Independent Random Variables Research Articles

On the Probabilities of Large Deviations of Combinatorial Sums of Independent Random Variables That Satisfy the Linnik Condition

On the Probabilities of Large Deviations of Combinatorial Sums of Independent Random Variables That Satisfy the Linnik Condition

МЕТОДИКА КРИТЕРІЇВ СУМ У ЗАДАЧАХ ТЕСТУВАННЯ НЕЗАЛЕЖНОСТІ ПОСЛІДОВНОСТЕЙ ВИПАДКОВИХ ЧИСЕЛ

Random and pseudo-random number generators (RNGs) were initially used to solve numerical integration problems (the Monte Carlo method). Currently, the RNGs are used in cryptography and simulation modeling. The latter one typically uses RNGs based on computer algorithms and programs. This article presents a method aimed at testing the independence of random numbers sequences (RNSs). The method is based on the sums properties of independent random variables. Algorithms based on this method operate fast. Here not only the instant statistics including correlation coefficients are analyzed, but also the properties of empirical functions of RNSs distributed sums. In this article, the analysis is limited only to the case of uniformly distributed RNSs. The calculations performed prove the high selective efficiency of the proposed criteria, which allows to reliably distinguish between dependent and independent RNSs. Due to the high operation speed, the proposed algorithms and criteria can be used for testing very long RNSs (especially in Big Data tasks).

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.

A Flexible Dispersed Count Model Based on Bernoulli Poisson–Lindley Convolution and Its Regression Model

Count data are encountered in real-life dealings. More understanding of such data and the extraction of important information about the data require some statistical analysis or modeling. One innovative technique to increase the modeling flexibility of well-known distributions is to use the convolution of random variables. This study examines the distribution that results from adding two independent random variables, one with the Bernoulli distribution and the other with the Poisson–Lindley distribution. The considered distribution is named as the two-parameter Bernoulli–Poisson–Lindley distribution. Many of its statistical properties are investigated, such as moments, survival and hazard rate functions, mode, dispersion behavior, mean deviation about the mean, and parameter inference based on the maximum likelihood method. To evaluate the effectiveness of the bias and mean square error of the produced estimates, a simulation exercise is carried out. Then, applications to two practical data sets are given. Finally, we construct a flexible count data regression model based on the proposed distribution with two practical examples.

Asynchronous secure control for singular nonhomogeneous Markov jump cyber‐physical systems against dual cyber‐attacks

AbstractIn this article, based on the dynamic event‐triggered transmission protocol, the issue of asynchronous secure control for singular nonhomogeneous Markov jump cyber‐physical systems (CPSs) is studied. The matrix that is governed by the time‐varying transition rates lies in a convex polyhedron. To further degrade the transmission percentage over the network, the dynamic event‐triggered transmission protocol is developed. Besides, owing to the vulnerability of the network, the replay attack and deception attack are both considered, the occurrences of which are modelled by mutual independent random variables satisfying Bernoulli distribution. By constructing a parameter dependent and an auxiliary dynamic variable dependent Lyapunov functional, sufficient conditions of stochastic admissible for singular nonhomogeneous Markov jump CPSs are derived by parameter dependent matrix inequalities, which can be converted into the finite linear matrix inequalities. A numerical example is provided to illustrate the feasibility and effectiveness of the theoretical results.

Strong convergence of peaks over a threshold

Abstract Extreme value theory plays an important role in providing approximation results for the extremes of a sequence of independent random variables when their distribution is unknown. An important one is given by the generalised Pareto distribution $H_\gamma(x)$ as an approximation of the distribution $F_t(s(t)x)$ of the excesses over a threshold t, where s(t) is a suitable norming function. We study the rate of convergence of $F_t(s(t)\cdot)$ to $H_\gamma$ in variational and Hellinger distances and translate it into that regarding the Kullback–Leibler divergence between the respective densities.

A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications

In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with uniform distribution in the interval (0, 1). We will call this new extension the slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients, and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, three applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution.

