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Note, for instance, that the probabilities given by Castellares et al. ([1]) in page 4 are the same than those shown in expression (9.115) in the book by Johnson et al. ([3]), with the change of parameters Graph HT ht and Graph HT ht Neyman type A (NTA) distribution is frequently used in Biology, Biodosimetry, Environmental Sciences, Epidemiology, etc. A count random variable I X i is said to follow a stopped-sum Poisson, compound Poisson, multiple Poisson or clustered-Poisson distribution, if it can be represented as Graph HT ht where I N i is a Poisson random variable with parameter Graph HT ht and Graph HT ht are independent, identically distributed random variables that are also independent of I N i . [Extracted from the article]

In this paper, the seven equivalent conditions of complete moment convergence and complete integral convergence for m -asymptotic negatively associated ( m -ANA, for short) random variables are established. The results obtained in the paper extend and improve some corresponding ones for negatively associated (NA, for short) random variables and negatively orthant dependent (NOD, for short) random variables. As an application of our main results, we present a result on complete consistency for the weighted estimator in a nonparametric regression model based on m-ANA errors. We perform a numerical simulation to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a flexible family of 2- and 3-parameter distributions whose support is the closed interval [0,1], with explicit density, cumulative density, and quantile functions. These distributions are suited to modeling quantiles, thereby expanding the toolbox of distributions for doubly-bounded random variables. The densities at the boundaries are determined by dispersion and skew parameters, and a third parameter exclusively influences location. The paper also discusses practical issues of parameter estimation and assesses estimation bias, Type I error-rate accuracy, and parameter-estimate collinearity. It also provides three examples of applications to real data using new packages implementing the distributions in R and Stata. [ABSTRACT FROM AUTHOR]

In this paper, the complete convergence for maximum of weighted sums of widely negative orthant dependent (WNOD) random variables are investigated. Some sufficient conditions for the convergence are provided and a relationship between the weight and the boundary function is revealed. Additionally, a Marcinkiewicz-Zygmund type strong law of large number for weighted sums of WNOD random variables is obtained. The results obtained in this paper generalize some corresponding ones for independent and some dependent random variables. As an application, the strong consistency for the weighted estimator in a non-parametric regression model is established. MR(2010) Subject Classification: 60F15; 62G05. [ABSTRACT FROM AUTHOR]

This paper is concerned with the consistency of nonparametric regression model. For the weighted estimator of unknown regression function, the strong consistency, the complete consistency and the convergence rate of the complete consistency are investigated under some mild conditions. These results extend or improve the corresponding ones of Yang et al. (2018) for extended negatively dependent (END, for short) random variables to asymptotically almost negatively associated random variables. Also, the simulation study of the finite samples provided in this paper shows the validity of our results. [ABSTRACT FROM AUTHOR]

Non-negative limited normal or gamma distributed random variables are commonly used to model physical phenomenon such as the concentration of compounds within gaseous clouds. This paper demonstrates that when a collection of random variables with limited normal or gamma distributions represents a stationary process for which the underlying variables have exponentially decreasing correlations, then a central limit theorem applies to the correlated random variables. [ABSTRACT FROM AUTHOR]

The strong limit theory is one of the most important problems in the probability theory. Some results on the convergence of ρ ˜ -mixing random sequences have been presented. In this paper, we study the almost sure convergence for ρ ˜ -mixing random sequences. As a result, we generalize partial results of Wu and Jiang [Wu, Q. Y., and Y. Y. Jiang. 2008. Some strong limit theorems for q ˜ -mixing sequences of random variables. Statistics & Probability Letters 78:1017–23.]. We obtain the mainstream and some corresponding conclusions by using truncation methods and the generalized three-series theorem. A known result is generalized in this paper. [ABSTRACT FROM AUTHOR]

