Independent Non-negative Random Variables Research Articles

Reliability assessment of degradation processes with random shocks considering recoverable shock damages

For the products that are exposed to random shocks in the field use, the continuous degradation processes of the products are often affected by random shocks. There are two types of shock effects typically: (1) a sudden change of the degradation signal; (2) an increment of the degradation rate. For some kinds of the products, such as the material with recovery properties and smart systems with self-recovery properties, the sudden change of degradation signals caused by shocks can be fully or partially recovered after the occurrence of the shocks. Thus, the recoverable shock damage should be considered in the degradation modeling and reliability prediction of such products. In this paper, a degradation model considering recoverable shock damage is established based on a Wiener process. Three transition zones are divided according to different shock effects. The arrival of the random shocks is expressed by a homogeneous Poisson process, and the shock magnitudes are non-negative independent random variables following a normal distribution. The shock effect is described as the cumulative shock damage function consisting of both the effect on the degradation signal which is expressed by an exponential function with/without a residual effect for a partial or full recovery, and on the degradation rate which is proportional to the shock magnitude when the shock magnitude exceeds the critical threshold. Parameter estimation and reliability prediction are discussed. Numerical examples are presented as demonstrations of the applications of the proposed model.

Phase Transitions and Percolation at Criticality in Enhanced Random Connection Models

We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal {P}_{\lambda }$ in $\mathbb {R}^{2}$ of intensity λ. In the homogeneous RCM, the vertices at x,y are connected with probability g(|x − y|), independent of everything else, where $g:[0,\infty ) \to [0,1]$ and |⋅| is the Euclidean norm. In the inhomogeneous version of the model, points of $\mathcal {P}_{\lambda }$ are endowed with weights that are non-negative independent random variables with distribution $P(W>w)= w^{-\beta }1_{[1,\infty )}(w)$ , β > 0. Vertices located at x,y with weights Wx,Wy are connected with probability $1 - \exp \left (- \frac {\eta W_{x}W_{y}}{|x-y|^{\alpha }} \right )$ , η,α > 0, independent of all else. The graphs are enhanced by considering the edges of the graph as straight line segments starting and ending at points of $\mathcal {P}_{\lambda }$ . A path in the graph is a continuous curve that is a subset of the union of all these line segments. The Poisson stick model consists of line segments of independent random lengths and orientation with the mid point of each segment located at a distinct point of $\mathcal {P}_{\lambda }$ . Intersecting lines form a path in the graph. A graph is said to percolate if there is an infinite connected component or path. We derive conditions for the existence of a phase transition and show that there is no percolation at criticality.

Geometric sums, size biasing and zero biasing

The geometric sum plays a significant role in risk theory and reliability theory [Kalashnikov (1997)] and a prototypical example of the geometric sum is Rényi’s theorem [Rényi (1956)] saying a sequence of suitably parameterised geometric sums converges to the exponential distribution. There is extensive study of the accuracy of exponential distribution approximation to the geometric sum [Sugakova (1995), Kalashnikov (1997), Peköz & Röllin (2011)] but there is little study on its natural counterpart of gamma distribution approximation to negative binomial sums. In this note, we show that a nonnegative random variable follows a gamma distribution if and only if its size biasing equals its zero biasing. We combine this characterisation with Stein’s method to establish simple bounds for gamma distribution approximation to the sum of nonnegative independent random variables, a class of compound Poisson distributions and the negative binomial sum of random variables.

Posted Price Mechanisms and Optimal Threshold Strategies for Random Arrivals

The classic prophet inequality states that, when faced with a finite sequence of nonnegative independent random variables, a gambler who knows the distribution and is allowed to stop the sequence at any time, can obtain, in expectation, at least half as much reward as a prophet who knows the values of each random variable and can choose the largest one. In this work, we consider the situation in which the sequence comes in random order. We look at both a nonadaptive and an adaptive version of the problem. In the former case, the gambler sets a threshold for every random variable a priori, whereas, in the latter case, the thresholds are set when a random variable arrives. For the nonadaptive case, we obtain an algorithm achieving an expected reward within at least a 0.632 fraction of the expected maximum and prove that this constant is optimal. For the adaptive case with independent and identically distributed random variables, we obtain a tight 0.745-approximation, solving a problem posed by Hill and Kertz in 1982. We also apply these prophet inequalities to posted price mechanisms, and we prove the same tight bounds for both a nonadaptive and an adaptive posted price mechanism when buyers arrive in random order.

