Statistical monitoring of a web server for error rates: a bivariate time-series copula-based modeling approach

The monitoring of web servers through statistical frameworks is of utmost important in order to verify possible suspicious anomalies in network traffic or misuse actions that compromise integrity, confidentiality, and availability of information. In this paper, by considering the Plackett copula function, we propose a bivariate beta-autoregressive moving average time-series model for proportion data over time, which is the case for variables present in web server monitoring such as error rates. To illustrate the proposed methodology, we monitor a Brazilian web server's rate of connection synchronization and rejection errors in a web system, with error logging rate in the past 10?min. In essence, the entire methodology may be generalized to any number of time-series of error rates.     

A bivariate model of claim frequencies and severities

Bivariate claim data come from a population that consists of insureds who may claim either one, both or none of the two types of benefits covered by a policy. In the present paper, we develop a statistical procedure to fit bivariate distributions of claims in presence of covariates. This allows for a more accurate study of insureds' choice and size in the frequency and severity of the two types of claims. A generalised logistic model is employed to examine the frequency probabilities, whilst the three parameter Burr distribution is suggested to model the underlying severity distributions. The bivariate copula model is exploited in such a way that it allows us to adjust for a range of frequency dependence structures; a method for assessing the adequacy of the fitted severity model is outlined. A health claims dataset illustrates the methods; we describe the use of orthogonal polynomials for characterising the relationship between age and the frequency and severity models.     

Modified inference function for margins for the bivariate clayton copula-based SUN Tobit Model

This paper extends the analysis of the bivariate Seemingly Unrelated Regression (SUN) Tobit model by modeling its nonlinear dependence structure through the Clayton copula. The ability to capture/model the lower tail dependence of the SUN Tobit model where some data are censored (generally, left-censored at zero) is an useful feature of the Clayton copula. We propose a modified version of the (classical) Inference Function for Margins (IFS) method by Joe and XP [H. Joe and J.J. XP, The estimation method of inference functions for margins for multivariate models, Tech. Rep. 166, Department of Statistics, University of British Columbia, 1996], which we refer to as Modified Inference Function for Margins (MIFF) method, to obtain the (point) estimates of the marginal and Clayton copula parameters. More specifically, we employ the (frequenting) data augmentation technique at the second stage of the IFS method (the first stage of the MIFF method is equivalent to the first stage of the IFS method) to generate the censored observations and then estimate the Clayton copula parameter. This process (data augmentation and copula parameter estimation) is repeated until convergence. Such modification at the second stage of the usual estimation method is justified in order to obtain continuous marginal distributions, which ensures the uniqueness of the resulting Clayton copula, as stated by Solar's [A. Solar, Fonctions de répartition à n dimensions et leurs marges, Publ. de l'Institut de Statistique de l'Université de Paris 8 (1959), pp. 229–231] theorem; and also to provide an unbiased estimate of the association parameter (the IFS method provides a biased estimate of the Clayton copula parameter in the presence of censored observations in both margins). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome in this case, we also propose the use of resampling procedures (bootstrap methods, such as standard normal and percentile, by Efron and Tibshirani [B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall, New York, 1993] to obtain confidence intervals for the model parameters.     

A bivariate extension of the beta generated distribution derived from copulas

In this paper, we introduce a new class of bivariate distributions whose marginals are beta-generated distributions. Copulas are employed to construct this bivariate extension of the beta-generated distributions. It is shown that when Archimedean copulas and convex beta generators are used in generating bivariate distributions, the copulas of the resulting distributions also belong to the Archimedean family. The dependence of the proposed bivariate distributions is examined. Simulation results for beta generators and an application to financial risk management are presented.     

Fitting competing risks data to bivariate Pareto models

This paper revisits two bivariate Pareto models for fitting competing risks data. The first model is the Frank copula model, and the second one is a bivariate Pareto model introduced by Sankaran and Nair (1993 Sankaran, P. G., and N. U. Nair. 1993. A bivariate Pareto model and its applications to reliability. Naval Research Logistics 40 (7):1013–1020. doi:10.1002/1520-6750(199312)40:7%3c1013::AID-NAV3220400711%3e3.0.CO;2-7.[Crossref], [Web of Science ®] , [Google Scholar]). We discuss the identifiability issues of these models and develop the maximum likelihood estimation procedures including their computational algorithms and model-diagnostic procedures. Simulations are conducted to examine the performance of the maximum likelihood estimation. Real data are analyzed for illustration.     

Weak convergence of bivariate sequence to bivariate normal

It is shown that under certain conditions the distributions of a bivariate sequence of random vectors converge weakly to that of a bivariate normal distribution.     

Concepts de dépendance et ordres stochastiques pour des lois bidimensionnelles

Yanagimoto and Okamoto (1969) introduced a stochastic ordering that generalizes a concept of monotone regression dependence introduced by Lehmann (1966). In this paper, we define and examine the properties of three new orderings which imply that of Yanagimoto and Okamoto. One of these orderings is seen to extend Shaked's (1977) notion of DTP(0, 1), and another includes Lehmann's concept of positive likelihood-ratio dependence as a special case. The proposed orderings are also compared with the TP₂ positive-dependence ordering defined by Kimeldorf and Sampson (1987).     

A software reliability model using mean residual quantile function

In this paper, we propose a class of distributions with the inverse linear mean residual quantile function. The distributional properties of the family of distributions are studied. We then discuss the reliability characteristics of the family of distributions. Some characterizations of the class of distributions are also discussed. The parameters of the class of distributions are estimated using the method of L-moments. The proposed class of distributions is applied to a real data set.     

