Mixture Of Exponential Distributions Research Articles

Generalized Ng–Kundu–Chan model of adaptive progressive Type‐II censoring and related inference

AbstractThe model of adaptive progressive Type‐II censoring introduced by Ng et al. (2009) (referred to as Ng–Kundu–Chan model) is extended to allow switching from a given initial censoring plan to any arbitrary given plan of the same length. In this generalized model, the joint distribution of the failure times and the corresponding likelihood function is derived. It is illustrated that the computation of maximum likelihood and Bayesian estimates are along the same lines as for standard progressive Type‐II censoring. However, the distributional properties of the estimators will usually be different since the censoring plan actually applied in the (generalized) Ng–Kundu–Chan model is random. As already mentioned in Cramer and Iliopoulos (2010), we directly show that the normalized spacings are independent and identically exponentially distributed. However, it turns out that the spacings themselves are generally dependent with mixtures of exponential distributions as marginals. These results are used to study linear estimators. Finally, we propose an algorithm for generating random numbers in the generalized Ng–Kundu–Chan model and present some simulation results. The results obtained also provide new findings in the original Ng–Kundu–Chan model; the corresponding implications are highlighted.

Markov reliability model of a wing farm

A Markov model of wind farm reliability based on the example of the station on the island of Anholt, Denmark, is constructed. Reliability indexes of equipment of one turbine as a function of wind velocity are calculated. Based on hourly measurements of wind speed and electricity consumption, the durations of periods of satisfied and unsatisfied demand are estimated. It is found that the distributions of these periods can be approximated by a mixture of exponential distributions. The plant operation process is approximated by a Markov process with 5 states and continuous time. As a result, estimates of non-stationary and stationary probabilities of electricity demand being met by wind power are obtained.

Generalized Mixtures of Exponential Distribution and Associated Inference

A new generalization of the exponential distribution, namely the generalized mixture of exponential distribution, is introduced. Some of its basic properties, such as hazard function, moments, order statistics, mean deviation, measures of uncertainly, and reliability probability, are studied. Three different estimation methods are investigated by the maximum likelihood estimator, least-square estimator, and weighted least-square estimator. The performances of the estimators are assessed by simulation studies. Real-world applications of the proposed distribution are explored, and data fitting results show that the new distribution performs better than its competitors.

Improved estimators for functions of scale parameters in mixture models

Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions.

Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach.

We analyze the structure and stickiness in the chaotic components of generic Hamiltonian systems with divided phase space. Following the method proposed recently in Lozej and Robnik [Phys. Rev. E 98, 022220 (2018)2470-004510.1103/PhysRevE.98.022220], the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid. We perform extensive numerical studies of three example systems: the Chirikov standard map, the family of Robnik billiards, and the family of lemon billiards. The filling of the cells is compared to the random model of chaotic diffusion, introduced in Robnik et al. [J. Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003] for the description of transport in the phase spaces of ergodic systems. The model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential. In generic systems the stickiness induces correlations in the cell visits. The distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping. We model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times. We introduce the variable S, which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness. We use S to globally assess the distributions of recurrence times. We find that in the bulk of the chaotic sea S=1, while S>1 in areas of stickiness. We present the results in the form of animated grayscale plots of the variable S in the largest chaotic component for the three example systems, included as supplemental material to this paper.

Predicting future lifetime for mixture exponential distribution

This paper studies the prediction problem of future order statistics based on a mixture of exponential distributions. We suggest different scenarios to construct predictive intervals and point predictions for the future observations in two cases. In the first case, we assume fixed sample size, while in the second case, the sample size is assumed to be positive integer–valued random variable independent of the observations.

On Using Conventional and TL moments for the Estimation of a Mixture of Exponential Distributions, A Theoretical Review

This paper focuses on derivation of a mixture of exponential distribution and Linear Moments (LM) and Trimmed Linear moments (TL) is derived for estimation of its parameters. A comparison was made between L-moments, TL-moments of two component mixture of exponential distributions and conventional moments. In intensive simulation study, empirical study results are in the favor of conventional moments, the reason is that the L and TL-moments provided sometimes nearly unbiased results; sometimes much biased. Due to this vast variation, admittedly this study is in the favor of the conventional moments even its results are not much accurate and efficient.

