Kernel Construction for Exploring Trends in Probability Distribution Development

In this paper, we provided new methods that improve modeling flexibility of probability distributions. The methods focus on the construction of kernels for possible development of new probability models from (root) variable components or arbitrary functions. These approaches are further grouped into two different categories including construction of kernels from existing probability functions or directly using mathematical deterministic functions. The Direct substitution approach, homogeneous and inhomogeneous interaction methods are captured under kernel development from probabilistic functions. Two distributions namely, Lindley-Sine Distribution (LSD) and Alpha Lindley Distribution (ALD) were developed from the variable component of the Lindley distribution. More so, the combinations of normal and arcsine distribution, and Gumbel and exponential distributions birthed the Double Censored Normal-ArcSine Distribution (DCNAD) and Left Censored Gumbel-Exponential Distribution (LCGED) respectively. Interesting unconventional trends including decreasing sinusoidal, bathtub, triangular and circular trends realized from these developments validates the relevance of the approaches in probability forecasting. Finally, the asymptotic stability of the parameters of the derived distributions was established through simulation study.