A random effects model for analyzing mixed rank and binomial data with considering the missing values is presented. Occurring of missing data is an important problem in all research fields. The most common approach to dealing with missing data is to delete cases containing missing observations. However, this approach reduces statistical power and mislead us to biased statistical results. This paper aims to prepare guidance for researchers facing missing data problems and to provide techniques for jointly modeling of binomial and rank responses. We compare the cognitive abilities of different primates based on their performance on 17 cognitive assessments obtained on either a rank or binomial scale using Bayesian latent variable with random effects models. Random effects are used to take into account the correlation between responses of the same individual. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of the Jajte weak law of large numbers for exchangeable random variables. We make a simulation to illustrate the asymptotic behavior in the sense of convergence in probability for weighted sums of exchangeable weighted random variables. [ABSTRACT FROM AUTHOR]

The random coefficient autoregressive (RCAR) processes are very useful to model time series in applications. It is commonly observed that the random autoregressive coefficient is assumed to be an independent identically distributed (i.i.d.) random variable sequence. To make the RCAR model more practical, this paper considers a new RCAR(1) model driven by explanatory variable and observations. We use the conditional least squares, the quantile regression and the conditional maximum likelihood methods to estimate the model parameters. The consistency and asymptotic normality of the proposed estimates are established. Simulation studies are conducted for the evaluation of the developed approaches and two applications to real-data examples are provided. The results show that the proposed procedures perform well for the simulations and application. [ABSTRACT FROM AUTHOR]

There has been much work on the distribution of independent or dependent random variables. But we are not aware of any work giving exact results for the distribution of the sum of randomly weighted random variables. In this paper, we derive exact results for the randomly weighted sum of two dependent random variables. The derived expressions are for the cumulative distribution function, conditional expectation, moment generating function, value at risk, expected shortfall and the limiting tail behavior of the randomly weighted sum of two dependent random variables. Two numerical illustrations are given. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the tail behavior of randomly weighted sums and their maxima of dependent subexponential random variables, in which the primary random variables X 1 , ... , X n are real-valued and dependent following two general dependence structures, respectively, and the random weights θ 1 , ... , θ n are another n positive and arbitrarily dependent random variables, but independent of X 1 , ... , X n.. Under some technical conditions, we derive some asymptotic formulas for the tail probability of the randomly weighted sums and their maxima, which coincide with some existing ones in the literature. The merit of our results is that unbounded supports for the random weights are allowed and the distributions of primary random variables can be different. [ABSTRACT FROM AUTHOR]

In this paper, we present some new ordering properties between two parallel systems comprising general independent heterogeneous components. More precisely, let X 1 , ⋯ , X n and Y 1 , ⋯ , Y n be independent non-negative random variables with X i ∼ F (x ; α i , β i) and Y i ∼ F (x ; θ i , λ i) , i = 1 , ⋯ , n , where F (.) is an absolutely continuous distribution function with reversed hazard rate function r ˜ (·). 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. [ABSTRACT FROM AUTHOR]

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 ∑ i = 1 n θ i X i , in which the weights θi i = 1 , 2 , ⋯ , n are positive random variables which are bounded above and the primary random variables Xi, i = 1 , 2 , ⋯ , n are conditionally independent and follow subexponential distributions. [ABSTRACT FROM AUTHOR]

Main purpose of this paper is to study a robust measure of estimating dependence between random variables that can be used as an alternative to classical covariance estimator. An efficient univariate nested L-estimator (repeated median) Sn with high breakdown point is used to define bivariate dispersion. Results regarding in the characteristics of proposed estimator is discussed through this paper. [ABSTRACT FROM AUTHOR]

This paper explores the properties of the gap of representative points (RPs) in the sense of minimum mean square error for various univariate statistical distributions. We illustrate the relationship between RPs and doubly truncated mean residual life (DMRL) as well as mean residual life (MRL), which are widely used in survival analysis. The limiting behavior of the gap between the largest two RPs is discussed. In addition, an upper bound of the optimal MSE is given when the univariate random variable X has a domain on finite interval. In simulation studies, the performance of RPs for various distributions is assessed in terms of moment estimation and resampling technique. A brief discussion about the relationship between the tail of the distribution and the gap of RPs is also given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complete convergence and complete moment convergence of asymptotically almost negatively associated random variables with random coefficients. The obtained results extend and improve the ones of Li, Rao, and Wang (1992), Characiejus and Račkauskas (2016) and Zhang, Chen, and Sung (2017) to the case of AANA random variables with random coefficients. [ABSTRACT FROM AUTHOR]

