• Title, Summary, Keyword: heavy tails

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On Tail Probabilities of Continuous Probability Distributions with Heavy Tails (두꺼운 꼬리를 갖는 연속 확률분포들의 꼬리 확률에 관하여)

  • Yun, Seokhoon
    • The Korean Journal of Applied Statistics
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    • v.26 no.5
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    • pp.759-766
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    • 2013
  • The paper examines several classes of probability distributions with heavy tails. An (asymptotic) expression for tail probability needs to be known to understand which class a given probability distribution belongs to. It is usually not easy to get expressions for tail probabilities since most absolutely continuous probability distributions are specified by probability density functions and not by distribution functions. The paper proposes a method to obtain asymptotic expressions for tail probabilities using only probability density functions. Some examples are given to illustrate the proposed method.

EMPIRICAL REALITIES FOR A MINIMAL DESCRIPTION RISKY ASSET MODEL. THE NEED FOR FRACTAL FEATURES

  • Christopher C.Heyde;Liu, S.
    • Journal of the Korean Mathematical Society
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    • v.38 no.5
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    • pp.1047-1059
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    • 2001
  • The classical Geometric Brownian motion (GBM) model for the price of a risky asset, from which the huge financial derivatives industry has developed, stipulates that the log returns are iid Gaussian. however, typical log returns data show a distribution with much higher peaks and heavier tails than the Gaussian as well as evidence of strong and persistent dependence. In this paper we describe a simple replacement for GBM, a fractal activity time Geometric Brownian motion (FATGBM) model based on fractal activity time which readily explains these observed features in the data. Consequences of the model are explained, and examples are given to illustrate how the self-similar scaling properties of the activity time check out in practice.

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A NOTE ON THE SEVERITY OF RUIN IN THE RENEWAL MODEL WITH CLAIMS OF DOMINATED VARIATION

  • Tang, Qihe
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.4
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    • pp.663-669
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    • 2003
  • This paper investigates the tail asymptotic behavior of the severity of ruin (the deficit at ruin) in the renewal model. Under the assumption that the tail probability of the claimsize is dominatedly varying, a uniform asymptotic formula for the tail probability of the deficit at ruin is obtained.

Asymptotic Properties of LAD Esimators of a Nonlinear Time Series Regression Model

  • Kim, Tae-Soo;Kim, Hae-Kyung;Park, Seung-Hoe
    • Journal of the Korean Statistical Society
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    • v.29 no.2
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    • pp.187-199
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    • 2000
  • In this paper, we deal with the asymptotic properties of the least absolute deviation estimators in the nonlinear time series regression model. For the sinusodial model which frequently appears in a time series analysis, we study the strong consistency and asymptotic normality of least absolute deviation estimators. And using the derived limiting distributions we show that the least absolute deviation estimators is more efficient than the least squared estimators when the error distribution of the model has heavy tails.

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Power t distribution

  • Zhao, Jun;Kim, Hyoung-Moon
    • Communications for Statistical Applications and Methods
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    • v.23 no.4
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    • pp.321-334
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    • 2016
  • In this paper, we propose power t distribution based on t distribution. We also study the properties of and inferences for power t model in order to solve the problem of real data showing both skewness and heavy tails. The comparison of skew t and power t distributions is based on density plots, skewness and kurtosis. Note that, at the given degree of freedom, the kurtosis's range of the power t model surpasses that of the skew t model at all times. We draw inferences for two parameters of the power t distribution and four parameters of the location-scale extension of power t distribution via maximum likelihood. The Fisher information matrix derived is nonsingular on the whole parametric space; in addition we obtain the profile log-likelihood functions on two parameters. The response plots for different sample sizes provide strong evidence for the estimators' existence and unicity. An application of the power t distribution suggests that the model can be very useful for real data.

