Random Variables Xi Research Articles

Strong laws for weighted sums of some dependent random variables and applications

Let {Xn, n ? 1} be a sequence of random variables satisfying a generalized Rosenthal type inequality and stochastically dominated by a random variable X. Let {ani, 1 ? i ? n, n ? 1} be an array of constants. We study the Marcinkiewicz-Zygmund type strong laws for weighted sums ?n i=1 aniXi under the condition that the exponential moment of the random variable X exists. These results are the interesting supplements for some known results. As statistical applications, we provide the strong consistency of LS estimators in simple linear EV regression models with widely orthant dependent random errors.

A Comparison Between the MEWMA and Mahalanobis Distance Control Chart

Statistical Process Control (SPC) is approaching that uses statistical techniques to monitor the process. The methods of quality control are widely used in controlling any strategy—the widely used control charts. A traditional variable control chart includes three lines: the Centre Line (CL), the Upper Control Limit (UCL), and the Lower Control Limit (LCL), all of which are represented by numeric values—represented by numeric values. A control chart illustrates the centerline of the average value of the quality feature under investigation.
 Depending on numeric observation values, a process is either "in control" or "out of control."
 Considering the current, there are no questions regarding the observations and their values during the production process. However, when these observations include human judgments, assessments, and choices, a continuous random variable (xi) of a manufacturing process should be made up of the variable.
 The MEWMA and Mahalanobis distance control chart techniques are used when more than one variable is involved.
 This paper aims to compare the MEWMA and Mahalanobis distance chart for three compounds for water drinking production in the ALA -Factory in the Sulaimani-Iraq region.
 So, linguistic terms can be used instead of the exact value of the continuous random variable. Today, quality control has become one of the most essential techniques for studying all variables to control production or consumption decision factors. Quality control's primary aim is to guarantee that the goods, services, or processes delivered satisfy particular standards and are reliable and satisfactory. The factor's direct approach is based on the product's quality.

Normal approximation for sum of random number of summands

Normal aproximationof sum Zt =ΣNti=1Xi of i.i.d. random variables (r.v.) Xi , i = 1, 2, . . . with mean EXi = μ and variance DXi = σ2 > 0 is analyzed taking into consideration large deviations. Here Nt is non-negative integer-valued random variable, which depends on t , but not depends at Xi , i = 1, 2, . . ..

The finite-time ruin probability of the nonhomogeneous Poisson risk model with conditionally independent subexponential claims

This paper obtains an asymptotic formula for the finite-time ruin probability of the compound nonhomogeneous Poisson risk model with a constant interest force, in which the claims are conditionally independent random variables with a common subexponential distribution. The paper also obtains some asymptotic relations of randomly weighted sums in which the weights θi are positive random variables which are bounded above and the primary random variables Xi , are conditionally independent and follow subexponential distributions.

Discrete uniform and binomial distributions with infinite support

We study properties of two probability distributions defined on the infinite set {0,1,2, ldots } and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses 1/textcircled {1} to all points in the set {0,1,ldots ,textcircled {1}-1}, where textcircled {1} denotes the grossone. For this distribution, we study the problem of decomposing a random variable xi with this distribution as a sum xi {mathop {=}limits ^mathrm{d}} xi _1 + cdots + xi _m, where xi _1 , ldots , xi _m are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin(textcircled {1},p) with p=c/textcircled {1}^{alpha } with 1/2<alpha le 1. The accuracy of this approximation is assessed using a numerical study.

Nonparametric detection for univariate and functional data

The problem under study is detecting the presence, and identifying the exact position, of a block Cn of “anomalous” observations in a finite sequence of independent random variables X1,…,Xn. By “anomalous” we mean that the distribution G of the observations in the block Cn is possibly different from the “under control” distribution F of the remaining Xi’s. We first propose a nonparametric approach, based on classical goodness of fit (Kolmogorov–Smirnov (KS) and Cramer–von Mises (CvM)) statistics, for the case of real random variables Xi. An application in stochastic geometry is outlined. Lastly, we focus on the case where the Xi are functional data, that is, trajectories of a stochastic process X=X(t), t∈[0,1]. The strategy we follow in this case is to take the functional data to the real line with an appropriate transformation and then using the nonparametric detection/identification methodologies mentioned above. The real valued transformation we propose is the Radon–Nikodym derivative of the “under control” distribution (that assumed for most observations) with respect to another suitably chosen reference distribution.

