On the Application of Generalized Beta-G Family of Distributions to Prices of Cereals

Generalized Beta-G family of distributions proposed has alternative distributions to unbounded distributions for modeling price returns. In contrast to Gaussian and other unbounded distributions that take values from ( ) , −∞ ∞ , Generalized Beta-G family of distributions takes values from [ ) 0, ∞ so as to properly contain only positive valued observations like that of price returns. In line with this, Nine (9) befitting candidates of the Generalized Beta-G family of distributions were proposed and subjected to monthly prices of cereals. Chen distributional random noise outstripped other candidates of the Generalized Beta-G family of distributions to produce minimum monthly standard deviations of 0.2686 (26.86%), 0.2572 (25.72%), 0.2404 (24.40%), 0.2267 (22.67%), 0.2257 (22.57%), 0.2544 (25.44%), 0.2343 (23.43%), 0.2391 (23.91%), 0.2273 (22.73%) and 0.2465 (24.65%) for prices of Rice, Maize, Sorghum, Millet, G-corn, Cowpea, Groundnut, Beans, Wheat and Cassava re-spectively. Chen and Loglogistic distributional random noises are the leading candidates among the Generalized Beta-G family of distributions in modelling price returns of the cereals, followed by Fréchet, Weibull and Birnbaum-Saunders random noises in order of significant. Lomax and Linear Failure Rate (LFR) are the ineffective random noises in modeling the price returns.


Introduction
Over the past few years, generalization of statistical distribution has attracted much attention. The attention can be classified based on range of values the distribution (s) and subjected matter (s) is/are defined for. When the range of val-tions is Birnbaum-Saunders, Chen, Weibull, Fréchet, F, Life Failure Rate (LFR), Log-logistic, lognormal and Lomax [13].
Among the few applications of the members of the Generalized Beta-G family of distributions to real life events was the application of the extended Birnbaum-Saunders distribution (Otherwise known as Marshall-Olkin extended Birnbaum-Saunders distribution) to reliability studies and fatigue failure times by [14]. Reference [15] also introduced a modified Burr III distribution called Beta-Burr III distribution and highlighted is importance in modeling problems related to actuarial science and survival analysis. They did not only derive the distribution's docile attributes like the moments (including its moment generating function), reliability, entropies and quantile functions, but also applied it to a survival data of acute myelogeneous Leukaemia of thirty-three (33) patients suffering from the disease.
Reference [16] propounded Beta Gumbel distribution and highlighted its ability to model accelerated life testing problems through earthquakes, flood frequency analysis, rainfall, sea currents, and wind speeds. Reference [17] extended the work of Reference [16] and introduced Beta modified Fréchet distribution called Beta Fréchet (BF) distribution as an extrapolation of Fréchet and Exponentiated Fréchet (EF) distributions. They applied the proposed BF distribution to two sets of data: the uncensored dataset that consist of hundred (100) observations of breaking stress of carbon fibres (in Gba); and used dataset by [18], the dataset that consist of strengths of 1.5 cm glass fibres measured at the National Physical Laboratory, England. They adopted the Maximum Likelihood method of estimation, and they were able to estimate the four embedded parameters with 95% confidence level that the BF distribution is an adequate model for modelling the two set of fibres. It is to be noted that Gumbel and Fréchet are two out of the three distributions of the Extreme-Value-Distributions (EVDs). The only notable application of Beta distribution to financial returns was when [19] presented a skewed distribution known as modified Beta distributions and applied it to Standard & Poor's/International Finance Corporation global daily price indices in United States dollars for South Africa with some inferences made. The statistical properties of the distributions were derived as well as the parameter estimation of the embedded parameters via Maximum Likelihood estimation technique. In light of this, none of the related members or real members of the Generalized Beta-G family of distributions has been applied to stock returns or price indices. The Generalized Beta-G family of distributions is a family of distributions that takes only positive values on the real number line against unbounded distributions that have been used in modeling price indices. The novelty of this work is the first ever application of the Beta-G family of distributions to financial returns of price of commodities, in contrast to its known application to survival analyzes and reliability studies. However, this piece of work will focus on the application of Generalized Beta-G family of distributions to wholesale prices of cereals in Kano state, Nigeria. The wholesale prices of the edible grains to be considered will be from 2007 to 2019. The members of the family of the Generalized Beta-G distributions to be considered are Birnbaum-Saunders, Burrxii, Chen, Gamma, Lognormal, Log-Logistic, Lomax, Weibull and Fréchet.

