Time series models associated with Mittag-Leffler type distributions and its properties

ABSTRACT

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order α(0 < α ? 1). A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors, and sample path properties are studied. Finally we introduce the q-Mittag-Leffler process associated with the q-Mittag-Leffler distribution.     

On the probability of ruin in a continuous risk model with two types of delayed claims

ABSTRACT

This article studies a risk model involving one type of main claims and two types of by-claims, which is an extension of the general risk model with delayed claims. We suppose that every main claim may not induce any by-claims or may induce one by-claim belonging to one of the two types of by-claims with a certain probability. In addition, assume that the by-claim and its associated main claim may occur at the same time and that the occurrence of the by-claim may be delayed. An integro-differential equation system for survival probabilities is derived by using two auxiliary risk models. The expression of the survival probability is obtained by applying Laplace transforms and Rouché theorem. Furthermore, we provide a method for solving the survival probability when the two by-claim amounts satisfy different exponential distributions. As a special case, an explicit expression of survival probability is given when all the claim amounts obey the same exponential distribution. Finally, numerical results are provided to examine the proposed method.     

Limit Distributions of Random Sums of Z+-Valued Random Variables

In this article, we formulate a transfer theorem in terms of probability generating functions and discuss two approaches to limit distributions of random sums of Z ₊-valued random variables. We then develop Z ₊-valued N-ID and ?-ID laws.     

On the two-parameter Bell–Touchard discrete distribution

Abstract

In this paper we introduce a new two-parameter discrete distribution which may be useful for modeling count data. Additionally, the probability mass function is very simple and it may have a zero vertex. We show that the new discrete distribution is a particular solution of a multiple Poisson process, and that it is infinitely divisible. Additionally, various structural properties of the new discrete distribution are derived. We also discuss two methods (moments and maximum likelihood) to estimate the model parameters. The usefulness of the proposed distribution is illustrated by means of real data sets to prove its versatility in practical applications.     

On Erlang(2) Risk Process Perturbed by Diffusion

ABSTRACT

In this article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek–Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.     

The multivariate Markov and multiple Chebyshev inequalities

Abstract

A sharp probability inequality named the multivariate Markov inequality is derived for the intersection of the survival functions for non-negative random variables as an extension of the Markov inequality for a single variable. The corresponding result in Chebyshev’s inequality is also obtained as a special case of the multivariate Markov inequality, which is called the multiple Chebyshev inequality to distinguish from the multivariate Chebyshev inequality for a quadratic form of standardized uncorrelated variables. Further, the results are extended to the inequalities for the union of the survival functions and those with lower bounds.     

Interval Estimation for the Correlation Coefficient

ABSTRACT

The correlation coefficient (CC) is a standard measure of a possible linear association between two continuous random variables. The CC plays a significant role in many scientific disciplines. For a bivariate normal distribution, there are many types of confidence intervals for the CC, such as z-transformation and maximum likelihood-based intervals. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is not well-developed. In this paper, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval for the CC when the underlying bivariate distribution is a normal distribution, and two empirical likelihood-based intervals for the CC when the underlying bivariate distribution is unknown. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.     

Distance Between Random Points in Two Rectangular Cities

ABSTRACT

The travel distance between two cities of rectangular shape is considered. Two uniformly distributed random points, one from each city, are taken. Their straight path travel distance is then measured. The explicit forms for the probability density function of this distance and its expected value are obtained. Numerical results of calculating the exact expected distance and the estimated distance as well as computer simulation are given for various cases. The integer moments of the distance are also discussed.     

A LINEAR PROGRAMMING APPROACH TO THE STEADY-STATE ANALYSIS OF REFLECTED BROWNIAN MOTION

Abstract

We consider a flow of data packets from one source to many destinations in a communication network represented by a random oriented tree. Multicast transmission is characterized by the ability of some tree vertices to replicate received packets depending on the number of destinations downstream. We are interested in characteristics of multicast flows on Galton–Watson trees and trees generated by point aggregates of a Poisson process. Such stochastic settings are intended to represent tree shapes arising in the Internet and in some ad hoc networks. The main result in the branching process case is a functional equation for the joint probability generating function of flow volumes through a given vertex and in the whole tree. We provide conditions for the existence and uniqueness of solution and a method to compute it using Picard iterations. In the point process case, we provide bounds on flow volumes using the technique of stochastic comparison from the theory of continuous percolation. We use these results to derive a number of random trees' characteristics and discuss their applications to analytical evaluation of the load induced on a network by a multicast session.     