Forecasting trade durations via ACD models with mixture distributions

Under Autoregressive Conditional Duration model framework, suitably modelling the positive valued innovations has been challenging. Most often, irregularly spaced trade duration data display a long right tail and a high density close to zero. Many subtle features and intricacies inherent in such time series demand flexible distributions to adequately capture these features. This paper introduces two mixture models by extending the mixture of exponential distributions of De Luca and Gallo [Mixture processes for financial intradaily durations. Stud. Nonlinear Dyn. Econ., 2004, 8(2), 1–20] to include more flexible and general distributions. In the initial extension, a Weibull distribution replaces one exponential component and, subsequently, a generalised beta of type 2 distribution is considered to further improve flexibility and extreme quantile estimation. Both these models are extended by incorporating dynamic mixture weights. Parameter estimation is done with a Bayesian methodology based on a Markov chain Monte Carlo sampling scheme. Simulation experiments are conducted to evaluate its performance and practical usefulness is assessed with empirical applications of trade durations from the Australian Securities Exchange. The performance of the proposed mixture models in comparison to several other competing distributions is evaluated in terms of model fit and forecast analysis via Time-at-Risk (TaR) quantiles and conditional expectation TaR (CTaR).

Renewal Function under Prefailure Lives Distributed as a Mixture of n Exponential Distributions: Obtaining the Parameters of Mixtures Using the Method of Moments

This paper deals with the problems of the theory of reliability of technical systems for the case when the prefailure lives of the restored (replaced) elements are distributed as a mixture of distributions. For a simple renewal process, when the prefailure lives are distributed as a mixture of exponential distributions, a method for obtaining the renewal function (average number of failures in the range from 0 to t) is presented. For the case n = 3, its explicit formula is written out. The case of n = 2 was considered in [1]. For mixtures of two and three distributions, such as exponential, Erlang, Rayleigh, and Maxwell distributions, an algorithm to obtain point estimates of the unknown parameters of the mixture in an explicit form is presented. In this work, the studies begun in [1, 2] are continued.

Consistency of the maximum likelihood estimator in seasonal hidden Markov models

In this paper, we introduce a variant of hidden Markov models in which the transition probabilities between the states, as well as the emission distributions, are not constant in time but vary in a periodic manner. This class of models, which we will call seasonal hidden Markov models (SHMMs), is particularly useful in practice, as many applications involve a seasonal behaviour. However, up to now, there is no theoretical result regarding this kind of model. We show that under mild assumptions, SHMMs are identifiable: we can identify the transition matrices and the emission distributions from the joint distribution of the observations on a period, up to state labelling. We also give sufficient conditions for the strong consistency of the maximum likelihood estimator (MLE). These results are applied to simulated data, using the EM algorithm to compute the MLE. Finally, we show how SHMM can be used in real-world applications by applying our model to precipitation data, with mixtures of exponential distributions as emission distributions.

Linear predictor of the discounted renewal aggregate claims with dependent inter-occurrence times

In this paper we derive the first two moments and a linear predictor of the compound discounted renewal aggregate claims when taking into account dependence within the inter-occurrence times. Using specific mixtures of exponential distributions to define the dependence structure between the inter-occurrence times, we compare the accuracy of the proposed linear predictor to the simulated value of that sum.Keywords: Discounted compound renewal aggregate sums; moments; Archimedean copula; random interest rate; linear predictor

Extension of the Class of Queueing Systems with Delay

We consider the problem of determining the characteristics of queuing systems with delay by the classical spectral decomposition method for the solution of the Lindley integral equation. As input distributions for the systems we choose mixtures of exponential distributions shifted to the right of the zero point, for which the spectral decomposition approach allows one to obtain a solution in closed form. We show that in such systems with delay, the average waiting time is shorter than in conventional systems.

A Characterization of Exponential Distribution in Risk Model

In the general risk model (or the Sparre-Andersen model), it is well-known that the following assertion holds: if the claim size is exponentially distributed then the non-ruin probability distribution is a mixture of exponential distributions. In this paper, under some general conditions, we prove that the converse statement of the previous assertion is also true. Besides, we define a new non-ruin measure associated with the aggregate logarithms of the claim-over-profit ratios and obtain a result on Pareto-type distributions.

Statistical analysis of the sustained lava dome emplacement and destruction processes at Popocatépetl volcano, Central México

Popocatepetl volcano reawakened in 1994 after nearly 70 years of quiescence. Between 1996 and 2015, a succession of at least 38 lava domes have been irregularly emplaced and destroyed, with each dome reaching particular volumes at specific emplacement rates. The complexity of this sequence is analyzed using statistical methods in an attempt to gain insight into the physics and dynamics of the lava dome emplacement and destruction process and to objectively assess the hazards related to that volcano. The time series of emplacements, dome residences, lava effusion lulls, and emplaced dome volumes and thicknesses are modeled using the simple exponential and Weibull distributions, the compound non-homogeneous generalized Pareto–Poisson process (NHPPP), and the mixture of exponentials distribution (MOED). The statistical analysis reveals that the sequence of dome emplacements is a non-stationary, self-regulating process most likely controlled by the balance between buoyancy-driven magma ascent and volatile exsolution crystallization. This balance has supported the sustained effusive activity for decades and may persist for an undetermined amount of time. However, the eruptive history of Popocatepetl includes major Plinian phases that may have resulted from a breach in that balance. Certain criteria to recognize such breaching conditions are inferred from this statistical analysis.

