A NOTE ON ASYMPTOTIC BEHAVIOR FOR NEGATIVE DRIFT RANDOM WALK WITH DEPENDENT HEAVY-TAILED STEPS AND ITS APPLICATION TO RISK THEORY

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process. Xn=-u＋∑j=-∞^∞ φn-jεj, where { ε, εn; -∞〈n〈＋∞} is a sequence of independent, identically distributed random variables with zero mean, u 〉 0 is a constant and the coefficients {φi; -∞〈i〈∞} satisfy 0〈 ∑j=-∞^∞ ｜jφj｜〈 ∞ . Under the conditions that the distribution function of ｜ε｜ has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{sup n≥0 （-qu＋∑j=-∞^∞ εj βnj）〉x} is discussed. Then the result is applied to ultimate ruin probability.

Distributed Estimator of Market Beta under Extreme Conditions

Market beta is a measure of the volatility or systematic risk of a security or portfolio compared to the market as a whole. This paper considers the distributed estimation of market beta in the case of massive data, and obtains the consistency and asymptotic normality of the estimator. Further, simulations show the finite sample properties of this estimator.

The Finite Time Ruin Probability with the Same Heavy-tailed Insurance and Financial Risks

This note complements a recent study in ruin theory with risky investment byestablishing the same asymptotic estimate for the finite time ruin probability under a weakerrestriction on the financial risks. In particular, our result applies to a critical case that theinsurance and financial risks have Pareto-type tails with the same regular index.

Bootstrap Test for Stationarity of Heavy-Tailed Series with Structural Breaks

The paper proposes a statistic to test stationarity of series with κ-stable innovations and structural breaks,obtains the asymptotical distribution of the statistic,and proves the consistency of the test.To obtain critic values for the test without the estimation of the index κ,the paper proposes the bootstrap procedures to approximate the distribution,and proves the consistency of the procedures.The simulations demonstrate that the bootstrap test is practical and powerful.

Nonparametric inferences for kurtosis and conditional kurtosis

Under the assumption of strictly stationary process, this paper proposes a nonparametric model to test the kurtosis and conditional kurtosis for risk time series. We apply this method to the daily returns of S＆P500 index and the Shanghai Composite Index, and simulate GARCH data for verifying the efficiency of the presented model. Our results indicate that the risk series distribution is heavily tailed, but the historical information can make its future distribution light-tailed. However the far future distribution＇s tails are little affected by the historical data.

A New Statistical Tool： Scalar Score Function

The basic inference function of mathematical statistics, the score function, is a vector function. The author has introduced the scalar score, a scalar inference function, which reflects main features of a continuous probability distribution and which is simple. Its simplicity makes it possible to introduce new relevant numerical characteristics of continuous distributions. The t-mean and score variance are descriptions of distributions without the drawbacks of the mean and variance, which may not exist even in cases of regular distributions. Their sample counterparts appear to be alternative descriptions of the observed data. The scalar score itself appears to be a new mathematical tool, which could be used in solving traditional statistical problems for models far from the normal one, skewed and heavy-tailed.

A Comparison of the Estimators of the Scale Parameter of the Errors Distribution in the L1 Regression

The L1 regression is a robust alternative to the least squares regression whenever there are outliers in the values of the response variable, or the errors follow a long-tailed distribution. To calculate the standard errors of the L1 estimators, construct confidence intervals and test hypotheses about the parameters of the model, or to calculate a robust coefficient of determination, it is necessary to have an estimate of a scale parameterτ. This parameter is such that τ2/n is the variance of the median of a sample of size n from the errors distribution. [1] proposed the use of , a consistent, and so, an asymptotically unbiased estimator of τ. However, this estimator is not stable in small samples, in the sense that it can increase with the introduction of new independent variables in the model. When the errors follow the Laplace distribution, the maximum likelihood estimator of τ, say , is the mean absolute error, that is, the mean of the absolute residuals. This estimator always decreases when new independent variables are added to the model. Our objective is to develop asymptotic properties of under several errors distributions analytically. We also performed a simulation study to compare the distributions of both estimators in small samples with the objective to establish conditions in which is a good alternative to for such situations.

FINITE-TIME RUIN PROBABILITY WITH NQD DOMINATED VARYING-TAILED CLAIMS AND NLOD INTER-ARRIVAL TIMES

In 2007,Chen and Ng investigated infinite-time ruin probability with constant interest forceand negatively quadrant dependent and extended regularly varying-tailed claims.Following this work,the authors obtain a weakly asymptotic equivalent formula for the finite-time and infinite-time ruinprobability with constant interest force,negatively quadrant dependent,and dominated varying-tailedclaims and negatively lower orthant dependent inter-arrival times.In particular,when the claims areconsistently varying-tailed,an asymptotic equivalent formula is presented.