In Holm and Alouini (2004), an explicit formula for the probability density function of λ 1 − 1 Z r + λ 2 − 1 Z ˜ r was obtained, where λ 1 ≠ λ 2 ≠ 0 , Z r , Z ˜ r are independent identical random variables with Γ (r , 1) distribution. In this paper, we use a recent technique from Gaunt (2018) involving Stein's method literature to obtain some simple and new proofs, which could also be used in other problems as a general method. [ABSTRACT FROM AUTHOR]

The local limit theorem (LLT) is one of the well-known limit theorems which can be used to estimate the probability at a particular point of a random variable. In this paper, we generalize weighted sums which are introduced by Giuliano and Weber and give the LLT with explicit error bounds. [ABSTRACT FROM AUTHOR]

Let { ξ i , i ∈ Z } be a negatively superadditive dependent (NSD) sequence of positive random variables. In this paper, under suitable conditions, we obtain the limit distribution for products of sums of partial sums of linear processes generated by { ξ i , i ∈ Z }. [ABSTRACT FROM AUTHOR]

Let { X n , n ≥ 1 } be a sequence of NSD random variables which is stochastically dominated by a random variable X satisfying that X is α − tailed, α > 0. Let { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of real numbers with ∑ i = 1 n a n i 2 = O (n). In this paper, the author proves the Chover-type law of the iterated logarithm for weighted partial sums of NSD random variables, and also gives some classical examples. The results obtained improve and generalize the known results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some sufficient conditions of Chung type almost sure convergence and strong law of large numbers of double array of arbitrarily dependent random variables. In the proofs, the techniques developed by Ghosal and Chandra are employed. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are presented for comparing the residual lifetimes and inactivity times of dependent random variables at different fixed times, with respect to hazard rate, mean residual life and likelihood ratio orders. The stochastic comparisons of residual lifetimes at different random times are discussed next. Several examples are presented to illustrate the established results. [ABSTRACT FROM AUTHOR]

The study of Shannon differential entropy function and its associated measures using quantile function are of recent interest among researchers. However, a bivariate extension of the same have not been considered so far. Recently, a new method of deriving bivariate quantile function and their properties in the context of reliability modeling is available in literature. Motivated by this, the present paper introduces and obtain some properties of quantile-based entropy function in the bivariate case. We also examine various properties of bivariate quantile entropy function for residual random variables. Finally, we illustrate the bivariate quantile-based residual entropy function using a suitable lifetime data. [ABSTRACT FROM AUTHOR]

In this paper, we consider the complete convergence for weighted sums of negatively superadditive-dependent (NSD) random variables without assumptions of identical distribution. Some sufficient and necessary conditions to prove the complete convergence for weighted sums of NSD random variables are presented, which extend and improve the corresponding ones of Naderi et al. As an application of the main results, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of NSD random variables is also achieved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the complete moment convergence andLrconvergence for maximal partial sums of asymptotically almost negatively associated random variables under some general conditions. The results obtained in the paper generalize some corresponding ones for negatively associated random variables. [ABSTRACT FROM AUTHOR]

Elasticity (or elasticity function) is a new concept that allows us to characterize the probability distribution of any random variable in the same way as characteristic functions and hazard and reverse hazard functions do. Initially defined for continuous variables, it was necessary to extend the definition of elasticity and study its properties in the case of discrete variables. A first attempt to define discrete elasticity is seen in Veres-Ferrer and Pavía (2014a). This paper develops this definition and makes a comparative study of its properties, relating them to the properties shown by discrete hazard and reverse hazard, as both defined in Chechile (2011). Similar to continuous elasticity, one of the most interesting properties of discrete elasticity focuses on the rate of change that this undergoes throughout its support. This paper centers on the study of the rate of change and develops a set of properties that allows us to carry out a detailed analysis. Finally, it addresses the calculation of the elasticity for the resulting variable obtained from discretizing a continuous random variable, distinguishing whether its domain is in real positives or negatives. [ABSTRACT FROM PUBLISHER]

