In a linear mean regression setting with repeated measurement errors, we develop asymptotic properties of a naive estimator to better clarify the effects of these errors. We then construct a group of unbiased estimating equations with independent repetitions and make use of these equations in two ways to obtain two estimators: a weighted averaging estimator and an estimator based on the generalized method of moments. The proposed estimators do not require any additional information about the measurement errors. We also prove the consistency and asymptotic normality of the two estimators. Our theoretical results are verified by simulation studies and a real data analysis.

In this article, we propose geometric vitality function introduced by Nair and Rajesh (IAPQR Trans 25(1):1–8, 2000) for the past lifetime of a random variable. This measure plays a vital role in analysing different characteristics of a system/component when it fails in the interval (0, t). The monotonic behaviour and some ordering properties in terms of the proposed measure were studied under certain conditions. Similar properties of the proposed measure were analysed for order statistics as well. Further, bounds were obtained for the past geometric vitality function of order statistics. Apart from this, characterizations of some lifetime probability distributions with respect to order statistics were also discussed.

In recent years, big datasets are often split into several subsets due to the storage requirements. We propose a parallel group Bayesian method for statistical inference in sparse big data. This method improves the existing methods in two aspects: the total datasets are also split into a data subset sequence and the parameter vector is divided into several sub-vectors. Besides, we add a weight sequence to optimize the sub-estimators when each of them has a different covariance matrix. We obtain several theoretical properties of the estimator. The results of numerical simulations show that our method is consistent with the theoretical results and is more effective than classic Markov chain Monte Carlo methods.

We discuss a new weighted likelihood method for robust parametric estimation. The method is motivated by the need for generating a simple estimation strategy which provides a robust solution that is simultaneously fully efficient when the model is correctly specified. This is achieved by appropriately weighting the score function at each observation in the maximum likelihood score equation. The weight function determines the compatibility of each observation with the model in relation to the remaining observations and applies a downweighting only if it is necessary, rather than automatically downweighting a proportion of the observations all the time. This allows the estimators to retain full asymptotic efficiency at the model. We establish all the theoretical properties of the proposed estimators and substantiate the theory developed through simulation and real data examples. Our approach provides an alternative to the weighted likelihood method of Markatou et al. (J Stat Plan Inference 57(2):215–232, 1997; J Am Stat Assoc 93(442):740–750, 1998).

For the special case of balanced one-way random effects ANOVA, it has been established that the generalized likelihood ratio test (LRT) and Wald’s test are largely equivalent in testing the variance component. We extend these results to explore the relationships between Wald’s F test, and the LRT for a much broader class of linear mixed models; the generalized split-plot models. In particular, we explore when the two tests are equivalent and prove that when they are not equivalent, Wald’s F test is more powerful, thus making the LRT test inadmissible. We show that inadmissibility arises in realistic situations with common number of degrees of freedom. Further, we derive the statistical distribution of the LRT under both the null and alternative hypotheses $$H_0$$ and $$H_1$$ where $$H_0$$ is the hypothesis that the between variance component is zero. Providing an exact distribution of the test statistic for the LRT in these models will help in calculating a more accurate p-value than the traditionally used p-value derived from the large sample chi-square mixture approximations.

In this paper, first-order response surface model incorporating neighbour effects from experimental units up to distance 2 has been discussed where experimental plots experience the neighbour effects from immediate left and right neighbouring units as well as from units at distance 2 assuming the plots are placed linearly without gaps. Conditions have been derived for the orthogonal estimation of the parameters of the model. A method of constructing designs for fitting first-order response surface model which ensures the constancy in the prediction variance after incorporating neighbor effects up to distance 2 has been developed and named as FORDDNE-2.

The gamma model is a generalized linear model for gamma-distributed outcomes. The model is widely applied in psychology, ecology or medicine. Recently, Gaffke et al. (J Stat Plan Inference 203:199–214, 2019) established a complete class and an essentially complete class of designs for gamma models to obtain locally optimal designs in particular when the linear predictor includes an intercept term. In this paper we extend this approach to gamma models having linear predictors without intercept. For a specific scenario sets of locally D- and A-optimal designs are established. It turns out that the optimality problem can be transformed to one under gamma models with intercept leading to a reduction in the dimension of the experimental region. On that basis optimality results can be transferred from one model to the other and vice versa. Additionally by means of the general equivalence theorem optimality can be characterized for multiple regression by a system of polynomial inequalities which can be solved analytically or by computer algebra. Thus necessary and sufficient conditions can be obtained on the parameter values for the local D-optimality of specific designs. The robustness of the derived designs with respect to misspecification of the initial parameter values is examined by means of their local D-efficiencies.

Binary time series can be derived from an underlying latent process. In this paper, we consider an ellipsoidal alpha mixing strictly stationary process and discuss the discriminant analysis and propose a classification method based on binary time series. Assume that the observations are generated by time series which belongs to one of two categories described by different spectra. We propose a method to classify into the correct category with high probability. First, we will show that the misclassification probability tends to zero when the number of observation tends to infinity, that is, the consistency of our discrimination method. Further, we evaluate the asymptotic misclassification probability when the two categories are contiguous. Finally, we show that our classification method based on binary time series has good robustness properties when the process is contaminated by an outlier, that is, our classification method is insensitive to the outlier. However, the classical method based on smoothed periodogram is sensitive to outliers. We also deal with a practical case where the two categories are estimated from the training samples. For an electrocardiogram data set, we examine the robustness of our method when observations are contaminated with an outlier.

A multi-sample set-up of sequential order statistics from Weibull distribution functions with known scale parameters and a common unknown shape parameter is considered. The respective likelihood equation may have multiple roots even in the single-sample case, which is demonstrated by a simple example and illustrated with a simulation study. Uniqueness of the root of the likelihood equation and of the maximum likelihood estimator is examined with respect to different models of ordered data, sufficient conditions for uniqueness are shown, and the distribution of the number of roots of the likelihood equation is seen to be independent of the unknown shape parameter.

Fractional factorial designs are widely used because of their various merits. Foldover or level permutation are usually used to construct optimal fractional factorial designs. In this paper, a novel method via foldover and level permutation, called quadrupling, is proposed to construct uniform four-level designs with large run sizes. The relationship of uniformity between the initial design and the design obtained by quadrupling is investigated, and new lower bounds of wrap-around $$L_2$$-discrepancy for such designs are obtained. These results provide a theoretical basis for constructing uniform four-level designs with large run sizes by quadrupling successively. Furthermore, the analytic connection between the initial design and the design obtained by quadrupling is presented under generalized minimum aberration criterion.