An improved method of estimation for the parameters of the Birnbaum–Saunders distribution

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution, frequently used for analysing lifetime data. In this paper, we propose a simple method of estimation for the parameters of the two-parameter BS distribution by making use of some key properties of the distribution. Compared with the maximum likelihood estimators and the modified moment estimators, the proposed method has smaller bias, but having the same mean square errors as these two estimators. We also discuss some methods of construction of confidence intervals. The performance of the estimators is then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.     

Value-at-risk forecasting based on Gaussian mixture ARMA–GARCH model

In this paper, we develop a new forecasting algorithm for value-at-risk (VaR) based on ARMA–GARCH (autoregressive moving average–generalized autoregressive conditional heteroskedastic) models whose innovations follow a Gaussian mixture distribution. For the parameter estimation, we employ the conditional least squares and quasi-maximum-likelihood estimator (QMLE) for ARMA and GARCH parameters, respectively. In particular, Gaussian mixture parameters are estimated based on the residuals obtained from the QMLE of GARCH parameters. Our algorithm provides a handy methodology, spending much less time in calculation than the existing resampling and bias-correction method developed in Hartz et al. [Accurate value-at-risk forecasting based on the normal-GARCH model, Comput. Stat. Data Anal. 50 (2006), pp. 3032–3052]. Through a simulation study and a real-data analysis, it is shown that our method provides an accurate VaR prediction.     

Extensions to the modified Gram–Schmidt strategy and its application in the steepest ascent method

The orthogonalization of undesigned experiments is introduced to increase statistical precision of the estimated regression coefficients. The goals are to minimize the covariance and the bias of the least squares estimator for estimating the path of the steepest ascent (SA) that leads the users toward the neighbour of the optimum response. An orthogonal design is established to decrease the inverse determinant of X′X and the angle between the true and the estimated SA paths. For orthogonalization of an undesigned matrix, our proposed solution is constructed on the modified Gram–Schmidt strategy relevant to the process of Gaussian elimination. The proposed solution offers an orthogonal basis, in full working accuracy, for the space spanned by the columns of the original matrix.     

A note on scheffé's method of deriving confidence sets for directions and ratios of normals

Scheffé (1970) introduced a method for deriving confidence sets for directions and ratios of normals. The procedure requires use of an approximation and Scheffé provided evidence that the method performs well for cases in which the variances of the random deviates are known. This paper extends Scheffé's numerical integrations to the case of unknown variances. Our results indicate that Scheffé's method works well when variances are unknown     

On a nonlinear Birnbaum–Saunders model based on a bivariate construction and its characteristics

Abstract

The Birnbaum-Saunders (BS) distribution is an asymmetric probability model that is receiving considerable attention. In this article, we propose a methodology based on a new class of BS models generated from the Student-t distribution. We obtain a recurrence relationship for a BS distribution based on a nonlinear skew–t distribution. Model parameters estimators are obtained by means of the maximum likelihood method, which are evaluated by Monte Carlo simulations. We illustrate the obtained results by analyzing two real data sets. These data analyses allow the adequacy of the proposed model to be shown and discussed by applying model selection tools.     

A new estimator of Kullback–Leibler information and its application in goodness of fit tests*

We propose here a general statistic for the goodness of fit test of statistical distributions. The proposed statistic is constructed based on an estimate of Kullback–Leibler information. The proposed test is consistent and the limiting distribution of the test statistic is derived. Then, the established results are used to introduce goodness of fit tests for the normal, exponential, Laplace and Weibull distributions. A simulation study is carried out for examining the power of the proposed test and to compare it with those of some existing procedures. Finally, some illustrative examples are presented and analysed, and concluding comments are made.     

An application of Lévy's inversion formula for higher order asymptotic expansion

The asymptotic expansions for the distribution of statistics are, in general, given by applying Lévy's inversion formula to the characteristic function. This paper shows an inversion formula for higher order asymptotic expansion of the distribution of a scalar valued function which contains dependent statistics. The usage of the formula is illustrated by derivation of third order asymptotic expansion of the distribution of Hotelling's T²-statistic under the elliptical distribution as an example.     

Ridge Regression and Generalized Maximum Entropy: An improved version of the Ridge–GME parameter estimator

In this article, the Ridge–GME parameter estimator, which combines Ridge Regression and Generalized Maximum Entropy, is improved in order to eliminate the subjectivity in the analysis of the ridge trace. A serious concern with the visual inspection of the ridge trace to define the supports for the parameters in the Ridge–GME parameter estimator is the misinterpretation of some ridge traces, in particular where some of them are very close to the axes. A simulation study and two empirical applications are used to illustrate the performance of the improved estimator. A MATLAB code is provided as supplementary material.     

Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ² (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1?α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1?α. When p→∞ this value of τ increases as O(p^1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations.     