The shape of a statistical distribution of data is important in such diverse areas as descriptive analysis, risk analysis, portfolio optimization and strategic decision making. If it is known (or believed) that the probability density function of a random variable is not symmetric, the question of its skewness becomes important. Various methods of assessing skewness have been formulated, but none are totally satisfactory. The classical measurement of skewness is based on higher moments of the random variable about its mean. However, it is well known that those measurements are sensitive to extreme outliers. The implication of the sensitivity is that mean-based metrics of skewness are inefficient, especially in small or medium size sample data. Those mean-based metrics are known as Pearson skewness, Quartile skewness and Octile skewness. All have been devised to try to accommodate for the presence of outliers. However, those test statistics are demonstrably inefficient in the presence of outliers. That inefficiency motivates the approach in this paper; the development of an efficient and more robust skewness metric we call Split Sample Skewness, hereafter referred to as SSS. In this context, efficiency means that the SSS metric requires fewer sample observations than the less efficient metrics cited above to achieve the same level of statistical robustness. The name reflects a methodology that partitions the sample into two subgroups at the median. This paper displays the findings of multiple simulation studies adducing evidence bearing on the efficiency of split ample skewness relative to other measures of skewness. [ABSTRACT FROM AUTHOR]

In this paper, we study the consistency of the P–C estimator in non parametric regression model based on m-END errors, and obtain the convergence rates of the complete consistency by using the inequalities for m-END sequence. Finally, some simulations are illustrated. [ABSTRACT FROM AUTHOR]

In this paper we develop a likelihood ratio test for independence between multivariate discrete random vectors whose components may be dependent or may have different marginal distributions. A special case that tests independence between two discrete random variables is also discussed. The model under the null hypothesis is constructed based on the marginal distributions of the random variables, while the model under the alternative is built by adding a common random effect which is not restricted to follow the same distribution as those of the other variables. The test statistic is shown to admit an asymptotic distribution of a weighted sum of chi-square random variables, each with one degree of freedom. In order to avoid the high computational cost of finding the parameters of the asymptotic distribution, a permutation-based procedure is introduced. It is shown in simulations that the permutation-based realization yields empirical level close to the nominal level, while achieving power comparable to or higher than that of existing competitors in the literature. The application of the test is illustrated with a real data set. [ABSTRACT FROM AUTHOR]

In certain applied areas such as reliability and mathematical neurobiology, it is often required to deal with shift-dependent measures of uncertainty. In this paper, we propose a shift-dependent uncertainty measure related to extropy that assigns higher weight to larger values of the random variable. Moreover, in order to describe the dynamic information of random lifetimes, we consider weighted dynamic survival and failure extropies along with some of its properties as well. Further, we have also established nonparametric estimators for the proposed measures. The methods are illustrated using simulated and real data sets. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new kind of strong law of large numbers in sublinear expectation spaces without the independent or identically distributed assumption. Instead, the strong law of large numbers is based on some restrictions on the sub-linear expectations of the partial sums. As an application, we give an interesting example in which the random variable sequence enjoys our moment restrictions, but does not enjoy the independent conditions introduced by Marinacci (1999), Peng (2009), Chen, Wu, and Li (2013), Zhang (2016b), nor Chen, Huang, and Wu (2019). [ABSTRACT FROM AUTHOR]

In this paper, we consider a special class of dependent random variables (APND) that contains negative dependent random variables classes(ND,PND,NUOD,NA,NLOD,) and some classes of positive dependent random variables. Then we generalize Kolmogrov–Feller weak law of large numbers for i.i.d. random variables to APND random variables. we further give corresponding forms of dependence for random elements taking values in separable Banach space. This development will be of use for obtaining Banach space weak law of large numbers. [ABSTRACT FROM AUTHOR]

