On Asymptotics of Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Statistical Modeling of Energy Harvesting in Hybrid PLC-WLC Channels

This work introduces statistical models for the energy harvested from the in-home hybrid power line-wireless channel in the frequency band from 0 to 100 MHz. Based on numerical analyses carried out over the data set obtained from a measurement campaign together with the use of the maximum likelihood value criterion and the adoption of five distinct power masks for power allocation, it is shown that the log-normal distribution yields the best model for the energies harvested from the free-of-noise received signal and from the additive noise in this setting. Additionally, the total harvested energy can be modeled as the sum of these two statistically independent random variables. Thus, it is shown that the energies harvested from this kind of hybrid channel is an easy-to-simulate phenomenon when carrying out research related to energy-efficient and self-sustainable networks.

A New Generalization of the Student’s t Distribution with an Application in Quantile Regression

In this work, we present a new generalization of the student’s t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution divided by its degrees of freedom. Thus, the new symmetric distribution has heavier tails than the student’s t distribution and extensions of the slash distribution. We develop a procedure to use quantile regression where the response variable or the residuals have high kurtosis. We give the density function expressed by an integral, we obtain some important properties and some useful procedures for making inference, such as moment and maximum likelihood estimators. By way of illustration, we carry out two applications using real data, in the first we provide maximum likelihood estimates for the parameters of the generalized student’s t distribution, student’s t, the extended slash distribution, the modified slash distribution, the slash distribution generalized student’s t test, and the double slash distribution, in the second we perform quantile regression to fit a model where the response variable presents a high kurtosis.

Numerical modeling of boundary value problems for differential equations with random coefficients

This paper deals with the numerical modeling of differential equations with coefficients in the form of random fields. Using the Karhunen-Lo´eve expansion, we approximate these coefficients as a sum of independent random variables and real functions. This allows us to use the computational probabilistic analysis. In particular, we apply the technique of probabilistic extensions to construct the probability density functions of the processes under study. As a result, we present a comparison of our approach with Monte Carlo method in terms of the number of operations and demonstrate the results of numerical experiments for boundary value problems for differential equations of the elliptic type.

A New Estimation Study of the Stress-Strength Reliability for the Topp–Leone Distribution Using Advanced Sampling Methods

In this manuscript, we investigate the estimation of the unknown reliability measure R = P [Y < X], in the case where Y and X are two independent random variables with Topp–Leone distributions. As the main contribution, various advanced sampling strategies are studied. The suggested strategies are simple random, ranked set, and median ranked set samplings. Firstly, based on the maximum likelihood, we give an efficient estimator of R when the observations of the two random variables are selected from the same simple random sample. Secondly, such an estimator is addressed when the observations of the two random variables are selected from the ranked set sampling method. Then, based on median ranked set sampling, the maximum likelihood estimator of R is addressed in all the four cases. When the observations from the two random variables are selected from the same set size, two cases are considered, while the other two cases are considered at different set sizes. A simulation research is developed to evaluate the behavior of the obtained estimates based on standard and median ranked set samplings with their simple random sampling equivalents. The ratio of mean square error is used to assess the effectiveness of these estimates.

Stochastic monotonicity of dependent variables given their sum

Given a finite set of independent random variables, assume one can observe their sum, and denote with s its value. Efron in 1965, and Lehmann in 1966, described conditions on the involved variables such that each of them stochastically increases in the value s, i.e., such that the expected value of any non-decreasing function of the variable increases as s increases. In this paper, we investigate conditions such that this stochastic monotonicity property is satisfied when the assumption of independence is removed. Comparisons in the stronger likelihood ratio order are considered as well.

The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.