PRECISE LARGE DEVIATIONS FOR AGGREGATE LOSS PROCESS IN A MULTI-RISK MODEL

  • Tang, Fengqin;Bai, Jianming
    • Journal of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.447-467
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    • 2015
  • In this paper, we consider a multi-risk model based on the policy entrance process with n independent policies. For each policy, the entrance process of the customer is a non-homogeneous Poisson process, and the claim process is a renewal process. The loss process of the single-risk model is a random sum of stochastic processes, and the actual individual claim sizes are described as extended upper negatively dependent (EUND) structure with heavy tails. We derive precise large deviations for the loss process of the multi-risk model after giving the precise large deviations of the single-risk model. Our results extend and improve the existing results in significant ways.

A Study on Statistical Approach for Nonlinear Image Denoising Algorithms (비선형 영상 잡음제거 알고리즘의 통계적 접근 방법에 관한 연구)

  • Hahn, Hee-Il
    • The Journal of the Institute of Internet, Broadcasting and Communication
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    • v.12 no.1
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    • pp.151-156
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    • 2012
  • In this paper robust nonlinear image denoising algorithms are introduced for the distribution which is Gaussian in the center and Laplacian in the tails. The distribution is known as the least favorable ${\epsilon}$-contaminated normal distribution that maximizes the asymptotic variance. The proposed filter proves to be the maximum likelihood estimator under the heavy-tailed Gaussian noise environments. It is optimal in the respect of maximizing the efficacy under the above noise environment. Another filter for reducing impulsive noise is proposed by mixing with the myriad filter to propose an amplitude-limited myriad filter. Extensive experiment is conducted with images corrupted with ${\alpha}$-stable noise to analyze the behavior and performance of the proposed filters.

Projected Circular and l-Axial Skew-Normal Distributions

  • Seo, Han-Son;Shin, Jong-Kyun;Kim, Hyoung-Moon
    • The Korean Journal of Applied Statistics
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    • v.22 no.4
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    • pp.879-891
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    • 2009
  • We developed the projected l-axial skew-normal(LASN) family of distributions for I-axial data. The LASN family of distributions contains the semicircular skew-normal(SCSN) and the circular skew-normal(CSN) families of distributions as special cases. The LASN densities are similar to the wrapped skew-normal densities for the small values of the scale parameter. However CSN densities have more heavy tails than those of the wrapped skew-normal densities on the circle. Furthermore the CSN densities have two modes as the scale parameter increases. The LASN distribution has very convenient mathematical features. We extend the LASN family of distributions to a bivariate case.

Robust Bayesian analysis for autoregressive models

  • Ryu, Hyunnam;Kim, Dal Ho
    • Journal of the Korean Data and Information Science Society
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    • v.26 no.2
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    • pp.487-493
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    • 2015
  • Time series data sometimes show violation of normal assumptions. For cases where the assumption of normality is untenable, more exible models can be adopted to accommodate heavy tails. The exponential power distribution (EPD) is considered as possible candidate for errors of time series model that may show violation of normal assumption. Besides, the use of exible models for errors like EPD might be able to conduct the robust analysis. In this paper, we especially consider EPD as the exible distribution for errors of autoregressive models. Also, we represent this distribution as scale mixture of uniform and this form enables efficient Bayesian estimation via Markov chain Monte Carlo (MCMC) methods.

CLOSURE PROPERTY AND TAIL PROBABILITY ASYMPTOTICS FOR RANDOMLY WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES WITH HEAVY TAILS

  • Dindiene, Lina;Leipus, Remigijus;Siaulys, Jonas
    • Journal of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1879-1903
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    • 2017
  • In this paper we study the closure property and probability tail asymptotics for randomly weighted sums $S^{\Theta}_n={\Theta}_1X_1+{\cdots}+{\Theta}_nX_n$ for long-tailed random variables $X_1,{\ldots},X_n$ and positive bounded random weights ${\Theta}_1,{\ldots},{\Theta}_n$ under similar dependence structure as in [26]. In particular, we study the case where the distribution of random vector ($X_1,{\ldots},X_n$) is generated by an absolutely continuous copula.