Abstract
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.
Highlights
The gamma distribution, in its simplest form, is characterized by two positive parameters known as the degrees of freedom or the shape parameter, k, and the rate or shape parameter
The ordinary differential equations obtained from quantile mechanics were solved explicitly and the product of quantile functions that are very close to the R software values, as shown in the error analysis
The approach used in this paper can be used for any given degrees of freedom of the gamma distribution
Summary
The gamma distribution, in its simplest form, is characterized by two positive parameters known as the degrees of freedom or the shape parameter, k, and the rate or shape parameter. Modeling by the direct use of gamma distribution fit via parameter estimation [22, 23, 24]. Approximating a phenomenon, for example, signal and interference powers [25, 26], co-channel interference [27], rendezvous time [28], simulation results in fading wireless channel conditions [29], action duration and inter-arrival [30] and resource requirements for traffic characteristics [31], and Derivation of closed forms of models that follows gamma distribution. The closed forms are useful in modelling wireless network systems and other related models as seen in beamforming [32], bit error rate [33, 34] and average bit error rate [35]
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