This study evaluates the robustness of Ordinary Least Squares (OLS) and Double Weighted M-Estimation (DWME) methods for predicting crude oil prices in Nigeria, focusing on predictive accuracy and generalization. Using 192 monthly data points (2006–2021) from the Central Bank of Nigeria (CBN) and Nigerian National Petroleum Corporation (NNPC), the dataset included crude oil prices, production, crude oil production, and exchange rates, with synthetic datasets simulated via multivariate normal distribution for varying dimensions (n = 10 to 1,000). The performance measures such as Mean Squared Error (MSE), Mean Absolute Error (MAE), and R-squared were assessed. Results showed comparable MSE values for training data, with OLS_TRAIN ranging from 172.85 to 694.56 and DWME_TRAIN from 173.03 to 699.27. Testing data revealed DWME's marginal superiority, with slightly lower MSE (e.g., DWME_TEST median 548.68 vs. OLS_TEST median 543.85). MAE trends indicated consistency for both methods, with DWME showing marginally better stability across dimensions. R-squared values highlighted improved generalization for smaller datasets, with DWME_TEST peaking at 0.7043 and OLS_TEST at 0.7544 for the 10x3 dimension. Both methods struggled with generalization as dimensions increased but exhibited stable training performance. In conclusion, DWME demonstrated slightly better robustness, especially in testing scenarios, affirming its suitability for predictive tasks involving economic and energy-related variables. Read more

The paper reviews creative ways to developing continuous probability models playing around integration method and the concept of normalization. It further projects a probability distribution realized by combining two symmetric probability models that differ in shape, to produce a perfect hemisphere or half-sun trend. Normal and arcsine distributions are the root distributions used for this development. At some values of the parameter, the distribution can be right skewed; where other moments and related measures, and estimation are studied as its properties, alongside simulation. Read more

This paper focused on the use of three probability distributions to justify the central limit theorem (CLT). The aim was to use the moment generating function (MGF) to prove (CLT) and also to portray the shape of different sample sizes (15, 30 and 100) of distributions of sample means on a histogram. The population distributions studied were: Normal, Gamma and Exponential distributions. In addition, sampling distribution of the mean table was constructed for better understanding of CLT. The study used simulation to simulate population distribution of Normal, Gamma and Exponential. Five hundred (500) distributions of sample means were drawn from each of the simulated population distributions at three different sample sizes (n): 15, 30 and 100. The shape of the simulated population distribution and sampling distribution of mean were presented on a histogram. The mean and standard deviation of each population distribution together with distribution of sample means at different sample sizes were also presented on the histogram plotted. The findings showed that under normal distribution, the sampling distribution of mean produced a shape like normal distribution irrespective of the sample size. Conversely, the shape of sampling distribution of mean under non-normal distributions gradually converges to normal distribution as sample size tends to infinity, while the variability of each sampling distribution decreases as the sample size increases. Therefore, CLT holds for large sample size (n ≥ 30). Read more

In this study, we considered various transformation problems for a left-truncated normal distribution recently announced by several researchers and then possibly seek to establish a unified approach to such transformation problems for certain type of random variable and their associated probability density functions in the generalized setting. The results presented in this research, actually unify, improve and as well trivialized the results recently announced by these researchers in the literature, particularly for a random variable that follows a left-truncated normal distribution. Furthermore, we employed the concept of approximation theory to establish the existence of the optimal value y_max in the interval denoted by (σ_a,σ_b) ((σ_p,σ_q)) corresponding to the so-called interval of normality estimated by these authors in the literature using the Monte carol simulation method. Read more