
systems described by the gamma distribution. This is due to the inability to find a suitable closed-form expression
for the inverse cumulative distribution function, commonly known as the quantile function (QF). This paper
adopted the Quantile mechanics approach to transform the probability density function of the gamma distribution
to second-order nonlinear ordinary differential equations (ODEs) whose solution leads to quantile approximation.
Closed-form expressions, although complex of the QF, were obtained from the solution of the ODEs for degrees of
freedom from one to five. The cases where the degree of freedom is not an integer were obtained, which yielded
values closed to the R software values via Monte Carlo simulation. This paper provides an alternative for simulating
gamma random variables when the degree of freedom is not an integer. The results obtained are fast,
computationally efficient and compare favorably with the machine (R software) values using absolute error and
Kullback–Leibler divergence as performance metrics.
Keywords
QA Mathematics