Inference on P(Y<X) in Bivariate Rayleigh Distribution

This paper deals with the estimation of reliability $R=P(Y<X)$ when $X$ is a random strength of a component subjected to a random stress $Y$ and $(X,Y)$ follows a bivariate Rayleigh distribution. The maximum likelihood estimator of $R$ and its asymptotic distribution are obtained. An asymptotic confidence interval of $R$ is constructed using the asymptotic distribution. Also, two confidence intervals are proposed based on Bootstrap method and a computational approach. Testing of the reliability based on asymptotic distribution of $R$ is discussed. Simulation study to investigate performance of the confidence intervals and tests has been carried out. Also, a numerical example is given to illustrate the proposed approaches.Comment: Accepted for publication. Communications in Statistics- Theory and Methods, 201

Bounds and approximations for sums of dependent log-elliptical random variables.

Dhaene, Denuit, Goovaerts, Kaas and Vyncke [Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke,D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math.Econom. 31 (1), 3-33; Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: Applications. Insurance Math. Econom. 31 (2), 133-161]have studied convex bounds for a sum of dependent random variables and applied these to sums of lognormal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum. In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case. Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed-form approximations for the different risk measures. Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed-form approximations.We also present numerical examples to show the accuracy of the proposed approximations.

Forecasting Agricultural Commodity Prices with Asymmetric-Error GARCH Models

The performance of a proposed asymmetric-error GARCH model is evaluated in comparison to the normal-error- and Student-t-GARCH models through three applications involving forecasts of U.S. soybean, sorghum, and wheat prices. The applications illustrate the relative advantages of the proposed model specification when the error term is asymmetrically distributed, and provide improved probabilistic forecasts for the prices of these commodities.GARCH, nonnormality, skewness, time-series forecasting, U.S. commodity prices, Demand and Price Analysis,

Portfolio Diversification and Value at Risk Under Thick-Tailedness

We present a unified approach to value at risk analysis under heavy-tailedness using new majorization theory for linear combinations of thick-tailed random variables that we develop. Among other results, we show that the stylized fact that portfolio diversification is always preferable is reversed for extremely heavy-tailed risks or returns. The stylized facts on diversification are nevertheless robust to thick-tailedness of risks or returns as long as their distributions are not extremely long-tailed. We further demonstrate that the value at risk is a coherent measure of risk if distributions of risks are not extremely heavy-tailed. However, coherency of the value at risk is always violated under extreme thick-tailedness. Extensions of the results to the case of dependence, including convolutions of a-symmetric distributions and models with common stochs are provided.

Estimation of Stress-Strength model in the Generalized Linear Failure Rate Distribution

In this paper, we study the estimation of $R=P [Y < X ]$, also so-called the stress-strength model, when both $X$ and $Y$ are two independent random variables with the generalized linear failure rate distributions, under different assumptions about their parameters. We address the maximum likelihood estimator (MLE) of $R$ and the associated asymptotic confidence interval. In addition, we compute the MLE and the corresponding Bootstrap confidence interval when the sample sizes are small. The Bayes estimates of $R$ and the associated credible intervals are also investigated. An extensive computer simulation is implemented to compare the performances of the proposed estimators. Eventually, we briefly study the estimation of this model when the data obtained from both distributions are progressively type-II censored. We present the MLE and the corresponding confidence interval under three different progressive censoring schemes. We also analysis a set of real data for illustrative purpose.Comment: 31 pages, 2 figures, preprin

ESTIMATION OF EFFICIENT REGRESSION MODELS FOR APPLIED AGRICULTURAL ECONOMICS RESEARCH

This paper proposes and explores the use of a partially adaptive estimation technique to improve the reliability of the inferences made from multiple regression models when the dependent variable is not normally distributed. The relevance of this technique for agricultural economics research is evaluated through Monte Carlo simulation and two mainstream applications: A time-series analysis of agricultural commodity prices and an empirical model of the West Texas cotton basis. It is concluded that, given non-normality, this technique can substantially reduce the magnitude of the standard errors of the slope parameter estimators in relation to OLS, GLS and other least squares based estimation procedures, in practice, allowing for more precise inferences about the existence, sign and magnitude of the effects of the independent variables on the dependent variable of interest. In addition, the technique produces confidence intervals for the dependent variable forecasts that are more efficient and consistent with the observed data. Key Words: Efficient regression models, partially adaptive estimation, non-normality, skewness, heteroskedasticity, autocorrelation.Efficient regression models, partially adaptive estimation, non-normality, skewness, heteroskedasticity, autocorrelation., Research Methods/ Statistical Methods,

The Distribution of Stochastic Shrinkage Parameters in Ridge Regression

In this article we derive the density and distribution functions of the stochastic shrinkage parameters of three well-known operational Ridge Regression estimators by assuming normality. The stochastic behavior of these parameters is likely to affect the properties of the resulting Ridge Regression estimator, therefore such knowledge can useful in the selection of the shrinkage rule. Some numerical calculations are carried out to illustrate the behavior of these distributions, throwing light on the performance of the different Ridge Regression estimators.

Bayesian Lower Bounds for Dense or Sparse (Outlier) Noise in the RMT Framework

Robust estimation is an important and timely research subject. In this paper, we investigate performance lower bounds on the mean-square-error (MSE) of any estimator for the Bayesian linear model, corrupted by a noise distributed according to an i.i.d. Student's t-distribution. This class of prior parametrized by its degree of freedom is relevant to modelize either dense or sparse (accounting for outliers) noise. Using the hierarchical Normal-Gamma representation of the Student's t-distribution, the Van Trees' Bayesian Cram\'er-Rao bound (BCRB) on the amplitude parameters is derived. Furthermore, the random matrix theory (RMT) framework is assumed, i.e., the number of measurements and the number of unknown parameters grow jointly to infinity with an asymptotic finite ratio. Using some powerful results from the RMT, closed-form expressions of the BCRB are derived and studied. Finally, we propose a framework to fairly compare two models corrupted by noises with different degrees of freedom for a fixed common target signal-to-noise ratio (SNR). In particular, we focus our effort on the comparison of the BCRBs associated with two models corrupted by a sparse noise promoting outliers and a dense (Gaussian) noise, respectively

Semiparametric Inference and Lower Bounds for Real Elliptically Symmetric Distributions

This paper has a twofold goal. The first aim is to provide a deeper understanding of the family of the Real Elliptically Symmetric (RES) distributions by investigating their intrinsic semiparametric nature. The second aim is to derive a semiparametric lower bound for the estimation of the parametric component of the model. The RES distributions represent a semiparametric model where the parametric part is given by the mean vector and by the scatter matrix while the non-parametric, infinite-dimensional, part is represented by the density generator. Since, in practical applications, we are often interested only in the estimation of the parametric component, the density generator can be considered as nuisance. The first part of the paper is dedicated to conveniently place the RES distributions in the framework of the semiparametric group models. The second part of the paper, building on the mathematical tools previously introduced, the Constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB) for the estimation of the mean vector and of the constrained scatter matrix of a RES distributed random vector is introduced. The CSCRB provides a lower bound on the Mean Squared Error (MSE) of any robust $M$-estimator of mean vector and scatter matrix when no a-priori information on the density generator is available. A closed form expression for the CSCRB is derived. Finally, in simulations, we assess the statistical efficiency of the Tyler's and Huber's scatter matrix $M$-estimators with respect to the CSCRB.Comment: This paper has been accepted for publication in IEEE Transactions on Signal Processin