The limit joint distributions of statistics of three tests of the NIST package

Abstract For sequences of independent random variables having a Bernoulli distribution the joint distribution and the limit joint distribution of statistics of three tests of the NIST statistical package («Monobit Test», «Frequency Test within a Block», and «Cumulative Sums Test») are obtained. In the case when two blocks are used in «Frequency Test within a Block», pairwise covariance of these statistics is given.

Partially Functional Linear Models with Linear Process Errors

In this paper, we focus on the partial functional linear model with linear process errors deduced by not necessarily independent random variables. Based on Mercer’s theorem and Karhunen–Loève expansion, we give the estimators of the slope parameter and coefficient function in the model, establish the asymptotic normality of the estimator for the parameter and discuss the weak convergence with rates of the proposed estimators. Meanwhile, the penalized estimator of the parameter is defined by the SCAD penalty and its oracle property is investigated. Finite sample behavior of the proposed estimators is also analysed via simulations.

A Probable Approach to Reliability Assessment of Reinforced Plates

A scheme for solving the problem of determining the probability of failure and probabilistic (statistical) characteristics of the failure loading magnitude of a composite material plate is considered. The plate structure is a flat homogeneous matrix with stochastically distributed rigid rod inclusions. The geometric parameters of inclusions are considered independent random variables with given probability distribution laws. The expressions for the failure loading distribution function, the probability of failure, the mean value, and the dispersion of the failure loading were received and presented. Their dependence on the type of stress state, the inclusion number, and matrix Poisson’s ratio were studied graphically.

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.

Complete convergence and complete moment convergence for maximal weighted sums of arrays of rowwise extended negatively dependent random variables with statistical applications

In this paper, complete convergence and complete moment convergence for maximal weighted sums of arrays of rowwise extended negatively dependent random variables are investigated, and some sufficient conditions for convergence are provided. Three ways of optimising the conditions for convergence, namely reducing the moment condition, narrowing the boundary function, and raising the weight function respectively, are considered under light weights. Besides, rather than assuming stochastic domination, we turn to a weaker assumption of uniformly bounded expectations to describe the moment conditions. The results obtained in the paper extend the corresponding ones for independent random variables and some dependent random variables. In addition, the results are applied to establish strong consistency for estimators in some statistical models.

PENENTUAN TARIF PREMI PADA ASURANSI KENDARAAN DENGAN BESAR KLAIM BERDISTRIBUSI EKSPONENSIAL DAN GAMMA

Calculation of vehicle insurance premium rates can be done using the aggregate claims model. The aggregate claims model consists of a combination of two independent random variables, namely the number of claims that occur and the amount of claims for each event. The research method is in the form of literature studies and case studies using secondary data in the form of data on the number of claims and the amount of claims collected from January 2013 to December 2019. Based on the data collected, there were 802 claims with the smallest claim being IDR 50,000 and the largest being IDR 211,715,000. Testing the hypothesis shows that the data on the number of claims has a Poisson distribution, and the amount of claims follows two types of distribution, namely the exponential and gamma distribution so that the aggregate claim distribution is a combination of Poisson-Exponential and Poisson-Gamma. Parameter estimation for each distribution is carried out by the moment method with available secondary data. This study concludes that the use of the pure premium principle provides the same premium rate for both distributions of aggregate claims, amounting to IDR 82,856.39 per month per person. While the use of the expected value principle provides a premium rate for the Poisson-Gamma aggregate claims distribution 8.76 times greater than the Poisson- Exponential aggregate claims distribution, namely IDR 1,920,019.55 per month and IDR 219,155.20 per month, respectively.

Multiple description encoding-decoding-based resilient filtering for complex networks under the round-Robin protocol

The problem of distributed filtering for a class of discrete-time nonlinear complex networks subject to the network bandwidth limitation is investigated. In order to reduce the network transmission burden, the round-robin protocol is employed. Under which, a quantization-based encoding-decoding mechanism is introduced to protect the security of measurements. Taking into account the quality of transmission data packets, the multiple description coding scheme is proposed. In order to describe the behavior of the packet dropouts induced by DoS attackers, two independent Bernoulli random variables are presented. Based on the round-robin protocol and multiple description encoding-decoding scheme, a distributed recursive filtering is derived to estimate the states of complex networks. Furthermore, the sufficient conditions to obtain the upper bound of estimation error covariance are given. Finally, an example is presented to certify the effectiveness of the proposed distributed recursive filtering.