The information based on entropy of uncertainty associated with the residual life of a random variable is implemented in various fields, e.g., information theory, statistical physics, reliability theory, statistics, demography and economics, etc. In this paper, we propose two new R ˜ -norm dynamic survival entropy (RNDSE) and R ˜ -norm weighted dynamic survival entropy (RNWDSE) based on the survival function and some results on stochastic ordering are studied in detail. Further, we have investigated the bounds on RNDSE and RNWDSE in terms of failure rate function. In addition, some relationships between RNDSE, RNWDSE and failure function have also been well established. Finally, in order to illustrate the applications of RNDSE and RNWDSE, some useful distributions like exponential, Rayleigh, Weibull, Pareto, finite range distributions and one new distribution called T-exponential have been characterized. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete moment convergence for the partial sum of linear processes with random coefficients to form { X t = ∑ j = − ∞ ∞ A j ε t − j , t ∈ Z } , where { ε i , i ∈ Z } is a sequence of random variables with zero means and stochastically dominated by a random variable ε and { A i , i ∈ Z } is a sequence of random variables, also { ε i , i ∈ Z } and { A i , i ∈ Z } satisfying the Rosenthal-type maximal inequality. [ABSTRACT FROM AUTHOR]

In this paper, we extend Kolmogorov–Feller weak law of large numbers for weighted random variables. In particular some results known so far for i.i.d. sequences are obtained for negatively superadditive random variables which are stochastically dominated. [ABSTRACT FROM AUTHOR]

In this paper, we essentially characterize the real power of the convolution set of X + Y, where X and Y ≥ 0 are two independent random variables which have respectively a Bernoulli distribution, with parameter r ∈ (0 , 1) , and an infinitely divisible probability distribution ν. The above problem is equivalent to finding the set of x , y ≥ 0 such that the mapping z ↦ (1 − r + r e z) x (E (e z Y)) y is a Laplace transform of some probability distribution. This class of real power of convolution x is provided and described. The obtained results generalize the case where Y is Gamma or Negative Binomial distributed. [ABSTRACT FROM AUTHOR]

In this paper, we study the semi-functional partial linear regression for spatial data with considering a both parametric and nonparametric modeling. In this case we obtain the asymptotic normality of the parametric component, and probability convergence with rate of the nonparametric component under spatial dependency. Finally, the performance of the parametric and nonparametric estimators, for finite spatial sample sizes, are given by using simulated and real data with comparison to the nonparametric kernel regression (FNR) model by using cross-validation and k nearest neighbor methods. [ABSTRACT FROM AUTHOR]

This paper develops a method to calculate the moments of statistical ratios as functionals of Bernoulli random variables via inverse moments of binomial distributions. We derive exact expressions for the mean, bias and variance formulas of several statistics, including the "modified" versions of the odds ratio and relative risk as well as the (positive) false discovery proportion (FDP), in which the case of zero cells is handled via indicator functions. Similar to the "positive" FDP, we introduce the "positive" odds ratio and relative risk statistics. We provide numerical examples to illustrate the biases and variances of these statistical ratios. [ABSTRACT FROM AUTHOR]

Let NK = {NKt, t ∈ [0, T]} be a filtered Poisson process defined on a probability space , and let θ ≔ (θt, t ∈ [0, T]) be a deterministic function which is the intensity of NK under a probability Pθ. In the present paper we prove that the natural maximum likelihood estimator (MLE) NK is an efficient estimator for θ under Pθ. Using Malliavin calculus we construct superefficient estimators of Stein type for θ which dominate, under the usual quadratic risk, the MLE NK. These superefficient estimators are given under the form where F is a random variable satisfying some assumptions and is the Malliavin derivative with respect to the compensated version of NK. [ABSTRACT FROM AUTHOR]

Consider the following nonparametric regression model: where xni are known fixed design points from for some positive integer d ⩾ 1, g(·) is an unknown regression function defined on A and ϵni are random errors. Under some suitable conditions, the asymptotic normality of the linear weighted estimator of g in the nonparametric regression model based on ρ-mixing errors is established. The key techniques used in the paper are the Rosenthal type inequality and the Bernstein's bigblock and small-block procedure. The result obtained in the paper generalizes the corresponding ones for some dependent sequences. [ABSTRACT FROM AUTHOR]

