Generalized method of moments for an extended gamma process

In reliability theory, a widely used process to model the phenomena of the cumulative deterioration of a system over time is the standard gamma process (SGP). Based on several restrictions, such as a constant variance-to-mean ratio, this process is not always a suitable choice to describe the deterioration. A way to overcome these restrictions is to use an extended version of the gamma process introduced by Cinlar (1980), which is characterized by shape and scale functions. In this article, the aim is to propose statistical methods to estimate the unknown parameters of parametric forms of the shape and scale functions. We here develop two generalized methods of moments (Hansen 1982 Hansen, L. P. 1982. Large sample properties of generalized method of moments estimators. Econometrica 50 (4):1029–54.[Crossref], [Web of Science ®] , [Google Scholar]), based either on the moments or on the Laplace transform of an extended gamma process. Asymptotic properties are provided and a Wald-type test is derived, which allows to test SGPs against extended ones with a specific parametric shape function. Also, the performance of the proposed estimation methods is illustrated on simulated and real data.     

A robust posterior preference multi-response optimization approach in multistage processes

Nowadays, many manufacturing and service systems provide products and services to their customers in several consecutive stages of operations, in each of which one or more quality characteristics of interest are monitored. In these environments, the final quality in the last stage not only depends on the quality of the task performed in that stage but also is dependent on the quality of the products and services in intermediate stages as well as the design parameters in each stage. In this paper, a novel methodology based on the posterior preference approach is proposed to robustly optimize these multistage processes. In this methodology, a multi-response surface optimization problem is solved in order to find preferred solutions among different non dominated solutions (NDSs) according to decision maker's preference. In addition, as the intermediate response variables (quality characteristics) may act as covariates in the next stages, a robust multi-response estimation method is applied to extract the relationships between the outputs and inputs of each stage. NDSs are generated by the ?-constraint method. The robust preferred solutions are selected considering some newly defined conformance criteria. The applicability of the proposed approach is illustrated by a numerical example at the end.     

Bayesian inference of three-parameter bathtub-shaped lifetime distribution

We investigate a Bayesian inference in the three-parameter bathtub-shaped lifetime distribution which is obtained by adding a power parameter to the two-parameter bathtub-shaped lifetime distribution suggested by Chen (2000). The Bayes estimators under the balanced squared error loss function are derived for three parameters. Then, we have used Lindley's and Tierney–Kadane approximations (see Lindley 1980; Tierney and Kadane 1986) for computing these Bayes estimators. In particular, we propose the explicit form of Lindley's approximation for the model with three parameters. We also give applications with a simulated data set and two real data sets to show the use of discussed computing methods. Finally, concluding remarks are mentioned.     

Measuring Nonlinear Granger Causality in Mean

We propose model-free measures for Granger causality in mean between random variables. Unlike the existing measures, ours are able to detect and quantify nonlinear causal effects. The new measures are based on nonparametric regressions and defined as logarithmic functions of restricted and unrestricted mean square forecast errors. They are easily and consistently estimated by replacing the unknown mean square forecast errors by their nonparametric kernel estimates. We derive the asymptotic normality of nonparametric estimator of causality measures, which we use to build tests for their statistical significance. We establish the validity of smoothed local bootstrap that one can use in finite sample settings to perform statistical tests. Monte Carlo simulations reveal that the proposed test has good finite sample size and power properties for a variety of data-generating processes and different sample sizes. Finally, the empirical importance of measuring nonlinear causality in mean is also illustrated. We quantify the degree of nonlinear predictability of equity risk premium using variance risk premium. Our empirical results show that the variance risk premium is a very good predictor of risk premium at horizons less than 6 months. We also find that there is a high degree of predictability at the 1-month horizon, that can be attributed to a nonlinear causal effect. Supplementary materials for this article are available online.     

WLAD-LASSO method for robust estimation and variable selection in partially linear models

This paper focuses on robust estimation and variable selection for partially linear models. We combine the weighted least absolute deviation (WLAD) regression with the adaptive least absolute shrinkage and selection operator (LASSO) to achieve simultaneous robust estimation and variable selection for partially linear models. Compared with the LAD-LASSO method, the WLAD-LASSO method will resist to the heavy-tailed errors and outliers in the parametric components. In addition, we estimate the unknown smooth function by a robust local linear regression. Under some regular conditions, the theoretical properties of the proposed estimators are established. We further examine finite-sample performance of the proposed procedure by simulation studies and a real data example.     

A bivariate distribution with Lomax and geometric margins

We develop a stochastic model describing the joint distribution of $(X,N)$, where $N$ has a geometric distribution while $X$ is the sum of $N$ dependent, heavy-tail Pareto components. Models of this form arise in many applications, ranging from hydro-climatology to finance and insurance. We present fundamental properties of this vector, including marginal and conditional distributions, moments, representations, and parameter estimation. We also include an example from finance, illustrating modeling potential of this new bivariate distribution.     

Estimating the Conditional Error Distribution in Non‐parametric Regression

Abstract. We consider a general non‐parametric regression model, where the distribution of the error, given the covariate, is modelled by a conditional distribution function. For the estimation, a kernel approach as well as the (kernel based) empirical likelihood method are discussed. The latter method allows for incorporation of additional information on the error distribution into the estimation. We show weak convergence of the corresponding empirical processes to Gaussian processes and compare both approaches in asymptotic theory and by means of a simulation study.     

Sample size determination for estimating multivariate process capability indices based on lower confidence limits

With the advent of modern technology, manufacturing processes have become very sophisticated; a single quality characteristic can no longer reflect a product's quality. In order to establish performance measures for evaluating the capability of a multivariate manufacturing process, several new multivariate capability (NMC) indices, such as NMC _p and NMC _pm , have been developed over the past few years. However, the sample size determination for multivariate process capability indices has not been thoroughly considered in previous studies. Generally, the larger the sample size, the more accurate an estimation will be. However, too large a sample size may result in excessive costs. Hence, the trade-off between sample size and precision in estimation is a critical issue. In this paper, the lower confidence limits of NMC _p and NMC _pm indices are used to determine the appropriate sample size. Moreover, a procedure for conducting the multivariate process capability study is provided. Finally, two numerical examples are given to demonstrate that the proper determination of sample size for multivariate process indices can achieve a good balance between sampling costs and estimation precision.     

Non Asymptotic Minimax Estimation of Functionals with Noisy Observations

The minimax estimation of functionals by a finite number of noisy observations is considered. A new way to formalize the problem that enables one to calculate non asymptotic optimal estimates is proposed. The calculations can be turned into and executed as a computer algorithm or carried out analytically under week assumptions on random variables. Some examples are considered.     

On Some Applications of Poisson Point Processes

Poisson point processes play important role in various domains of Probability Theory and Mathematical Statistics. In this article, we investigate only two applications of Poisson point processes: a generated white noise problem and parameters estimation problem. This work continues the investigations started in paper Egorov and Kondybaev (2009 Egorov , V. A. , Kondybaev , N. S. ( 2009 ). On the estimation of a signal covered by background om Poisson noise . Methods and programs of data processing 4 : 75 – 81 . [Google Scholar]).