Approximations for the inverse cumulative distribution function of the gamma distribution used in wireless communication

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.

Earthquake Magnitude Probability and Gamma Distribution

“…Using the M-A method, Figure 3(a) shows the simulated magnitude probability distribution along with the observed one around the region of Taiwan (Figure 2). Compared to the observation, the two look like in good agreement, but actually the normalized differences (i.e., (observation -simulation)/observation) are quite substantial, in the range of -12% -35% as shown in Figure 3 Like the normal distribution that is commonly used in statistical and probabilistic studies, the gamma distribution has also been utilized in a variety of studies (e.g., Okagbue et al, 2020 [9]). For example, in medical science, it was found that the sum of basal and menstrual iron losses in women can be modeled by the gamma distribution satisfactorily (Yokoi, 2020 [10]).…”

Sampling Inspection Plan to Test Daily COVID-19 Cases Using Gamma Distribution under Indeterminacy Based on Multiple Dependent Scheme

“…In recent years, more researchers studied the applications of gamma distribution in statistical quality control, reliability, queuing theory, survival analysis, and communication engineering. For more details, refer to [43]. The present research is motivated by the idea of neutrosophic statistics given by [26] and extensive studies by Aslam from 2018 onwards on various neutrosophic and indeterminacy probability distributions in different sampling and control chart schemes; some citations are given in the introduction section.…”