Poisson approximation in terms of the Gini–Kantorovich distance

It is long known that the distribution of a sum Sn of independent non-negative integer-valued random variables can often be approximated by a Poisson law: Sn≈πλ, where . The problem of evaluating the accuracy of such approximation has attracted a lot of attention in the past six decades. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, it provides insights into Kolmogorov’s problem. Among popular metrics considered in the literature is the Gini–Kantorovich distance dG. The task of establishing an estimate of dG(Sn;πλ) with correct (the best possible) constant at the leading term remained open for a long while. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We show that the accuracy of approximation can be considerably better if the random variables obey an extra condition involving the first two moments. A sharp estimate of the accuracy of shifted (translated) Poisson approximation is established as well.

Usual stochastic and reversed hazard orders of parallel systems with independent heterogeneous components

In this paper, we present some new ordering properties between two parallel systems comprising general independent heterogeneous components. More precisely, let and be independent non-negative random variables with and where is an absolutely continuous distribution function with reversed hazard rate function In this paper, under certain conditions, by using the concept of vector majorization, unordered order, p-majorization and related orders, we discuss stochastic comparisons between parallel systems in the sense of usual stochastic and reversed hazard rate orders. The results developed in this paper generalize some known results in the literature.

Small deviations of sums of independent random variables

A well-known discovery of Feige's is the following [3]: Let X1,…,Xn be nonnegative independent random variables, with E[Xi]≤1∀i, and let X=∑i=1nXi. Then for any n,Pr⁡[X<E[X]+1]≥α>0, for some α≥1/13. This bound was later improved to 1/8 by He, Zhang, and Zhang [5]. By a finer consideration of the first four moments, we further improve the bound to approximately .14. The conjectured true bound is 1/e≃.368.

Recent developments in prophet inequalities

The classic prophet inequality states that, when faced with a finite sequence of non-negative independent random variables, a gambler who knows their distribution and is allowed to stop the sequence at any time, can obtain, in expectation, at least half as much reward as a prophet who knows the values of each random variable and can choose the largest one. Following this classic theorem from the 70s, many results have been obtained for several related optimal stopping problems. Moreover, the recently uncovered connection between prophet inequalities and posted price mechanisms, has given the area a new surge. We survey some new developments and highlight some compelling open problems.

Estimates for Order Statistics in Terms of Quantiles

Let X1, . . .,Xn be independent nonnegative random variables with cumulative distribution functions F1, F2, . . . , Fn satisfying certain (rather mild) conditions. We show that the median of kth smallest order statistic of the vector (X1, . . . , Xn) is equivalent to the quantile of order (k − 1/2)/n with respect to the averaged distribution $$ F=\frac{1}{n}\sum \limits_{i=1}^n{F}_i $$ .

Poisson Approximation for Call Function via Stein–Chen Method

Let V be a sums of independent nonnegative integer-valued random variables and $$h_z$$ be a call function defined by $$h_{z}(v)=(v-z)^+$$ for $$v\ge 0, z \ge 0$$ where $$(v-z)^+=\max \{v-z,0\}$$. In this paper, we give bounds of Poisson approximation for $${E}[h_{z}(V)]$$. These bounds improve the results of Jiao and Karoui (Finance Stoch 13(2):151–180, 2009). The technique used is Stein–Chen method with the zero bias transformation. One example of applications for a call function in finance is the standard collateralized debt obligation (CDO) tranche pricing. The CDO is a security backed by a diversified pool of debt obligation such as bounds, loans and credit default swaps.

