Tolerance Limits Based on the Multivariate MaxMin Chart

In a previous paper, we have showed how to obtain sequences of number proved random. With this aim, we used sequences of noises y_n such that the conditional probabilities have Lipschitz coefficients not too large. We transformed them using Fibonacci congruences. Then, we obtained sequences x_n which admit the IID model for correct model. This method consisted to value the work of Marsaglia in order to build his CD-ROM. But we did not use Rap Music (as Marsaglia), but texts files. This method also uses an extractor and at the same time the notion of correct models. In this paper, we apply this method to numbers provided by machines or chips. Unfortunately, it is less sure than they have Lipschtiz coefficient not too large. But we can solve this problem: it suffices to use the Central Limit Theorem. We do it modulo 1. In this case, we use a new limit theorem, the XOR Limit theorem : asymptotic distribution of sum of random vectors modulo 1 are asymptotically independent. Then Lipschtiz coefficient of associated sequences are not too large and we can obtain IID sequences by using Fibonacci congruences.     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

Copulas Constructed from Horizontal Sections

In analogy with the study of copulas whose diagonal sections have been fixed, we study the set _h of copulas for which a horizontal section h has been given. We first show that this set is not empty, by explicitly writing one such copula, which we call horizontal copula. Then we find the copulas that bound both below and above the set _h. Finally, we determine the expressions for Kendall's tau and Spearman's rho for the horizontal and the bounding copulas.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Large deviation for negatively dependent random variables under sublinear expectation

Abstract

In this article, first we give the definition of negatively dependent sequence of random variables under sublinear expectation then we establish large deviation principle for this kind of sequence. Moreover, we obtain the upper bound of moderate deviation principle.     

Asymptotic distribution of least square estimators for linear models with dependent errors

In this paper, we consider the usual linear regression model in the case where the error process is assumed strictly stationary. We use a result from Hannan (Central limit theorems for time series regression. Probab Theory Relat Fields. 1973;26(2):157–170), who proved a Central Limit Theorem for the usual least squares estimator under general conditions on the design and on the error process. Whatever the design satisfying Hannan's conditions, we define an estimator of the covariance matrix and we prove its consistency under very mild conditions. As an application, we show how to modify the usual tests on the linear model in this dependent context, in such a way that the type-I error rate remains asymptotically correct, and we illustrate the performance of this procedure through different sets of simulations.     

Analysis of non-repairable cold-standby systems in Bayes theory

ABSTRACT

In this paper, we seek to analyse the reliability of k-out-of-n cold-standby system with components having Weibull time-to-failure distribution in view of Bayes theory. At first, we review the existing methods exhaustively and find that all these methods have not considered Bayes theory. Then we modify the simplest method and propose new methods based on Monte Carlo simulation. Next, we combine all the information to derive the posterior distribution of Weibull parameters. A robust and universal sample-based method is proposed according to the Monte Carlo Markov Chain method to draw the sample of parameters to obtain the Bayes estimate of reliability. The drawn samples are proved to be rather satisfactory. Conducting a simulation study to compare all the methods in terms of accuracy and computational time, we have presented some useful recommendations from the simulation results. These conclusions would provide insight on the application for k-out-of-n cold-standby system.     

On the property of multivariate generalized hyperbolic distribution and the Stein-type inequality

This note consists of two parts . In the first part, we provide a pedagogic review on the multivariate generalized hyperbolic (MGH) distribution. We show that this probability family is close under margining, conditioning, and linear transforms; however, such property does not hold for its subclasses. In the second part, we obtain the Stein-type inequality in the context of MGH distribution. Moreover, we apply the Stein-type inequality to prove a lower bound for Var[h(X)]. Particularly, we present examples when X belongs to some well-known subclasses in MGH family.     

Shrinkage and LASSO strategies in high-dimensional heteroscedastic models

ABSTRACT

In this paper, we consider the estimation problem of the parameter vector in the linear regression model with heteroscedastic errors. First, under heteroscedastic errors, we study the performance of shrinkage-type estimators and their performance as compared to theunrestricted and restricted least squares estimators. In order to accommodate the heteroscedastic structure, we generalize an identity which is useful in deriving the risk function. Thanks to the established identity, we prove that shrinkage estimators dominate the unrestricted estimator. Finally, we explore the performance of high-dimensional heteroscedastic regression estimator as compared to classical LASSO and shrinkage estimators.     

On tests detecting difference in means and variances simultaneously under normality

In this paper, we propose several tests for detecting difference in means and variances simultaneously between two populations under normality. First of all, we propose a likelihood ratio test. Then we obtain an expression of the likelihood ratio statistic by a product of two functions of random quantities, which can be used to test the two individual partial hypotheses for differences in means and variances. With those individual partial tests, we propose a union-intersection test. Also we consider two optimal tests by combining the p-values of the two individual partial tests. For obtaining null distributions, we apply the permutation principle with the Monte Carlo approach. Then we compare efficiency among the proposed tests with well-known ones through a simulation study. Finally, we discuss some interesting features related to the simultaneous tests and resampling methods as concluding remarks.     