Approximating M/G/1 Waiting Time Tail Probabilities

Abstract

We propose a new approximation formula for the waiting time tail probability of the M/G/1 queue with FIFO discipline and unlimited waiting space. The aim is to address the difficulty of obtaining good estimates when the tail probability has non-exponential asymptotics. We show that the waiting time tail probability can be expressed in terms of the waiting time tail probability of a notional M/G/1 queue with truncated service time distribution plus the tail probability of an extreme order statistic. The Cramér–Lundberg approximation is applied to approximate the tail probability of the notional queue. In essence, our technique extends the applicability of the Cramér–Lundberg approximation to cases where the standard Lundberg condition does not hold. We propose a simple moment-based technique for estimating the parameters of the approximation; numerical results demonstrate that our approximation can yield very good estimates over the whole range of the argument.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

Valuation of contingent claims with stochastic interest rate and mortality driven by Lévy processes

Abstract

In this paper, we consider a model with stochastic interest rate and stochastic mortality, which is driven by a Lévy process. Under the assumption that the stochastic mortality and interest rate are dependent, we discuss the valuation of life insurance contracts. Employing the method of change of measure together with the Bayes’ rule, we present the pricing formulas in closed form for the survival and death benefit models. Finally, numerical experiments illustrate the effects of some parameters.     

Exponentiated power Lindley power series class of distributions: Theory and applications

ABSTRACT

We introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley power series (EPLPS) distribution. The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the minimum lifetime value among all risks. The distribution exhibits a variety of bathtub-shaped hazard rate functions. It contains as particular cases several lifetime distributions. Various properties of the distribution are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Expressions for the Rényi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Finally, two data applications are given showing flexibility and potentiality of the EPLPS distribution.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

Exact calculation of minimum sample size for estimating a Poisson parameter

Abstract

We develop an exact approach for the determination of the minimum sample size for estimating a Poisson parameter such that the pre-specified levels of relative precision and confidence are guaranteed. The exact computation is made possible by reducing infinitely many evaluations of coverage probability to finitely many evaluations. The theory for supporting such a reduction is that the minimum of coverage probability with respect to the parameter in an interval is attained at a discrete set of finitely many elements. Computational mechanisms have been developed to further reduce the computational complexity. An explicit bound for the minimum sample size is established.     

KERNEL ESTIMATORS OF PROBABILITY DENSITY FUNCTIONS BY RANKED-SET SAMPLING

ABSTRACT

Kernel estimation of probability density functions is considered when ranked-set samples are available. The properties of the resulting estimators are derived for small and large samples, while performance with respect to the usual simple random sample estimators is investigated for a range of probability density models.     

The sample monomode and an associated test for discrete monomodality

Abstract

The notions of (sample) mean, median and mode are common tools for describing the central tendency of a given probability distribution. In this article, we propose a new measure of central tendency, the sample monomode, which is related to the notion of sample mode. We also illustrate the computation of the sample monomode and propose a statistical test for discrete monomodality based on the likelihood ratio statistic.     

Bayesian predictive distribution for a Poisson model with a parametric restriction

Abstract

Predictive probability estimation for a Poisson distribution is addressed when the parameter space is restricted. The Bayesian predictive probability against the prior on the restricted space is compared with the non-restricted Bayes predictive probability. It is shown that the former predictive probability dominates the latter under some conditions when the predictive probabilities are evaluated by the risk function relative to the Kullback-Leibler divergence. This result is proved by first showing the corresponding dominance result for estimating the restricted parameter and then translating it into the framework of predictive probability estimation.     

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT

Let {y_t } be a Poisson-like process with the mean μ _t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.     

Theoretical results on the discrete Weibull distribution of Nakagawa and Osaki

In this paper, we discuss some theoretical results and properties of the discrete Weibull distribution, which was introduced by Nakagawa and Osaki [The discrete Weibull distribution. IEEE Trans Reliab. 1975;24:300–301]. We study the monotonicity of the probability mass, survival and hazard functions. Moreover, reliability, moments, p-quantiles, entropies and order statistics are also studied. We consider likelihood-based methods to estimate the model parameters based on complete and censored samples, and to derive confidence intervals. We also consider two additional methods to estimate the model parameters. The uniqueness of the maximum likelihood estimate of one of the parameters that index the discrete Weibull model is discussed. Numerical evaluation of the considered model is performed by Monte Carlo simulations. For illustrative purposes, two real data sets are analyzed.