Solving Optimization Problems with Diseconomies of Scale via Decoupling

We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as x q , q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems and then show that the integrality gap of the relaxation is A q , where A q is the q -th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ║∑ i =1 n X i ║ q ≤ C q ║∑ i =1 n Y i ║ q , where X i are independent nonnegative random variables, Y i are possibly dependent nonnegative random variables, and each Y i has the same distribution as X i . The inequality was proved by de la Peña in 1990. De la Peña, Ibragimov, and Sharakhmetov showed that C q ≤ 2 for q ∈(1,2) and C q ≤ A q 1/ q for q ≥ 2. We show that the optimal constant is C q = A q 1/ q for any q ≥ 1. We then prove a more general inequality: For every convex function φ, E[φ(∑ i =1 n X i )] ≤ E[φ ( P ∑ i =1 n Y i )], and, for every concave function ψ, E[ψ (∑ i =1 n X i )] ≥ E[ψ(P∑ i =1 n Y i )], where P is a Poisson random variable with parameter 1 independent of the random variables Y i .

Weak laws of large numbers for nonnegative independent random variables without finite means

For a sequence of nonnegative independent random variables with where are slowly varying functions, we present some mild conditions on under which in probability, where is a norming sequence of positive numbers with When the distributions of the random variables are specified more precisely, the norming sequence can be obtained more concretely. As special cases, we obtain the results of Adler (2012, 2017) and Nakata (2016).

Weak and one-sided strong laws for random variables with infinite mean

Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn−1Sn(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.

On computing distributions of products of non-negative independent random variables

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.

Interval Estimation of a P(X&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt; &amp;lt; X&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;) Model for Variables having General Inverse Exponential Form Distributions with Unknown Parameters

In this article the interval estimation of a P(X₁ < X₂) model is discussed when X₁ and X₂are non-negative independent random variables, having general inverse exponential form distributions with different unknown parameters. Different interval estimators are derived, by applying different approaches. A simulation study is performed to compare the estimators obtained. The comparison is carried out on basis of average length, average coverage, and tail errors. The results are illustrated, using inverse Weibull distribution as an example of the general inverse exponential form distribution.

One sided strong laws for random variables with infinite mean

Abstract This paper establishes conditions that secure the almost sure upper and lower bounds for a particular normalized weighted sum of independent nonnegative random variables. These random variables do not possess a finite first moment so these results are not typical. These mild conditions allow us to show that the almost sure upper limit is infinity while the almost sure lower bound is one.

STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X1, …, Xn be a set of non-negative independent random variables with Xi, i=1, …, p, having common distribution function F(λ1x), and Xj, j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let Xi:n(p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s1≤s2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

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.

Approximations to inverse moments of double-indexed weighted sums

In this paper, we investigate some approximations to inverse moments of double-indexed weighted sums of random variables. Let {Zn} be a sequence of nonnegative independent random variables and {wni} be a triangular array of nonnegative non-random weights. With the finite first moment, we establish that E(a+∑i=1nwniZi)−α∼(a+∑i=1nwniEZi)−α holds for all a>0 and α>0. By the finite r-th moment (r>2), the convergence rate of approximation between E(a+∑i=1nwniZi)−α and (a+∑i=1nwniEZi)−α is presented. On the other hand, we obtain some approximations for the estimators of E(∑i=1nwniZia+∑i=1nZi), E(∑i=1nwniZia+∑i=1nZi)2 and Var(∑i=1nwniZia+∑i=1nZi) for all a>0. Finally, some examples and simulations are illustrated.

Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

Exact upper bounds on the generalized moments $\operatorname{\mathsf {E}} f(S_n)$ of sums $S_n$ of independent nonnegative random variables $X_i$ for certain classes $\mathcal{F} $ of nonincreasing functions $f$ are given in terms of (the sums of) the first two moments of the $X_i$’s. These bounds are of the form $\operatorname{\mathsf {E}} f(\eta )$, where the random variable $\eta $ is either binomial or Poisson depending on whether $n$ is fixed or not. The classes $\mathcal{F} $ contain, and are much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $\operatorname{\mathsf {P}} (S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $\operatorname{\mathsf {E}} \exp \{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.