On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics

Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 &lt; α &lt; 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| &gt; t) = o(t−p(α)) when r &lt; p(α) or E|X1|r &lt; ∞ when r ≥ p(α). As an application we show that for every α &lt; 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p &lt; ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α &gt; 1∕2 − 1∕p with an appropriate normalization.

Global reliability sensitivity estimation based on failure samples

Global reliability sensitivity analysis (RSA) can help to assess the effects of input random variables X on the probability of failure Pr(F) of an engineering system. Conventionally, this requires repeated evaluations of the conditional failure probability Pr(F|Xi = xi) for multiple values of the input random variable Xi and for all Xi of interest. Such a solution is straightforward but computationally expensive. In this paper, we propose a new method to perform global RSA, which requires as an input only samples of X that fall in the failure domain. Such samples are a by-product of many sampling-based reliability analysis methods. The proposed method constructs the Pr(F|Xi = xi) by application of Bayes’ rule, based on the probability density function (PDF) of X conditioned on system failure F. This conditional PDF is approximated with a kernel density estimation from the failure samples. In this way, the reliability sensitivities of all the input random variables can be computed following a sampling-based reliability analysis with no additional computation cost. The approach is investigated on numerical examples in conjunction with crude Monte Carlo simulation, importance sampling and subset simulation. The results demonstrate the computational advantages over existing single-loop sampling methods for global RSA.

SA105A GENOME-WIDE ASSOCIATION STUDY OF TREATMENT-RESISTANT DEPRESSION AND META-ANALYSIS OF THREE INDEPENDENT SAMPLES

The problem under study is detecting the presence, and identifying the exact position, of a block Cn of “anomalous” observations in a finite sequence of independent random variables X1,…,Xn. By “anomalous” we mean that the distribution G of the observations in the block Cn is possibly different from the “under control” distribution F of the remaining Xi’s. We first propose a nonparametric approach, based on classical goodness of fit (Kolmogorov–Smirnov (KS) and Cramer–von Mises (CvM)) statistics, for the case of real random variables Xi. An application in stochastic geometry is outlined. Lastly, we focus on the case where the Xi are functional data, that is, trajectories of a stochastic process X=X(t), t∈[0,1]. The strategy we follow in this case is to take the functional data to the real line with an appropriate transformation and then using the nonparametric detection/identification methodologies mentioned above. The real valued transformation we propose is the Radon–Nikodym derivative of the “under control” distribution (that assumed for most observations) with respect to another suitably chosen reference distribution.

Slow convergence in generalized central limit theorems

We study the central limit theorem in the non-normal domain of attraction to symmetric α-stable laws for 0<α≤2. We show that for i.i.d. random variables Xi, the convergence rate in L∞ of both the densities and distributions of ∑inXi/(n1/αL(n)) is at best logarithmic if L is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using convergence of ∑i=1nXi/nlog⁡n to Gaussian distributions. Our result implies that such asymptotic laws are accurate only for exponentially large n.

Impact of the Mathematical Model Description on the Assessment of the Reliability of Structural Elements

The study presents a probabilistic approach to the problems of static analysis of structural elements. A steel and reinforced concrete elements were analysed. Structural design parameters were defined as deterministic values and random variables. The latter were not correlated. The criterion of structural failure is expressed by limit functions related to the ultimate limit state. In the performed analyses explicit form of the random variables function were used. The Hasofer-Lind index was used as a reliability measure. In the description of random variables were used the different types of probability distribution appropriate to the nature of the variable. Sensitivity of reliability index to the random variables was defined. If the reliability index sensitivity due to the random variable Xi is low when compared with other variables, it can be stated that the impact of this variable on failure probability is small. Therefore, in successive computations it can be treated as a deterministic parameter. Sensitivity analysis leads to simplify the description of the mathematical model, determine the new limit functions and values of the Hasofer-Lind reliability index. Besides the effect of the assumed level of the variation coefficient of selected random variables on value of the reliability index was determined. The primary research method is the FORM method. In order to verify the correctness of the calculation SORM, Monte Carlo and Importance Sampling methods were used. In the examples of reliability analysis the NUMPRESS program was used. In the considered issues the time was not taken into account explicitly.

