This paper develops the estimation method of mean and covariance functions of functional data with additional covariate information. With the strength of both local linear smoothing modeling and general weighing scheme, we are able to explicitly characterize the mean and covariance functions with incorporating covariate for irregularly spaced and sparsely observed longitudinal data, as typically encountered in engineering technology or biomedical studies, as well as for functional data which are densely measured. Theoretically, we establish the uniform convergence rates of the estimators in the general weighing scheme. Monte Carlo simulation is conducted to investigate the finite-sample performance of the proposed approach. Two applications including the children growth data and white matter tract dataset obtained from Alzheimer's Disease Neuroimaging Initiative study are also provided.

In this paper, we compare the entropy of the original distribution and its corresponding compound distribution. Several results are established based on convex order and relative log-concave order. The necessary and sufficient condition for a compound distribution to be log-concave is also discussed, including compound geometric distribution, compound negative binomial distribution and compound binomial distribution.

In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$, where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$, for $i=1,\ldots, n$, which are coupled by Archimedean copulas having different generators. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$, in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of [5] (Barmalzan, G., Ayat, S.M., Balakrishnan, N., & Roozegar, R. (2020). Stochastic comparisons of series and parallel systems with dependent heterogeneous extended exponential components under Archimedean copula. Journal of Computational and Applied Mathematics 380: Article No. 112965).

Omega ratio, a risk-return performance measure, is defined as the ratio of the expected upside deviation of return to the expected downside deviation of return from a predetermined threshold described by an investor. Motivated by finding a solution protected against sampling errors, in this paper, we focus on the worst-case Omega ratio under distributional uncertainty and its application to robust portfolio selection. The main idea is to deal with optimization problems with all uncertain parameters within an uncertainty set. The uncertainty set of the distribution of returns given characteristic information, including the first two orders of moments and the Wasserstein distance, can handle data problems with uncertainty while making the calculation feasible.

We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.

In this paper, a new point process is introduced. It combines the nonhomogeneous Poisson process with the generalized Polya process (GPP) studied in recent literature. In reliability interpretation, each event (failure) from this process is minimally repaired with a given probability and GPP-repaired with the complementary probability. Characterization of the new process via the corresponding bivariate point process is presented. The mean numbers of events for marginal processes are obtained via the corresponding rates, which are used for considering an optimal replacement problem as an application.

Information generating functions (IGFs) have been of great interest to researchers due to their ability to generate various information measures. The IGF of an absolutely continuous random variable (see Golomb, S. (1966). The information generating function of a probability distribution. IEEE Transactions in Information Theory, 12(1), 75–77) depends on its density function. But, there are several models with intractable cumulative distribution functions, but do have explicit quantile functions. For this reason, in this work, we propose quantile version of the IGF, and then explore some of its properties. Effect of increasing transformations on it is then studied. Bounds are also obtained. The proposed generating function is studied especially for escort and generalized escort distributions. Some connections between the quantile-based IGF (Q-IGF) order and well-known stochastic orders are established. Finally, the proposed Q-IGF is extended for residual and past lifetimes as well. Several examples are presented through out to illustrate the theoretical results established here. An inferential application of the proposed methodology is also discussed

We provide general expressions for the joint distributions of the k most significant b-ary digits and of the k leading continued fraction (CF) coefficients of outcomes of arbitrary continuous random variables. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the jth significant digit, which is the counterpart of the general convergence law of the distribution of the jth CF coefficient (Gauss-Kuz’min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among several other results, including the analogue of Benford’s law for CFs. The particularisation for Pareto variables—which include Benford variables as a special case—is especially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading CF coefficients of real data. Our results may find practical application in all areas where Benford’s law has been previously used.

We consider the problem of optimally maintaining an offshore wind farm in which major components progressively degrade over time due to normal usage and exposure to a randomly varying environment. The turbines exhibit both economic and stochastic dependence due to shared maintenance setup costs and their common environment. Our aim is to identify optimal replacement policies that minimize the expected total discounted setup, replacement, and lost power production costs over an infinite horizon. The problem is formulated using a Markov decision process (MDP) model from which we establish monotonicity of the cost function jointly in the degradation level and environment state and characterize the structure of the optimal replacement policy. For the special case of a two-turbine farm, we prove that the replacement threshold of one turbine depends not only on its own state of degradation but also on the state of degradation of the other turbine in the farm. This result yields a complete characterization of the replacement policy of both turbines by a monotone curve. The policies characterized herein can be used to optimally prescribe timely replacements of major components and suggest when it is most beneficial to share costly maintenance resources.

Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function F, say that the $n^{\rm \scriptsize}$th observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize}$th observation sets a record. There are many interesting questions to address concerning pn and multivariate records more generally, but this short paper focuses on how pn varies with F, particularly if, under F, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping pn on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any continuous F governing independent coordinates.

In this study, we consider a class of multiple-drawing opposite-reinforcing urns with time-dependent replacement rules. The class has the symmetric property of a Friedman-type urn. We divide the class into a small-increment regime and a large-increment regime. For small-increment schemes, we prove almost-sure convergence and a central limit theorem for the proportion of white balls by stochastic approximation. For large-increment schemes, by assuming the affinity condition, we show almost-sure convergence of the proportion of white balls by martingale theory and present a way to identify the limit distribution of the proportion of white balls.