An efficient nonparametric test for bivariate two-sample location problem

We propose a new test for the two-sample bivariate location problem. The proposed test statistic has a U-statistic representation with a degenerate kernel. The limiting distribution is found for the proposed test statistic. The power of the test is compared using Monte Carlo simulation to the tests of Blumen [I. Blumen, A new bivariate sign-test for location, Journal of the American Statistical Association 53 (1958) 448–456], Mardia [K.V. Mardia, A non-parametric test for the bivariate two-sample location problem, Journal of the Royal Statistical Society, Series B 29 (1967) 320–342], Peters and Randles [D. Peters, R.H. Randles, A bivariate signed-rank test for the two-sample location problem, Journal of the Royal Statistical Society, Series B 53 (1991) 493–504], LaRocque, Tardif and van Eeden [D. LaRocque, S. Tardif, C. van Eeden, An affine-invariant generalization of the Wilcoxon signed-rank test for the bivariate location problem, Australian and New Zealand Journal of Statistics 45 (2003) 153–165], and Baringhaus and Franz [L. Baringhaus, C. Franz, On a new multivariate two-sample test, Journal of Multivariate Analysis 88 (2004) 190–206]. Under the bivariate normal and bivariate t distributions the proposed test was more powerful than the competitors for almost every change in location. Under the other distributions the proposed test reached the desired power of one at a faster rate than the other tests in the simulation study. Application of the test is presented using bivariate data from a synthetic and a real-life data set.     

Note on estimation of reliability under bivariate pareto stress-strength model

In this paper, we discuss the problem of estimating reliability (R) of a component based on maximum likelihood estimators (MLEs). The reliability of a component is given byR=P[Y<X]. Here X is a random strength of a component subjected to a random stress(Y) and (X,Y) follow a bivariate pareto(BVP) distribution. We obtain an asymptotic normal(AN) distribution of MLE of the reliability(R).     

A bivariate generalized linear model with an application to meteorological data analysis

A bivariate generalized linear model is developed as a mixture distribution with one component of the mixture being discrete with probability mass only at the origin. The use of the proposed model is illustrated by analyzing local area meteorological measurements with constant correlation structure that incorporates predictor variables. The Monte Carlo study is performed to evaluate the inferential efficiency of model parameters for two types of true models. These results suggest that the estimates of regression parameters are consistent and the efficiency of the inference increases for the proposed model for ρ≥0.50 especially in larger samples. As an illustration of a bivariate generalized linear model, we analyze a precipitation monitoring data of adjacent local stations for Tokyo and Yokohama.     

Comparison of hazard rates for dependent random variables

In this paper, we define and study a new notion for the comparison of the hazard rates of two random variables taking into account their mutual dependence. Properties, applications and the comparison for a data set are given.     

Estimation of the cumulative baseline hazard function for dependently right-censored failure time data

In this paper, we study the properties of a special class of frailty models when the frailty is common to several failure times. The models are closely linked to Archimedean copula models. We establish a useful formula for cumulative baseline hazard functions and develop a new estimator for cumulative baseline hazard functions in bivariate frailty regression models. Based on our proposed estimator, we present a graphical model checking procedure. We fit a leukemia data set using our model and end our paper with some discussions.     

Modelling lifetimes by quantile functions using Parzen's score function

The present paper introduces methods of constructing quantile functions as models of lifetimes with monotone and nonmonotone hazard functions. This is accomplished on the basis of the relationships the hazard quantile function has with the score function introduced by Parzen in connection with the tail heaviness of probability distributions. Three models illustrated here contain several existing models as particular cases. The appropriateness of the models in real situations is also demonstrated.     

Alternative definitions of bivariate equilibrium distributions

The concept of equilibrium distribution plays an important role in survival analysis, reliability and insurance studies. If we consider the problem of extending this concept to higher dimensions, we do not have a unique solution. In this paper, alternative definitions of bivariate equilibrium distributions are studied and proposed. The Navarro et al. (2006) proposal is considered and some new results are given. We continue with the Gupta and Sankaran's (1998) definition. Necessary and sufficient conditions for its existence are stated and a characterization theorem is given. As a third alternative, a new definition based on conditional specification is introduced and several results are obtained. Reliability properties of the different versions are studied.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

Cure rate model with bivariate interval censored data

A mixture model is proposed to analyze a bivariate interval censored data with cure rates. There exist two types of association related with bivariate failure times and bivariate cure rates, respectively. A correlation coefficient is adopted for the association of bivariate cure rates and a copula function is applied for bivariate survival times. The conditional expectation of unknown quantities attributable to interval censored data and cure rates are calculated in the E-step in ES (Expectation-Solving algorithm) and the marginal estimates and the association measures are estimated in the S-step through a two-stage procedure. A simulation study is performed to evaluate the suggested method and a real data from HIV patients is analyzed as a real data example.     

An introduction to hypergeometric functions for economists

Hypergeometric functions are a generalization of exponential functions. They are explicit, computable functions that can also be manipulated analytically. The functions and series we use in quantitative economics are all special cases of them. In this paper, a unified approach to hypergeometric functions is given. As a result, some potentially useful general applications emerge in a number of areas such as in econometrics and economic theory. The greatest benefit from using these functions stems from the fact that they provide parsimonious explicit (and interpretable) solutions to a wide range of general problems.