Robust Linear Regression Models:Use of a Stable Distribution for the Response Data

In this paper, we study some robustness aspects of linear regression models of the presence of outliers or discordant observations considering the use of stable distributions for the response in place of the usual normality assumption. It is well known that, in general, there is no closed form for the probability density function of stable distributions. However, under a Bayesian approach, the use of a latent or auxiliary random variable gives some simplification to obtain any posterior distribution when related to stable distributions. To show the usefulness of the computational aspects, the methodology is applied to two examples: one is related to a standard linear regression model with an explanatory variable and the other is related to a simulated data set assuming a 23 factorial experiment. Posterior summaries of interest are obtained using MCMC (Markov Chain Monte Carlo) methods and the OpenBugs software.

Consistency of change point estimators for symmetrical stable distribution with parameters shift

Assume that the characteristic index α of stable distribution satisfies 1 < α < 2, and that the distribution is symmetrical about its mean. We consider the change point estimators for stable distribution with α or scale parameter β shift. For the one case that mean is a known constant, if α or β changes, then density function will change too. To this end, we suppose the kernel estimation for a change point. For the other case that mean is an unknown constant, we suppose to apply empirical characteristic function to estimate the change-point location. In the two cases, we consider the consistency and strong convergence rate of estimators. Furthermore, we consider the mean shift case. If mean changes, then corresponding characteristic function will change too. To this end, we also apply empirical characteristic function to estimate change point. We obtain the similar convergence rate. Finally, we consider its application on the detection of mean shift in financial market.

An Improved Result in Almost Sure Central Limit Theory for Products of Partial Sums with Stable Distribution

Consider a sequence of i.i.d.positive random variables with the underlying distribution in the domain of attraction of a stable distribution with an exponent in (1,2].A universal result in the almost sure limit theorem for products of partial sums is established. Our results significantly generalize and improve those on the almost sure central limit theory previously obtained by Gonchigdanzan and Rempale and by Gonchigdanzan.In a sense,our results reach the optimal form.

PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS

We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small a-stable noises, observed at n regularly spaced time points ti = i/n, i = 1,...,n on [0, 1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator （LSE） when a small dispersion parameter ε→0 and n →∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.

Positive Stable Frailty Approach in the Construction of Dependence Life-Tables

Dependence may arise in insurance when the insureds are clustered into groups e.g. joint-life annuities. This dependence may be produced by sharing a common risk acting on mortality of members of the group. Various dependence models have been considered in literature; however, the focus has been on either the lower-tail dependence alone or upper-tail dependence alone. This article implements the frailty dependence approach to life insurance problems where most applications have been within medical setting. Our strategy is to use the conditional independence assumption given an observed association measure in a positive stable frailty approach to account for both lower and upper-tail dependence. The model is calibrated on the association of Kenyan insurers 2010 male and female published rates. The positive stable model is then proposed to construct dependence life-tables and generate life annuity payment streams in the competitive Kenyan market.

Applications of Fractional Lower Order Time-frequency Representation to Machine Bearing Fault Diagnosis

The machinery fault signal is a typical non-Gaussian and non-stationary process. The fault signal can be described by SaS distribution model because of the presence of impulses.Time-frequency distribution is a useful tool to extract helpful information of the machinery fault signal. Various fractional lower order(FLO) time-frequency distribution methods have been proposed based on fractional lower order statistics, which include fractional lower order short time Fourier transform(FLO-STFT), fractional lower order Wigner-Ville distributions(FLO-WVDs), fractional lower order Cohen class time-frequency distributions(FLO-CDs), fractional lower order adaptive kernel time-frequency distributions(FLO-AKDs) and adaptive fractional lower order time-frequency auto-regressive moving average(FLO-TFARMA) model time-frequency representation method.The methods and the exiting methods based on second order statistics in SaS distribution environments are compared, simulation results show that the new methods have better performances than the existing methods. The advantages and disadvantages of the improved time-frequency methods have been summarized.Last, the new methods are applied to analyze the outer race fault signals, the results illustrate their good performances.

A Law of the Iterated Logarithm for Moving Average Processes

Chover type law of iterated logarithm for moving average process of stable random variables with exponent α(0<α<1) was obtained.