Heavy tail and light tail of Cox-Ingersoll-Ross processes with regime-switching

This study aims to investigate the tail behavior of Cox-Ingersoll-Ross(CIR) processes with regime switching. An essential difference shown in this study between CIR processes with and without regime switching is that the stationary distribution of those with regime switching may be heavy-tailed. We first provide sharp criteria to justify the existence of a stationary distribution for the CIR process with regime switching, which is applied to study the long-term returns of interest rates. Then, we provide a criterion to identify whether this distribution is heavy-tailed. Our results provide theoretical evidence of the existence of regime switching for interest-rate models based on empirical evidence of a heavy-tailed distribution.

Randomly Weighted Sums for Negatively Associated Random Variables with Heavy Tails

In 2003, Tang Qihe et al. obtained a simple asymptotic formula for independent identically distributed (i.i.d.) random variables with heavy tails. In this paper, under certain moment conditions, we establish a formula as the same as Tang’s, when random variables are negatively associated (NA).

Flexible Factor Model for Handling Missing Data in Supervised Learning

This paper presents an extension of the factor analysis model based on the normal mean-variance mixture of the Birnbaum-Saunders in the presence of nonresponses and missing data.This model can be used as a powerful tool to model non-normal features observed from data such as strongly skewed and heavy-tailed noises.Missing data may occur due to operator error or incomplete data capturing therefore cannot be ignored in factor analysis modeling.We implement an EM-type algorithm for maximum likelihood estimation and propose single imputation of possible missing values under a missing at random mechanism.The potential and applicability of our proposed method are illustrated through analyzing both simulated and real datasets.

Nearly optimal Bayesian shrinkage for high-dimensional regression

During the past decade,shrinkage priors have received much attention in Bayesian analysis of high-dimensional data.This paper establishes the posterior consistency for high-dimensional linear regression with a class of shrinkage priors,which has a heavy and flat tail and allocates a sufficiently large probability mass in a very small neighborhood of zero.While enjoying its efficiency in posterior simulations,the shrinkage prior can lead to a nearly optimal posterior contraction rate and the variable selection consistency as the spike-and-slab prior.Our numerical results show that under the posterior consistency,Bayesian methods can yield much better results in variable selection than the regularization methods such as LASSO and SCAD.This paper also establishes a BvM-type result,which leads to a convenient way of uncertainty quantification for regression coefficient estimates.

The sum of two independent polynomially-modified hyperbolic secant random variables with application in computational finance

In this paper,by reshaping the hyperbolic secant distribution using Hermite polynomial,we devise a polynomially-modified hyperbolic secant distribution which is more flexible than secant distribution to capture the skewness,heavy-tailedness and kurtosis of data.As a portfolio possibly consists of multiple assets,the distribution of the sum of independent polynomially-modified hyperbolic secant random variables is derived.In exceptional cases,we evaluate risk measures such as value at risk and expected shortfall(ES)for the sum of two independent polynomially-modified hyperbolic secant random variables.Finally,using real datasets from four international computers stocks,such as Adobe Systems,Microsoft,Nvidia and Symantec Corporations,the effectiveness of the proposed model is shown by the goodness of Gram–Charlier-like expansion of hyperbolic secant law,for performance of value at risk and ES estimation,both in and out of the sample period.

Study on Statistical Out lier Det ection and Labelling

Outliers accompany control engineers in their real life activity.Indus trial reality is much richer than eleme ntary linear,quadratic,Gaussian assumptions.Outliers appear due to various and varying,often unknown,reasons.They meet research interest in statistical and regression analysis and in data mining.There are a lot of interesting algorithms and approaches to outlier detection,labelling,filtering and finally interpretation.Unfortunately,their impact on control systems has not been found sufficient attention in research.Their influence is frequently unnoticed,ignored or not mentioned.This work focuses on the subject of outlier detection and labelling in the cont ext of control system performance analysis.Selec ted statistical data-driven approaches are analyzed,as t hey can be easily implemented with limited a priori knowledge.The study consists of a simulation study followed by the analysis of real control data.Differe nt generation mechanisms are Simula ted,like overlapping Gaussian processes,symmetric and asymmetric,artificially shifted points and fat-tailed distributions.Simulation observations are confronted with industrial control loops datasets.The work concludes with a practical procedure,which should help practitioners in dealing with outliers in control engineering temporal data.

Least Absolute Deviation Estimation of Autoregressive Conditional Duration Model

This paper studies the least absolute deviation estimation of the high frequency financial autoregressive conditional duration （ACD） model. The asymptotic properties of the estimator are studied given mild regularity conditions. Furthermore, we develop a Wald test statistic for the linear restriction on the parameters. A simulation study is conducted for the finite sample properties of our estimator. Finally, we give an empirical study of financial duration.