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.     

New approximations to the exact distribution of the kruskal-wallis test statistic

Several approximations to the exact distribution of the Kruskal-Wallis test' statistic presently exist. There approximations can roughly be grouped into two classes: (i) computationally difficult with good accuracy, and (ii) easy to compute but not as accurate as the first class. The purpose of this paper is to introduce two nev approximations (one in the latter class and one which is computationally more involved)y and to compare these with other popular approximations. These comparisons use exact probabilities where available and Monte Carlo simulation otherwise.     

A New Estimator of Population Mean in Stratified Sampling

Kadilar and Cingi (2005 Kadilar , C. , Cingi , H. ( 2005 ). A new ratio estimator in stratified sampling . Comm. Statist. Theory Meth. 34 : 1 – 6 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) have suggested a new ratio estimator in stratified sampling. The efficiency of this estimator is compared with the traditional combined ratio estimator on the basis of mean square error (MSE). We propose another estimator by utilizing a simple transformation introduced by Bedi (1996 Bedi , P. K. ( 1996 ). Efficient utilization of auxiliary information at estimation stage . Biomet. J. 38 ( 8 ): 973 – 976 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The proposed estimator is found to be more efficient than the traditional combined ratio estimator as well as the Kadilar and Cingi (2005 Kadilar , C. , Cingi , H. ( 2005 ). A new ratio estimator in stratified sampling . Comm. Statist. Theory Meth. 34 : 1 – 6 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) ratio estimator.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

Bayes Estimation of Shift Point in Geometric Sequence

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,… X _n were observed from geometric population with parameter q ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in parameter q ₂. The Bayes estimates of m, q ₁, q ₂, reliability R ₁ (t) and R ₂ (t) at time t are derived for symmetric and asymmetric loss functions under informative and non informative priors. A simulation study is carried out.     

The Superiorities of Bayes Linear Minimum Risk Estimation in Linear Model

In this article, the Bayes linear minimum risk estimator (BLMRE) of parameters is derived in linear model. The superiorities of the BLMRE over ordinary least square estimator (LSE) is studied in terms of the mean square error matrix (MSEM) criterion and Pitman closeness (PC) criterion.     

Preliminary Estimation in Gaussian Stationary Processes

Preliminary estimation of the kth Lag autocorrelation function in the Gaussian stationary processes is considered. An estimation procedure is derived from the ratio of the sum filter and the difference filter. The performance of this estimator is compared to the sample estimator through a Monte Carlo study.     

Estimation in the Three-Parameter Gamma Distribution Based on the Profile Log-Likelihood Function

In this article, we express the profile log-likelihood function for the three-parameter gamma distribution in terms of the location parameter only and we study its properties. The behavior of the profile function is examined as the location parameter tends to the boundary values, i.e., to ? ∞ and to the minimum value of the sample. As a result, we obtain that if the log-likelihood function has a local maximum then it has another stationary value which is a saddle point. The results are supported with the use of simulation results.     

Rate of Strong Consistency of Maximum Quasi-Likelihood Estimator in Multivariate Generalized Linear Models

In this article, we consider a multivariate generalized linear model with adaptive design matrix and general link function. The asymptotic singularity of design matrix of adaptive design presents challenge when establishing the large sample properties of the maximum quasi-likelihood estimate (MQLE). Under some mild conditions, we obtain a strong convergence rate of the MQLE.     

Asymptotic Normality for Regression Function Estimate Under Truncation and α-Mixing Conditions

In this article we establish pointwise asymptotic normality of nonparametric kernel estimator of regression function for a left truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimation of the covariable's density is considered. As a by-product, we obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and truncated distributions under dependence, which is interesting independently. Finite sample behavior of the estimator of the regression function is investigated as well.