Nonparametric Estimation of Replacement Rates

Let Xi, i ≥ 1, describe the lifetimes of items with finite mean μ = E (Xi) which are successively placed in service. In order to estimate the replacement rate 1/μ or related quantities, the random variables Xi are usually assumed to be independent and identically distributed. It is shown that a nonparametric estimation of the replacement rate and other reciprocal functions of renewal theory is possible while using a delta method with weakened requirements upon the global growth of f which also allows dependent observations and respects the unboundedness of the analyzed reciprocal functions. Moreover, results on the moments and, furthermore, on corresponding simulations are included.

On large deviation rates for sums associated with Galton‒Watson processes

Abstract Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 log Z1&lt;∞ and X1 is in the domain of attraction of an α-stable law with 0&lt;α&lt;2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞.

Two-sided moment estimates for a class of nonnegative chaoses

We derive two-sided bounds for moments of random multi-linear forms (random chaoses) with nonnegative coefficients generated by independent nonnegative random variables Xi which satisfy the following condition on the growth of moments: ‖Xi‖2p≤A‖Xi‖p for any i and p≥1. The estimates are deterministic and exact up to multiplicative constants which depend only on the order of chaos and the constant A in the moment assumption.

A game-theoretic model of TV show “The Voice”

This paper proposes a game-theoretic model of the two-player best-choice problem with incomplete information. The players (experts) choose between objects by observing their quality in the form of two components forming a sequence of random variables (xi, yi), i = 1,..., n. By assumption, the first quality component xi is known to the players and the second one yi is hidden. A player accepts or declines an object based on the first quality component only. A player with the maximal sum of the components becomes the winner in the game. The optimal strategies are derived in the cases of independent and correlated quality components.

Some mixing properties of conditionally independent processes

ABSTRACTIn this paper, we consider conditionally independent processes with respect to some dynamic factor. More precisely, we assume that for all n, random variables (Xi)i ⩽ n are conditionally independent given . We study the mixing properties of the process and we provide an asymptotic result. Our work is motivated by some examples related to risk theory.

Uniform approximation of the tail probability of weighted sums of subexponential random variables

ABSTRACTFor subexponential random variables Xi, 1 ⩽ i ⩽ n, the uniform asymptotic resultis established for 0 < an(x) ⩽ ci ⩽ bn(x), 1 ⩽ i ⩽ n, where an(x) decreases to 0 and b(x) goes to infinity as x tends to infinity. It extends the current result when the subexponential random variables Xi’s are independent and gives the first uniform asymptotic result when Xi’s are dependent.

The Comparison of Methods of Assessing the Reliability of the Steel Frame

The study presents a probabilistic approach to the problems of static analysis of a steel building. Structural design parameters were defined as deterministic values and random variables. The latter were not correlated. The criterion of structural failure is expressed by limit functions related to the ultimate and serviceability limit state. The description of limit functions by the Mathematica program was generated. The Hasofer-Lind index was used as a reliability measure. In the description of random variables were used the normal distribution and, for comparison, different types of probability distribution appropriate to the nature of the variable. Sensitivity of reliability index to the random variables was defined. If the reliability index sensitivity due to the random variable Xi is low when compared with other variables, it can be stated that the impact of this variable on failure probability is small. Therefore, in successive computations it can be treated as a deterministic parameter. Sensitivity analysis leads to simplify the description of the mathematical model, determine the new limit functions and values of the Hasofer-Lind reliability index. The primary research method is the FORM method. In order to verify the correctness of the calculation SORM, Monte Carlo and Importance Sampling methods were used. In the examples of reliability analysis the STAND program was used.

An Estimate of the Probability Density Function of the Sum of a Random Number N of Independent Random Variables

A new estimate of the probability density function (PDF) of the sum of a random number of independent and identically distributed (IID) random variables Xi (i=1, 2, …, N) is shown. The sum PDF is represented as a sum of normal PDFs weighted according to the N PDF. The analytical model is verified by numerical simulations. The comparison is made by the Chi-Square Goodness-of-Fit test.

The Job-search Problem with Incomplete Information

This paper considers a best-choice game with incomplete information associated with the job search problem. Players (experts) observe a sequence of independent identically distributed random variables (xi, yi), i = 1. . ., n, which represent the quality of incoming objects. The first component is announced to the players and the other one is hidden. The players choose an object based on known information about it. The winner is the player having a higher sum of the quality components (total quality) than the opponent. And finally, the optimal strategies of the players are derived.