Stable Central Limit Theorems for Super Ornstein–Uhlenbeck Processes,Ⅱ

This paper is a continuation of our recent paper(Electron.J.Probab.,24(141),(2019))and is devoted to the asymptotic behavior of a class of supercritical super Ornstein-Uhlenbeck processes(X_(t))t≥0 with branching mechanisms of infinite second moments.In the aforementioned paper,we proved stable central limit theorems for X_(t)(f)for some functions f of polynomial growth in three different regimes.However,we were not able to prove central limit theorems for X_(t)(f)for all functions f of polynomial growth.In this note,we show that the limiting stable random variables in the three different regimes are independent,and as a consequence,we get stable central limit theorems for X_(t)(f)for all functions f of polynomial growth.

Limiting Behavior of Delayed Sums of φ-Mixing under a Non-Identical Distribution Setup

We present an integral test to determine the limiting behavior of delayed sums under a non-identical distribution setup for φ-mixing sequence, and deduce Chover-type laws of the iterated logarithm for them. These complement and extend the results of Vasudeva and Divanji and Chen et al.

Stable single-mode operation of a distributed feedback quantum cascade laser integrated with a distributed Bragg reflector

We report an index-coupled distributed feedback quantum cascade laser by employing an equivalent phase shift(EPS) of quarter-wave integrated with a distributed Bragg reflector(DBR) at λ~5.03 μm. The EPS is fabricated through extending one sampling period by 50% in the center of a sampled Bragg grating. The key EPS and DBR pattern are fabricated by conventional holographic exposure combined with the optical photolithography technology, which leads to improved flexibility, repeatability, and cost-effectiveness. Stable single-mode emission can be obtained by changing the injection current or heat sink temperature even under the condition of large driving pulse width.

Influence of haze pollution on water-soluble chemical species in PM2.5 and size-resolved particles at an urban site during fall

To investigate the influence of haze on the chemical composition and formation processes of ambient aerosol particles,PM_（2.5） and size-segregated aerosol particles were collected daily during fall at an urban site of Gwangju,Korea.During the study period,the total concentration of secondary ionic species（SIS） contributed an average of 43.9% to the PM_（2.5） ,whereas the contribution of SIS to the PM_（2.5） during the haze period was 62.3%.The NO_3 and SO^（2-）_4 concentrations in PM_（2.5） during the haze period were highly elevated,being 13.4 and 5.0 times higher than those during non-haze period,respectively.The PM,NO^-_3,SO^（2-）_4,oxalate,water-soluble organic carbon（WSOC）,and humic-like substances（HULIS） had tri-modal size distributions peaks at 0.32,1.0,and 5.2 μm during the non-haze and haze periods.However,during the non-haze period they exhibited dominant size distributions at the condensation mode peaking at 0.32 μm,while on October 21 when the heaviest haze event occurred,they had predominant droplet mode size distributions peaking at 1.00 μm.Moreover,strong correlations of WSOC and HULIS with SO^（2-）_4,oxalate,and K＋at particle sizes of 〈 1.8 μm indicate that secondary processes and emissions from biomass burning could be responsible for WSOC and HULIS formations.It was found that the factors affecting haze formation could be the local stable synoptic conditions,including the weak surface winds and high surface pressures,the long-range transportation of haze from eastern China and upwind regions of the Korean peninsula,as well as the locally emitted and produced aerosol particles.

A Markov-switching predator-prey model with Allee effect for preys

This paper concerns with a Markov-switching predator-prey model with Allee effect for preys.The conditions under which extinction of predator and prey populations occur have been established.Sufficient conditions are also given for persistence and global attractivity in mean.In addition,stability in the distribution of the system under con-sideration is derived under some assumptions.Finally,numerical simulations are carried out to illustrate theoretical results.

Markov branching processes with killing and resurrection

In this paper, we consider Markov branching processes with killing and resurrection. We first show that the Markov branching process with killing and stable resurrection is just the Feller minimum process which is honest and thus unique. We then further show that this honest Feller minimum process is not only positive recurrent but also strongly ergodic. The generating function of the important stationary distribution is explicitly expressed. For the interest of comparison and completeness, the results of the Markov branching processes with killing and instantaneous resurrection are also briefly stated. A new result regarding strong ergodicity of this difficult case is presented. The birth and death process with killing and resurrection together with another example is also analyzed.