Distribution of squared sum of products of independent Nakagami-m random variables

The need for the distribution of combination of random variables (RVs) arises in many areas of sciences and engineering. In this paper, closed-form approximations for the distribution of squared sum of products of independent Nakagami-m RVs are derived. Three different approaches (central limit theorem, Edgeworth expansion, and a one-term gamma approximation) are considered. The Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises statistics are used as a quantitative metrics to compare the derived distribution forms with the empirical distribution obtained from a simulation study. Furthermore, an application for the derived distributions in wireless communication field is presented. As a result, it is shown that the most accurate and simplest closed-form expression is the one obtained by a one-term gamma approximation.

On a characterization theorem in the space [formula omitted]

By Heyde's theorem, the class of Gaussian distributions on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We prove an analogue of this theorem for two independent random vectors taking values in the space Rn. The obtained class of distributions consists of convolutions of Gaussian distributions and a distribution supported in a subspace, which is determined by coefficients of the linear forms.

A New Family of Modified Slash Distributions with Applications

This article presents a new family of symmetric heavy-tailed distributions. This model is based on the ratio of two independent random variables; one with a normal distribution in the numerator and another with a Birnbaum–Saunders distribution in the denominator. The result is a new slash-like distribution capable of modeling high levels of kurtosis, so it can be considered as a viable alternative to other heavy-tailed distributions in the literature. Fundamental properties such as density and raw moments are derived. Parameter estimation is performed using the moment and maximum likelihood methods. A simulation study to evaluate the behavior of the estimators is carried out. Finally, the utility of the new distribution is illustrated by fitting two real datasets.

Law of large numbers for weak Fubini-independent random variables under capacities without sub-additivity

This article attempts to establish the law of large numbers for random variables in Choquet expectation spaces where the Choquet integral is not sub-additive. As a basis, the Fubini-independence of random variables is generalized to a weaker version. Then the strong and weak law of large numbers are proved for weak-Fubini independent random variables under capacities which are not sub-additive.

A decomposition algorithm for multi‐item production planning with independent random demand

AbstractProduction planning in a multiproduct setting where the demands for different items are independent random variables that are also featured with a dynamic behavior over the planning horizon is a challenging task. With a particular focus on maintenance facilities, this study proposes a multistage stochastic programming (MSP) model for operations planning under independent random demand of faulty components in the modular‐structured devices (e.g., gas turbines) received for repair and overhaul services. A Lagrangian relaxation‐based decomposition heuristic is also developed to efficiently solve the problem for real‐size instances. This heuristic relies on decomposing the MSP model into submodels corresponding to component STs and coordinating them via a subgradient algorithm to obtain a high‐quality feasible solution. Our numerical experiments conducted on a range of problem instances endorse the significant value of incorporating demand uncertainty and the effectiveness of the proposed solution methodology in overcoming computational complexity.

RIN Reduction in Gain-Switched InAs-InP(113)B Quantum Dot Laser Based on Multi-Population Rate Equations

For the first time, relative intensity noise (RIN) of an InAs-InP(113)B quantum dot (Q-Dot) laser is examined theoretically under the gain-switching condition with the application of a Gaussian pulse beam (GPB) to excited state (Exs) of the laser. The multi-population rate equations considering nonlinear gain are solved by the Runge–Kutta method. Noises are added to rate equations as Langevin noise sources and a different method is applied here for the addition of these noise sources to equations. In this method, new rate equations are defined to eliminate the cross-correlations between the noise sources to make them independent and simulate with independent white Gaussian random variables. Obtained results showed that RIN decreases with the increasing ac peak current and the increasing inhomogeneous broadening. It was also observed that further RIN reduction of about 30 dB/Hz and gain-switched short pulses with a high power due to Exs emission being obtained with the application of GPB to Exs of the laser. Moreover, it was demonstrated that while carrier noise generated from Exs and ground state (Grs) affect the RIN spectrum, there is no effect of wetting layer (Wly) carrier noise.