Let { Y n , n ≥ 1 } be sequence of random variables with EYn = 0. Assume that the Kolmogrov type exponential inequality is satisfied for { Y x j = − x I { Y j < − x } + Y j I { | Y j | ≤ x } + x I { Y j > x } , j ≥ 1 } with any x > 0. In this paper, the law of the logarithm and law of the iterated logarithm for a class of random variables satisfying the Kolmogrov type exponential inequality with non-identical distributions are obtained. The main results not only extend the known results from NA case to a class of random variables satisfying Kolmogrov type exponential inequality, but also weaken the moment assumptions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish complete convergence theorems for an array of row-wise END random variables under sub-linear expectation space. As applications of the exponential inequalities, we extend some complete convergence theorems from the traditional probability space to the sub-linear expectation space and our results generalize corresponding results obtained by Hu. [ABSTRACT FROM AUTHOR]

Exponentiated models have been widely used in modeling various types of data such as survival data and insurance claims data. However, the exponentiated composite distribution models have not been explored yet. In this paper, we introduce an improvement of the one-parameter Inverse Gamma-Pareto composite model by exponentiating the random variable associated with the one-parameter Inverse Gamma-Pareto composite distribution function. The goodness-of-fit of the exponentiated Inverse Gamma-Pareto was assessed using three different insurance data sets. The two-parameter exponentiated Inverse Gamma-Pareto model outperforms the one-parameter Inverse Gamma-Pareto model in terms of goodness-of-fit measures for all datasets. In addition, the proposed exponentiated composite Inverse Gamma-Pareto model provides a very good fit with some well-known insurance datasets. [ABSTRACT FROM AUTHOR]

In this paper, some results on complete moment convergence of moving average process generated by m-WOD random variables are established. The results improve and extend some corresponding ones in the literature. As corollaries, two new results on Marcinkiewicz-Zygmund type strong law of large numbers are established for moving average processes generated by m-WOD random variables and respectively, m-END random variables. [ABSTRACT FROM AUTHOR]

We prove the Rosnethal's inequality of m-dependent random variables under the sub-linear expectation in this paper. Furthermore, we use this inequality to investigate the central limit theorem for linear processes generated by m-dependent random variables under sub-linear expectations. This article use the basic definitions of sub-linear expectation space, Kronecker lemma, Cr inequality etc. to demonstrate the main conclusion. [ABSTRACT FROM AUTHOR]

Let X 1 , ... , X n be independent and identically distributed continuous nonnegative random variables and m n = min { X 1 , ... , X n } be the sample minimum. In the present paper, we find the best linear predictors E (m 1 + ... + m n | m j = x) and E (m 1 + ... + m n | m 1 = x 1 , ... , m j = x j) and then analyze their behavior by using simulation in the cases of the standard exponential and uniform distributions. [ABSTRACT FROM AUTHOR]

In this paper, the tail properties of the beta-normal distribution and an asymptotic Mills-type ratio are studied. Meanwhile, two applications are presented. The first application obtains the asymptotic behavior of the ratio of probability density functions and the ratio of the tails of the beta-normal and Student's t distributions. The other one considers the asymptotic distribution of the partial maximum of i.i.d. random variables with the beta-normal distribution. [ABSTRACT FROM AUTHOR]

In the presented paper, we introduce a partial functional linear model, where a scalar response variable is explained by a multivariate random variable and a functional random variable, and the relationship between the scalar response and both of the predictors is linear. Besides, the model has autoregressive errors. To estimate the model, we first expand the functional predictor and functional regression parametric on the functional principal component basis, and then estimate the coefficients for multivariate and functional regression parametric by a generalized least squares method. Theoretical properties are presented including the asymptotical normality for the multivariate coefficient and the optimal convergence rate for the functional regression parametric. Simulation studies are used to illustrate these characteristics. The proposed method is also applied on the power forecasting of photovoltaic systems data set. [ABSTRACT FROM AUTHOR]

Stochastic comparisons and ageing properties of residual life (inactivity time) of a random variable X at random time Y (RLRT/ITRT) taking X and Y independent have been investigated in the literature. In this article, we consider X and Y not necessarily independent and obtain stochastic comparison results for RLRT and ITRT based on variance residual life (VRL) order. We also study various ageing properties of RLRT/ITRT based on the associated classes of life distributions of VRL. An application of one of the results derived in this paper is also illustrated. [ABSTRACT FROM AUTHOR]