On a dual risk model perturbed by diffusion with dividend threshold

In the dual risk model, the surplus process of a company is a Levy process with sample paths that are skip-free downwards. In this paper, the authors assume that the surplus process is the sum of a compound Poisson process and an independent Wiener process. The dual of the jump-diffusion risk model under a threshold dividend strategy is discussed. The authors derive a set of two integro-differential equations satisfied by the expected total discounted dividend until ruin. The cases where profits follow an exponential or mixtures of exponential distributions are solved. Applying the key method of the Laplace transform, the authors show how the integro-differential equations are solved. The authors also discuss the conditions for optimality and show how an optimal dividend threshold can be calculated as well.

Bayesian Analysis of the 3-Component Mixture of Exponential Distribution Assuming the Non-Informative Priors

Bayesian analysis of the 3-component mixture of an Exponential distribution under type-I right censoring scheme is considered in this paper. The Bayes estimators and posterior risks for the unknown parameters are derived under squared error loss function, precautionary loss function and DeGroot loss function assuming the non-informative (uniform and Jeffreys') priors. The Bayes estimators and posterior risks are viewed as a function of the test termination time. A simulation study is given to highlight and compare the properties of the Bayes estimates.

On the ruin probability for nonhomogeneous claims and arbitrary inter-claim revenues

Recently, Raducan et al. (2015) obtained recursive formulas for the ruin probability of a surplus process at or before claim instants under the assumptions that the claim sizes are independent, nonhomogeneous Erlang distributed, and independent of the inter-claim times (i.e., the times between two successive claims), which are assumed to be independent, identically distributed (i.i.d.), following an Erlang or a mixture of exponentials distribution. In this paper, we extend these formulas to the more general case when the inter-claim times are i.i.d. nonnegative random variables following an arbitrary distribution. We also present numerical results based on the new recursions, discuss some computational aspects and state a conjecture that connects the ordering of the claims arrival with the magnitude of the corresponding ruin probabilities.

Rainfall derivatives pricing with an underlying semi-Markov model for precipitation occurrences

Weather derivatives represent a new and particular kind of contingent claim which shares a specific underlying weather index. These derivatives are written for different temperature indices, hurricanes, frost, snowfall and rainfall, and they are available for several cities. Our paper focuses on rainfall derivatives. In order to price this kind of derivatives, we have to model daily rainfall sequences at a specific location. For this purpose, we adopt a non-homogeneous parametric semi-Markov model to describe the rainfall occurrences, and a mixture of exponential distributions for rainfall amounts. The underlying Markov process has the obvious two states: dry and wet. In addition, dry and wet sequences are estimated by using best-fitting techniques. The model parameters are determined thanks to classical log-likelihood maximization. We finally price some rainfall contracts issued by the Chicago Mercantile Exchange through Monte Carlo simulation. The numerical applications and the parameter estimations are carried out using real data.

Call centers with hyperexponential patience modeling

An important feature in call center modeling is the presence of impatient customers. In this paper we show, using real data, that we can realistically model the patience distribution by the hyperexponential distribution. Since the hyperexponential distribution is a mixture of exponential distributions, an analytical Markov chain analysis is performed. A framework is developed in order to compute all kinds of practical service levels. This framework utilizes the recursive relation between the queue lengths at successive service completion epochs. Our approach shows overall better performance compared to current algorithms. Moreover, the computation times are short and our approach can therefore readily be applied in practice.

A Hierarchical Bayesian Model for Calibrating Estimates of Species Divergence Times

In Bayesian divergence time estimation methods, incorporating calibrating information from the fossil record is commonly done by assigning prior densities to ancestral nodes in the tree. Calibration prior densities are typically parametric distributions offset by minimum age estimates provided by the fossil record. Specification of the parameters of calibration densities requires the user to quantify his or her prior knowledge of the age of the ancestral node relative to the age of its calibrating fossil. The values of these parameters can, potentially, result in biased estimates of node ages if they lead to overly informative prior distributions. Accordingly, determining parameter values that lead to adequate prior densities is not straightforward. In this study, I present a hierarchical Bayesian model for calibrating divergence time analyses with multiple fossil age constraints. This approach applies a Dirichlet process prior as a hyperprior on the parameters of calibration prior densities. Specifically, this model assumes that the rate parameters of exponential prior distributions on calibrated nodes are distributed according to a Dirichlet process, whereby the rate parameters are clustered into distinct parameter categories. Both simulated and biological data are analyzed to evaluate the performance of the Dirichlet process hyperprior. Compared with fixed exponential prior densities, the hierarchical Bayesian approach results in more accurate and precise estimates of internal node ages. When this hyperprior is applied using Markov chain Monte Carlo methods, the ages of calibrated nodes are sampled from mixtures of exponential distributions and uncertainty in the values of calibration density parameters is taken into account.