Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).     

Factors effecting the model performance measures area under the ROC curve,net reclassification improvement and integrated discrimination improvement

ABSTRACT

The aim of this study is to investigate the impact of correlation structure, prevalence and effect size on the risk prediction model by using the change in the area under the receiver operating characteristic curve (ΔAUC), net reclassification improvement (NRI), and integrated discrimination improvement (IDI). In simulation study, the dataset is generated under different correlation structures, prevalences and effect sizes. We verify the simulation results with the real-data application. In conclusion, the correlation structure between the variables should be taken into account while composing a multivariable model. Negative correlation structure between independent variables is more beneficial while constructing a model.     

Nonparametric Estimation for Risk in Value-at-Risk Estimator

Abstract

Value-at-Risk (VaR) has become the standard tool used by many financial institutions to measure market risk. However, the performance of a VaR estimator may be affected by sample variation or estimation risk caused from heavy-tailed distributions. After surveying several existing procedures proposed by Jorin (Jorion, P. (1996 Jorion, P. 1996. Risk²—measuring the risk in value at risk. Financial Analysis Journal, 52: 47–56. [Taylor & Francis Online] , [Google Scholar]). Risk²—Measuring the risk in value at risk. Financial Analysis Journal 52:47–56), Huschens (Huschens, S. (1997 Huschens, S. 1997. “Confidence intervals for the value-at-risk”. In Risk Measurement, Econometrics and Neural Networks Edited by: Bol, G., Nakhaeizadeh, G. and Vollmer, K. H. 233–244. Heidelberg: Physica-Verlag.  [Google Scholar]). Confidence intervals for the value-at-risk. In: Bol, G., Nakhaeizadeh, G., Vollmer, K. H., eds. Risk Measurement, Econometrics and Neural Networks. Heidelberg: Physica-Verlag, pp. 233–244), and Ridder (Ridder, T. (1997 Ridder, T. 1997. “Basics of statistical VaR–estimation”. In Risk Measurement, Econometrics and Neural Networks Edited by: Bol, G., Nakhaeizadeh, G. and Vollmer, K. H. 161–187. Heidelberg: Physica-Verlag.  [Google Scholar]). Basics of statistical VaR-estimation. In: Bol, G., Nakhaeizadeh, G., Vollmer, K. H., eds. Risk Measurement, Econometrics and Neural Networks. Heidelberg: Physica-Verlag, pp. 161–187) etc., this article strives to propose several new estimators in measuring the risk involved in VaR estimation. We compare the performance of these VaR models through Monte Carlo simulation studies. We find that the newly proposed methods provide better accuracy and robustness in the estimation of the risk in VaR estimator.     

The Use of Ratios to Compare Volatilities and VaRs

The riskiness of two investments can be compared by looking at the ratio of the respective Value-at-Risk's (VaRs) or the ratio of volatilities. The exact distribution of the ratio of two volatilities calculated from normal observations and an asymptotic confidence interval for the ratio of two VaRs is derived. A simulation study shows good coverage rates for ratios of VaRs calculated from observations from distributions commonly used to model logarithmic returns.     

Evaluating the Risk in Sample Size Determination

This article considers experimental costs, besides power evaluation, in order to determine the sample size of an experiment. We focus on the use of standard tools of decision theory in the context of sample size determination. The loss function is defined, from the perspective of an experimenter which adopts the classical frequentist approach, and the risk function is computed. Then, we show the behavior of the risk function in the two-sample t-test, for a small sample experimental setting, with a medium-sized sample, and with large samples. Moreover, an objective criterion for a convenient sample size choice is introduced. Finally, a practical example of sample size determination, which also considers risk computation, is shown.     

Generalized EGARCH Random Effect Models Application to Financial Time Series

In this article, we propose a simple alternative model to analyze the volatility of the financial time series. In the applications, the performance of this model is compared with the performance of the GARCH type models. Using GARCH, EGARCH, and the proposed models, we analyze the time series of the Bovespa and Dow Jones Industrial Average indexes. In the applications we can see that the proposed models have good performance compared with the usual GARCH type model.     

A Stochastic Simulation Approach to Model Selection for Stochastic Volatility Models

Stochastic volatility models have been widely appreciated in empirical finance such as option pricing, risk management, etc. Recent advances of Markov chain Monte Carlo (MCMC) techniques made it possible to fit all kinds of stochastic volatility models of increasing complexity within Bayesian framework. In this article, we propose a new Bayesian model selection procedure based on Bayes factor and a classical thermodynamic integration technique named path sampling to select an appropriate stochastic volatility model. The performance of the developed procedure is illustrated with an application to the daily pound/dollar exchange rates data set.     

Estimation of Mittag-Leffler Parameters

We propose a procedure for estimating the parameters of the Mittag-Leffler (ML) and the generalized Mittag-Leffler (GML) distributions. The algorithm is less restrictive, computationally simple, and necessary to make these models usable in practice. A comparison with the fractional moment estimator indicated favorable results for the proposed method.     

A Study on the Robustness of G-Chart to Non-Normality

The G-chart is proposed by Riaz and Saghirr (2007 Riaz, M., Saghirr, A. (2007). Monitoring Process Variability using Gini's mean difference. Quality Technology and Quantitative Management 4:439–454.[Taylor & Francis Online] , [Google Scholar]) for monitoring the changes in the process variability more efficiently. This study investigates the influence of the parent population on the coefficients of the G-Chart. The appropriate values of the coefficients (b₂ and b₃) has been determined when the distributions of the processes are the normal, exponential, uniform, Laplace, and triangular. The false alarm rate and risk of type II error as performance measures of the G-chart are obtained. The robustness of the G-chart relative to Q- and R_Q-charts is studied. The efficiency of the G-chart relative to those of R-chart for detecting a shift in the process dispersion is examined. The results reveal that the proper attention to the parent population leads to the better performances of the G-chart. The G-chart is more robust than Q-chart and is slightly less efficient than R_Q-chart for detecting outliers. The G-chart is more efficient than R-chart to detect a shift for normal and non-normal processes.     

论中国经济改革发展30年的三个空间层面

中国经济改革开放发展30年来,所进行的经济体制改革与经济发展变革从空间路径来讲,主要是从宏观层面的创新社会主义市场经济体制,中观层面的实施区域经济发展战略,微观层面的创建现代企业制度三条路径来进行。这三个层面肾密相联,互相促进,铸就了当代中国改革开放的主战场。当前这三个空间层面的变革仍在深化中。