Mathematical Pro-Forma of Price Framework
Let 0 p denote the initial price for any commodity/stock returns assuming further that the evolution or time varying for such prices is via the horizon p = one year or p = one month. If the price of such commodity at p is denoted by t p , a random variable, such that, The p in Equation (1) is also known as growth rate.
Assuming G is a well-defined function on +  with Cumulative Distribution Function (CDF). Let F be another well-defined CDF positioned on G to be the sphere of an increasing function in an enclosed Beta function in the following form: and Ω being the parameter space of the welldefined G function. The CDF and Probability Density Function (PDF) of the Generalized Beta-G family of distributions can then be defined as: for "r" in the range of ( ) Ω is the universal parameter space of the Generalized Beta-G family of distributions with induced shape of denotes the incomplete beta function ratio. According to [5] and [20], among the few candidates of the Generalized Beta-G family of distributions, that is, the number of independent and identically distributed random variables whose PDF follows ( ) , g p Ω are: For r s µ Ω = ∋ + Ω ∈ ℜ ∀ shape, rate, and location parameters respectively.
B is stands for the Beta function defined above, for r s µ Ω = ∋ + Ω ∈ ℜ ∀ shape, scale and location parameters respectively.

Numerical Analysis
The monthly-harmonized wholesale prices (in naira (#)) of cereals in Kano state, Nigeria from 2007 to 2019 would be subjected to the Generalized Beta-G family of distributions. The cereals include-rice, maize, sorghum, millet, gcorn, cowpea, groundnut, beans, wheat and cassava. The time series dataset was obtained from the Ministry of Agriculture and Natural Resources (MANR), Kano state, Nigeria.
The dataset was a monthly uniform time-varying harmonized and regulated price of the edible grains by the ministry (Figure 1). The median value (that is the black line between the whiskers) for all the cereal prices except for the one of groundnut for the edible grains are more closer to their bottom boxes, with their whiskers shorter on the lower part of their boxes, this suggested an extremely positively skewed distribution (rightly skewed) for all. However, the groundnut possessed the same traits, but not to the extreme like others because the whisker (black line) for the groundnut boxplot was not at the basement of the wall of the plot. In other words, groundnut's Author's Computation (2021). whisker is in between the median (50 th percentile or second quartile) and first quartile (25 th percentile), in contrast to others that their whiskers leveled with the first quartile. Overall, it indicated that all the prices of the edible grains are affected by frequent modestly sized deviations that would surely affect estimates if model with Gaussian distribution or unbound distributions.
From Table 1  . From the Anderson-Darling estimate of 9.5843, that is greater than the critical value of 0.7752, we fail to accept that the data came from normal distribution. Additionally, since the Kolomogorov-Smirnov statistic is 0.2701 with its p-value = 0.00011 < 0.05 there is no sufficient evidence that the rice price sample came from normal distribution. . The Anderson-Darling of 7.1804 > 0.7752 shows that the strength of the price of the maize edible grains can be adequately described by the Generalized Beta-G family of distribution. However, since the Kolmogorov-Smirnov statistic is 0.2038 with its p-value = 0.0000 < 0.05, it is obvious that price of maize price did not emanate from Gaussian distribution. Fréchet and Loglogistic distributional random noises jointly produced ideal performance for sorghum with AIC = 2420.474; CAIC = 2421.038; BIC = 2438.773; HQIC = 2427.906, but with different parameters of ( )

Conclusion
In conclusion, Lomax and Linear Failure Rate (LFR) out of the Generalized Beta-G family of distributions are ineffective in modelling the prices of all the cereals studied. This might be due to the fact that LFR is peculiar to survival,