Inferences on stress–strength parameter based on GLD5 distribution

Let X and Y be independent random variables distributed as generalized Lindley distribution type 5 (GLD5). This article deals with the estimation of the stress–strength parameter R = P(Y < X), which plays an important role in reliability analysis. For this purpose, the maximum likelihood and the uniformly minimum variance unbiased estimators are presented in the explicit form. Moreover, considering Arnold and Strauss’ bivariate Gamma distribution as an informative prior and Jeffreys’ as noninformative prior, the Bayes estimators are derived. Various bootstrap confidence intervals are also proposed and, finally, the presented methods are compared using a simulation study.     

Constants on a Uniform Berry-Esseen Bound on Some Borel Sets in ℝ k via Stein's Method

For each n, k ∈ ?, let Y _i  = (Y _i1, Y _i2,…, Y _ik ), 1 ≤ i ≤ n be independent random vectors in ? ^k with finite third moments and Y _ij are independent for all j = 1, 2,…, k. In this article, we use the Stein's technique to find constants in uniform bounds for multidimensional Berry-Esseen inequality on a closed sphere, a half plane and a rectangular set.     

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 Note on “Shaved Dice” Inference

Two dice are rolled repeatedly, only their sum is registered. Have the two dice been “shaved,” so two of the six sides appear more frequently? Pavlides and Perlman discussed this somewhat complicated type of situation through curved exponential families. Here, we contrast their approach by regarding data as incomplete data from a simple exponential family. The latter, supplementary approach is in some respects simpler, it provides additional insight about the relationships among the likelihood equation, the Fisher information, and the EM algorithm, and it illustrates the information content in ancillary statistics.     

A note on Morris’ bound for search designs

A lower bound is given for the number of experimental runs required in search designs for two-level orthogonal array of strength one.     

Goodness-of-fit tests based on Verma Kullback–Leibler information

In this article, a new consistent estimator of Veram’s entropy is introduced. We establish the entropy test based on the new information namely Verma Kullback–Leibler discrimination methodology. The results are used to introduce goodness-of-fit tests for normal and exponential distributions. The root of mean square errors, critical values, and powers for some alternatives are obtained by simulation. The proposed test is compared with other tests.     

Some contributions on the multivariate Poisson–Skellam probability distribution

In this article, we introduce a new form of distribution whose components have the Poisson or Skellam marginal distributions. This new specification allows the incorporation of relevant information on the nature of the correlations between every component. In addition, we present some properties of this distribution. Unlike the multivariate Poisson distribution, it can handle variables with positive and negative correlations. It should be noted that we are only interested in modeling covariances of order 2, which means between all pairs of variables. Some simulations are presented to illustrate the estimation methods. Finally, an application of soccer teams data will highlight the relationship between number of points per season and the goal differential by some covariates.     

A shift parameter estimation based on smoothed Kolmogorov–Smirnov

A new procedure of shift parameter estimation in the two-sample location problem is investigated and compared with existing estimators. The proposed procedure smooths the empirical distribution functions of each random sample and replaces empirical distribution functions in the two-sample Kolmogorov–Smirnov method. The smoothed Kolmogorov–Smirnov is minimized with respect to an arbitrary shift variable in order to find an estimate of the shift parameter. The proposed procedure can be considered the smoothed version of a very little known method of shift parameter estimation from Rao-Schuster-Littell (RSL) [Rao et al., Estimation of shift and center of symmetry based on Kolmogorov–Smirnov statistics, Ann. Stat. 3(4) (1975), pp. 862–873]. Their estimator will be discussed and compared with the proposed estimator in this paper. An example and simulation studies have been performed to compare the proposed procedure with existing shift parameter estimators such as Hodges–Lehmann (H–L) and least squares in addition to RSL's estimator. The results show that the proposed estimator has lower mean-squared error as well as higher relative efficiency against RSL's estimator under normal or contaminated normal model assumptions. Moreover, the proposed estimator performs competitively against H–L and least-squares shift estimators. Smoother function and bandwidth selections are also discussed and several alternatives are proposed in the study.     

Comment on “recent developments in bootstrapping time series”

The paper develops a general framework for identification, estimation, and hypothesis testing in cointegrated systems when the cointegrating coefficients are subject to (possibly) non-linear and cross-equation restrictions, obtained from economic theory or other relevant a priori information. It provides a proof of the consistency of the quasi maximum likelihood estimators (QMLE), establishes the relative rates of convergence of the QMLE of the short-run and the long-run parameters, and derives their asymptotic distributions; thus generalizing the results already available in the literature for the linear case. The paper also develops tests of the over-identifying (possibly) non-linear restrictions on the cointegrating vectors. The estimation and hypothesis testing procedures are applied to an Almost Ideal Demand System estimated on U.K. quarterly observations. Unlike many other studies of consumer demand this application does not treat relative prices and real per capita expenditures as exogenously given.     

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.     

Robust multivariate control charts based on Birnbaum–Saunders distributions

Multivariate control charts are powerful and simple visual tools for monitoring the quality of a process. This multivariate monitoring is carried out by considering simultaneously several correlated quality characteristics and by determining whether these characteristics are in control or out of control. In this paper, we propose a robust methodology using multivariate quality control charts for subgroups based on generalized Birnbaum–Saunders distributions and an adapted Hotelling statistic. This methodology is constructed for Phases I and II of control charts. We estimate the corresponding parameters with the maximum likelihood method and use parametric bootstrapping to obtain the distribution of the adapted Hotelling statistic. In addition, we consider the Mahalanobis distance to detect multivariate outliers and use it to assess the adequacy of the distributional assumption. A Monte Carlo simulation study is conducted to evaluate the proposed methodology and to compare it with a standard methodology. This study reports the good performance of our methodology. An illustration with real-world air quality data of Santiago, Chile, is provided. This illustration shows that the methodology is useful for alerting early episodes of extreme air pollution, thus preventing adverse effects on human health.     

Simulation Study on Improved Shapiro–Wilk Tests for Normality

The article concerns tests for normality based on the Shapiro–Wilk W statistic. The constants in the test statistic are recalculated as those given in Shapiro and Wilk are incorrect. The empirical significance levels and power of improved tests have been evaluated in simulation study and compared to original ones. The improved tests were also applied to the multivariate case. In this case, we consider two implementations of the W statistic, the first one proposed by Srivastava and Hui and the other by Hanusz and Tarasinska. Empirical size of tests and their power have been compared to the Henze–Zirkler test.