Non-homogeneous Pólya-Aeppli process

Abstract

In this paper we suppose that the intensity parameter of the Pólya-Aeppli process is a function of time t and call the resulting process a non-homogeneous Pólya-Aeppli process (NHPAP). The NHPAP can be represented as a compound non-homogeneous Poisson process with geometric compounding distribution as well as a pure birth process. For this process we give two definitions and show their equivalence. Also, we derive some interesting properties of NHPAP and use simulation the illustrate the process for particular intensity functions. In addition, we introduce the standard risk model based on NHPAP, analyze the ruin probability for this model and include an example of the process under exponentially distributed claims.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

On bounds for the efficiency of block designs for comparing test treatments with a control

In this paper we study the class of augmented balanced incomplete block designs, which are used for comparing a control treatment with a set of test treatments. Under the A- criterion we establish a condition that enables us to determine the most efficient augmented design and we suggest some methods to compute a lower bound for the efficiency of these designs. For 3≤k≤10, v≥k, we list the parameters of the most efficient designs with a lower bound for their efficiency or, if known, mention their optimality.     

The analysis of MX/M/1 queue with two-stage vacations policy

Abstract

In this article, we consider a batch arrival M^X/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by M^X/M/1/SWV?+?MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace–Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples.     

Simulation of the p-generalized Gaussian distribution

In this paper, we introduce the p-generalized polar methods for the simulation of the p-generalized Gaussian distribution. On the basis of geometric measure representations, the well-known Box–Muller method and the Marsaglia–Bray rejecting polar method for the simulation of the Gaussian distribution are generalized to simulate the p-generalized Gaussian distribution, which fits much more flexibly to data than the Gaussian distribution and has already been applied in various fields of modern sciences. To prove the correctness of the p-generalized polar methods, we give stochastic representations, and to demonstrate their adequacy, we perform a comparison of six simulation techniques w.r.t. the goodness of fit and the complexity. The competing methods include adapted general methods and another special method. Furthermore, we prove stochastic representations for all the adapted methods.     

Nonlinear filters based on taylor series expansions ∗

The nonlinear filters based on Taylor series approximation are broadly used for computational simplicity, even though their filtering estimates are clearly biased. In this paper, first, we analyze what is approximated when we apply the expanded nonlinear functions to the standard linear recursive Kalman filter algorithm. Next, since the state variable α_t and α_t-t are approximated as a conditional normal distribution given information up to time t - 1 (i.e., I_t-1) in approximation of the Taylor series expansion, it might be appropriate to evaluate each expectation by generating normal random numbers of α_t and α_t-1 given I_t-1 and those of the error terms θ and η_t. Thus, we propose the Monte-Carlo simulation filter using normal random draws. Finally we perform two Monte-Carlo experiments, where we obtain the result that the Monte-Carlo simulation filter has a superior performance over the nonlinear filters such as the extended Kalman filter and the second-order nonlinear filter.     

A memory reduction method in pricing American options

This paper is concerned with the pricing of American options by simulation methods. In the traditional methods, in order to determine when to exercise, we have to store the simulated asset prices at all time steps on all paths. If N time steps and M paths are used, then the storage requirement is O(MN). In this paper, we present a simulation method for pricing American options where the number of storage required only grows like O(M). The only additional computational cost is that we have to generate each random number twice instead of once. For machines with limited memory, we can now use a larger N to improve the accuracy in pricing the options.     

Positive-rule stein-type almost unbiased ridge estimator in linear regression model

Abstract

In this article, when it is suspected that regression coefficients may be restricted to a subspace, we discuss the parameter estimation of regression coefficients in a multiple regression model. Then, in order to improve the preliminary test almost ridge estimator, we study the positive-rule Stein-type almost unbiased ridge estimator based on the positive-rule stein-type shrinkage estimator and almost unbiased ridge estimator. After that, quadratic bias and quadratic risk values of the new estimator are derived and compared with some relative estimators. And we also discuss the option of parameter k. Finally, we perform a real data example and a Monte Carlo study to illustrate theoretical results.     

Central limit theorems of range-based estimators for diffusion models

Abstract

In this article, we consider non parametric range-based estimation procedure for diffusion processes and propose a instantaneous volatility estimator. Under some weak conditions, we certify that the proposed estimator has convergence in probability. Adding some necessary conditions, we prove a central limit theorem. By inference, we reach a conclusion that, with high frequency data in hand, the proposed estimator is more precise than those pure realized instantaneous volatility ones. Numerical simulation illustrates the finite sample properties of the proposed estimator.     

Skewness Reduction Approach in Multi-Attribute Process Monitoring

Since the product quality of many industrial processes depends upon more than one dependent variable or attribute, they are either multivariate or multi-attribute in nature. Although multivariate statistical process control is receiving increased attention in the literature, little work has been done to deal with multi-attribute processes. In this article, we develop a new methodology to monitor multi-attribute processes. To do this, first we transform multi-attribute data in a way that their marginal probability distributions have almost zero skewness. Then, we estimate the transformed covariance matrix and apply the well-known T ² control chart. In order to illustrate the proposed method and evaluate its performance, we use two simulation experiments and compare the results with the ones from both MNP chart and the χ² control chart.