This paper studies a bi-dimensional compound risk model with quasi-asymptotically independent and consistently varying-tailed random numbers of claims and establishes an asymptotic formula for the finite-time sum-ruin probability. Additionally, some results related to tail probabilities of random sums are presented, which are of significant interest in their own right. Some numerical studies are carried out to check the accuracy of the asymptotic formula.

We consider a two-stage service system with two types of servers, namely subordinates who perform the first-stage service and supervisors who have their own responsibilities in addition to collaborating with the subordinates on the second-stage service. Rewards are earned when first- or second-stage service is completed and when supervisors finish one of their own responsibilities. Costs are incurred when impatient customers abandon without completing the second-stage service. Our problem is to determine how the supervisors should distribute their time between their joint work with the subordinates and their own responsibilities. Under the assumptions that service times at both stages are exponentially distributed and that the customers waiting for second-stage service abandon after an exponential amount of time, we prove that one of two policies will maximize the long-run average profit. Namely, it is optimal for supervisors to start collaborating with subordinates either when subordinates can no longer serve new customers or as soon as there is a customer ready for second-stage service. Furthermore, we show that the optimality condition is a simple threshold on the system parameters. We conclude by proving that pooling supervisors (and their associated subordinates) improves system performance, but with limited returns as more supervisors are pooled.

Let $(X_{1},\ldots,X_{n})$ be a random vector distributed according to a time-transformed exponential model. This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for $1\leq i\leq n$, $X_{i:n}$ denote the corresponding ith-order statistic. We consider the problem of comparing the strength of dependence between any pair of Xi’s with that of the corresponding order statistics. It is in particular proved that for $m=2,\ldots,n$, the dependence of $X_{2:m}$ on $X_{1:m}$ is more than that of X2 on X1 according to more stochastic increasingness (positive monotone regression) order, which in turn implies that $(X_{1:m},X_{2:m})$ is more concordant than $(X_{1},X_{2})$. It will be interesting to examine whether these results can be extended to other exchangeable models.

The purpose of this paper is twofold. The first part is to introduce relative-$\chi_{\alpha}^{2}$, Jensen-$\chi_{\alpha}^{2}$ and (p, w)-Jensen-$\chi_{\alpha}^2$ divergence measures and then examine their properties. In addition, we also explore possible connections between these divergence measures and Jensen–Shannon entropy measure. In the second part, we introduce $(p,\eta)$-mixture model and then show it to be an optimal solution to three different optimization problems based on $\chi_{\alpha}^{2}$ divergence measure. We further study the relative-$\chi_{\alpha}^{2}$ divergence measure for escort and arithmetic mixture densities. We also provide some results associated with relative-$\chi_{\alpha}^{2}$ divergence measure of mixed reliability systems. Finally, to demonstrate the usefulness of the Jensen-$\chi_{\alpha}^{2}$ divergence measure, we apply it to a real example in image processing and present some numerical results. Our findings in this regard show that the Jensen-$\chi_{\alpha}^{2}$ is an effective criteria for quantifying the similarity between two images.

In this paper, we study the estimation of a scale parameter from a sample of lifetimes of coherent systems with a fixed structure. We assume that the components are independent and identically distributed having a common distribution which belongs to a scale parameter family. Some results are obtained as well for dependent (exchangeable) components. To this end, we will use the representations for the distribution function of a coherent system based on signatures. We prove that the efficiency of the estimators depends on the structure of the system and on the scale parameter family. In the dependence case, it also depends on the baseline copula function.

This paper studies an M/M/1 retrial queue with negative customers, passive breakdown, and delayed repairs. Assume that the breakdown behavior of the server during idle periods is different from that during busy periods. Passive breakdowns may occur when the server is idle, due to the lack of monitoring of the server during idle periods. When the passive breakdown occurs, the server does not get repaired immediately and enters a delayed repair phase. Negative customers arrive during the busy period, which will cause the server to break down and remove the serving customers. Under steady-state conditions, we obtain explicit expressions of the probability generating functions for the steady-state distribution, together with some important performance measures for the system. In addition, we present some numerical examples to illustrate the effects of some system parameters on important performance measures and the cost function. Finally, based on the reward-cost structure, we discuss Nash equilibrium and socially optimal strategy and numerically analyze the influence of system parameters on optimal strategies and optimal social benefits.

Commodity spot prices tend to revert to some long-term mean level and most commodity derivatives are based on futures prices, not on spot prices. So, we consider spread options on futures instead of spot or spot index, where the log spot price follows a mean-reverting process. The volatility of the mean-reverting process is driven by two different (fast and slow) scale factors. We use asymptotic analysis to obtain a closed-form approximation of the futures prices and a closed-form formula for the approximate prices of spread options on the futures. The overall improvement of our analytic formula over the classical Kirk–Bjerksund–Sternsland (KBS) formula is discussed via numerical experiments.

The principle of maximum entropy is a well-known approach to produce a model for data-generating distributions. In this approach, if partial knowledge about the distribution is available in terms of a set of information constraints, then the model that maximizes entropy under these constraints is used for the inference. In this paper, we propose a new three-parameter lifetime distribution using the maximum entropy principle under the constraints on the mean and a general index. We then present some statistical properties of the new distribution, including hazard rate function, quantile function, moments, characterization, and stochastic ordering. We use the maximum likelihood estimation technique to estimate the model parameters. A Monte Carlo study is carried out to evaluate the performance of the estimation method. In order to illustrate the usefulness of the proposed model, we fit the model to three real data sets and compare its relative performance with respect to the beta generalized Weibull family.