Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.

Random Variables and Probability

The chapter “Random Variables and Probability” serves as both a review and a reference on probability. The random variable is the core concept in understanding probability, parameter estimation, and model selection. This chapter reviews the basic idea of a random variable and discusses the two main kinds of random variables: discrete random variables and continuous random variables. It covers the distinction between discrete and continuous random variables and outlines the most common probability mass or density functions used in ecology. Advanced sections cover distributions such as the gamma distribution, Student’s t-distribution, the beta distribution, the beta-binomial distribution, and zero-inflated models.

Gaussian Quadrature GQ Used to Accurately Approximate the Relative Weights of Reserves, Contingent Resources, and Prospective Resources Through A Cumulative Distribution Function

The objective of this work is to numerically estimate the fraction of Reserves assigned to each Reserves category of the PRMS matrix through a cumulative distribution function. We selected 38 wells from a Permian Basin dataset available to Texas A&M University. Previous work has shown that Swanson's Mean, which relates the Reserves categories through a cdf of a normal distribution, is an inaccurate method to determine the relationship of the Reserves categories with asymmetric distributions. Production data are lognormally distributed, regardless of basin type, thus cannot follow the SM concept. The Gaussian Quadrature (GQ) provides a methodology to accurately estimate the fraction of Reserves that lie in 1P, 2P, and 3P categories – known as the weights. Gaussian Quadrature is a numerical integration method that uses discrete random variables and a distribution that matches the original data. For this work, we associate the lognormal cumulative distribution function (CDF) with a set of discrete random variables that replace the production data, and determine the associated probabilities. The production data for both conventional and unconventional fields are lognormally distributed, thus we expect that this methodology can be implemented in any field. To do this, we performed probabilistic decline curve analysis (DCA) using Arps’ Hyperbolic model and Monte Carlo simulation to obtain the 1P, 2P, and 3P volumes, and calculated the relative weights of each Reserves category. We performed probabilistic rate transient analysis (RTA) using a commercial software to obtain the 1P, 2P, and 3P volumes, and calculated the relative weights of each Reserves category. We implemented the 3-, 5-, and 10-point GQ to obtain the weight and percentiles for each well. Once this was completed, we validated the GQ results by calculating the percent-difference between the probabilistic DCA, RTA, and GQ results. We increase the standard deviation to account for the uncertainty of Contingent and Prospective resources and implemented 3-, 5-, and 10-point GQ to obtain the weight and percentiles for each well. This allows us to also approximate the weights of these volumes to track them through the life of a given project. The probabilistic DCA, RTA and Reserves results indicate that the SM is an inaccurate method for estimating the relative weights of each Reserves category. The 1C, 2C, 3C, and 1U, 2U, and 3U Contingent and Prospective Resources, respectively, are distributed in a similar way but with greater variance, incorporated in the standard deviation. The results show that the GQ is able to capture an accurate representation of the Reserves weights through a lognormal CDF. Based on the proposed results, we believe that the GQ is accurate and can be used to approximate the relationship between the PRMS categories. This relationship will aid in booking Reserves to the SEC because it can be recreated for any field. These distributions of Reserves and resources other than Reserves (ROTR) are important for planning and for resource inventorying. The GQ provides a measure of confidence on the prediction of the Reserves weights because of the low percent difference between the probabilistic DCA, RTA, and GQ weights. This methodology can be implemented in both conventional and unconventional fields.

On subcopula estimation for discrete models

PurposeTo discuss subcopula estimation for discrete models.Design/methodology/approachThe convergence of estimators is considered under the weak convergence of distribution functions and its equivalent properties known in prior works.FindingsThe domain of the true subcopula associated with discrete random variables is found to be discrete on the interior of the unit hypercube. The construction of an estimator in which their domains have the same form as that of the true subcopula is provided, in case, the marginal distributions are binomial.Originality/valueTo the best of our knowledge, this is the first time such an estimator is defined and proved to be converged to the true subcopula.

A parameter-less algorithm for tensor co-clustering

The majority of the data produced by human activities and modern cyber-physical systems involve complex relations among their features. Such relations can be often represented by means of tensors, which can be viewed as generalization of matrices and, as such, can be analyzed by using higher-order extensions of existing machine learning methods, such as clustering and co-clustering. Tensor co-clustering, in particular, has been proven useful in many applications, due to its ability of coping with n-modal data and sparsity. However, setting up a co-clustering algorithm properly requires the specification of the desired number of clusters for each mode as input parameters. This choice is already difficult in relatively easy settings, like flat clustering on data matrices, but on tensors it could be even more frustrating. To face this issue, we propose a new tensor co-clustering algorithm that does not require the number of desired co-clusters as input, as it optimizes an objective function based on a measure of association across discrete random variables (called Goodman and Kruskal’s $$\tau$$ τ ) that is not affected by their cardinality. We introduce different optimization schemes and show their theoretical and empirical convergence properties. Additionally, we show the effectiveness of our algorithm on both synthetic and real-world datasets, also in comparison with state-of-the-art co-clustering methods based on tensor factorization and latent block models.

Second-Order Weak Approximations of CKLS and CEV Processes by Discrete Random Variables

In this paper, we construct second-order weak split-step approximations of the CKLS and CEV processes that use generation of a three−valued random variable at each discretization step without switching to another scheme near zero, unlike other known schemes (Alfonsi, 2010; Mackevičius, 2011). To the best of our knowledge, no second-order weak approximations for the CKLS processes were constructed before. The accuracy of constructed approximations is illustrated by several simulation examples with comparison with schemes of Alfonsi in the particular case of the CIR process and our first-order approximations of the CKLS processes (Lileika– Mackevičius, 2020).