L-moments of the Birnbaum–Saunders distribution and its extreme value version: estimation,goodness of fit and application to earthquake data

Understanding patterns in the frequency of extreme natural events, such as earthquakes, is important as it helps in the prediction of their future occurrence and hence provides better civil protection. Distributions describing these events are known to be heavy tailed and positive skew making standard distributions unsuitable for modelling the frequency of such events. The Birnbaum–Saunders distribution and its extreme value version have been widely studied and applied due to their attractive properties. We derive L-moment equations for these distributions and propose novel methods for parameter estimation, goodness-of-fit assessment and model selection. A simulation study is conducted to evaluate the performance of the L-moment estimators, which is compared to that of the maximum likelihood estimators, demonstrating the superiority of the proposed methods. To illustrate these methods in a practical application, a data analysis of real-world earthquake magnitudes, obtained from the global centroid moment tensor catalogue during 1962–2015, is carried out. This application identifies the extreme value Birnbaum–Saunders distribution as a better model than classic extreme value distributions for describing seismic events.     

Norm-preserving constraint in the Fisher–Rao registration and its application in signal estimation

Registration of temporal observations is a fundamental problem in functional data analysis. Various frameworks have been developed over the past two decades where registrations are conducted based on optimal time warping between functions. Comparison of functions solely based on time warping, however, may have limited application, in particular when certain constraints are desired in the registration. In this paper, we study registration with norm-preserving constraint. A closely related problem is on signal estimation, where the goal is to estimate the ground-truth template given random observations with both compositional and additive noises. We propose to adopt the Fisher–Rao framework to compute the underlying template, and mathematically prove that such framework leads to a consistent estimator. We then illustrate the constrained Fisher–Rao registration using simulations as well as two real data sets. It is found that the constrained method is robust with respect to additive noise and has superior alignment and classification performance to conventional, unconstrained registration methods.     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.     

Robust variable selection for the varying coefficient model based on composite L 1–L 2 regression

The varying coefficient model (VCM) is an important generalization of the linear regression model and many existing estimation procedures for VCM were built on L ₂ loss, which is popular for its mathematical beauty but is not robust to non-normal errors and outliers. In this paper, we address the problem of both robustness and efficiency of estimation and variable selection procedure based on the convex combined loss of L ₁ and L ₂ instead of only quadratic loss for VCM. By using local linear modeling method, the asymptotic normality of estimation is driven and a useful selection method is proposed for the weight of composite L ₁ and L ₂. Then the variable selection procedure is given by combining local kernel smoothing with adaptive group LASSO. With appropriate selection of tuning parameters by Bayesian information criterion (BIC) the theoretical properties of the new procedure, including consistency in variable selection and the oracle property in estimation, are established. The finite sample performance of the new method is investigated through simulation studies and the analysis of body fat data. Numerical studies show that the new method is better than or at least as well as the least square-based method in terms of both robustness and efficiency for variable selection.     

General treatment of goodness-of-fit tests based on Kullback–Leibler information

This paper introduces a general goodness-of-fit test based on the estimated Kullback–Leibler information. The test uses the Vasicek entropy estimate. Two special cases of the test for location–scale and shape families are discussed. The results are used to introduce goodness-of-fit tests for the uniform, Laplace, Weibull and beta distributions. The critical values and powers for some alternatives are obtained by simulation.     

Non-nested model selection based on the quantiles and it’s application in time series

We consider the problem of model selection based on quantile analysis and with unknown parameters estimated using quantile leasts squares. We propose a model selection test for the null hypothesis that the competing models are equivalent against the alternative hypothesis that one model is closer to the true model. We follow with two applications of the proposed model selection test. The first application is in model selection for time series with non-normal innovations. The second application is in model selection in the NoVas method, short for normalizing and variance stabilizing transformation, forecast. A set of simulation results also lends strong support to the results presented in the paper.     

On the Bahadur representation of sample quantiles for ψ-mixing sequences and its application

In this paper, the Bahadur representation of sample quantiles for ψ-mixing sequences is obtained under the given conditions. As its application, the uniformly asymptotic normality is derived.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

A class of one sample tests based on the Mann–Whitney–Wilcoxon functional

The main idea behind the proposed class of tests is rooted on an extension of the technique used in the derivation of the Mann–Whitney–Wilcoxon test. Just like the case of two-sample rank-based tests, the new class consists of tests defined through score functions. When properly selected, these score functions lead to consistent and often more powerful tests compared with classical goodness-of-fit tests. Theoretical results are supported by an extensive simulation study.     

On a generalization to Marshall–Olkin scheme and its application to Burr type XII distribution

Marshall–Olkin semi-Burr and Marshall–Olkin Burr distributions are introduced and studied. Their various characteristics in reliability analysis are derived. Applications in time series analysis are discussed.     

Kullback–Leibler information of a censored variable and its applications

In this paper, we study the Kullback–Leibler (KL) information of a censored variable, which we will simply call it censored KL information. The censored KL information is shown to have the necessary monotonicity property in addition to inherent properties of nonnegativity and characterization. We also present a representation of the censored KL information in terms of the relative risk and study its relation with the Fisher information in censored data. Finally, we evaluate the estimated censored KL information as a goodness-of-fit test statistic.