Efficiency of ranked set sampling in tests for normality

In this article, we consider the ranked set sampling (RSS) and investigate seven tests for normality under RSS. Each test is described and then power of each test is obtained by Monte Carlo simulations under various alternatives. Finally, the powers of the tests based on RSS are compared with the powers of the tests based on the simple random sampling and the results are discussed.     

Some contributions to sequential Monte Carlo methods for option pricing

Pricing options is an important problem in financial engineering. In many scenarios of practical interest, financial option prices associated with an underlying asset reduces to computing an expectation w.r.t. a diffusion process. In general, these expectations cannot be calculated analytically, and one way to approximate these quantities is via the Monte Carlo (MC) method; MC methods have been used to price options since at least the 1970s. It has been seen in Del Moral P, Shevchenko PV. [Valuation of barrier options using sequential Monte Carlo. 2014. arXiv preprint] and Jasra A, Del Moral P. [Sequential Monte Carlo methods for option pricing. Stoch Anal Appl. 2011;29:292–316] that Sequential Monte Carlo (SMC) methods are a natural tool to apply in this context and can vastly improve over standard MC. In this article, in a similar spirit to Del Moral and Shevchenko (2014) and Jasra and Del Moral (2011), we show that one can achieve significant gains by using SMC methods by constructing a sequence of artificial target densities over time. In particular, we approximate the optimal importance sampling distribution in the SMC algorithm by using a sequence of weighting functions. This is demonstrated on two examples, barrier options and target accrual redemption notes (TARNs). We also provide a proof of unbiasedness of our SMC estimate.     

Shrinkage estimation in lognormal regression model for censored data

We introduce in this paper, the shrinkage estimation method in the lognormal regression model for censored data involving many predictors, some of which may not have any influence on the response of interest. We develop the asymptotic properties of the shrinkage estimators (SEs) using the notion of asymptotic distributional biases and risks. We show that if the shrinkage dimension exceeds two, the asymptotic risk of the SEs is strictly less than the corresponding classical estimators. Furthermore, we study the penalty (LASSO and adaptive LASSO) estimation methods and compare their relative performance with the SEs. A simulation study for various combinations of the inactive predictors and censoring percentages shows that the SEs perform better than the penalty estimators in certain parts of the parameter space, especially when there are many inactive predictors in the model. It also shows that the shrinkage and penalty estimators outperform the classical estimators. A real-life data example using Worcester heart attack study is used to illustrate the performance of the suggested estimators.     

Critical values improvement for the standard normal homogeneity test by combining Monte Carlo and regression approaches

The distribution of the test statistics of homogeneity tests is often unknown, requiring the estimation of the critical values through Monte Carlo (MC) simulations. The computation of the critical values at low α, especially when the distribution of the statistics changes with the series length (sample cardinality), requires a considerable number of simulations to achieve a reasonable precision of the estimates (i.e. 10⁶ simulations or more for each series length). If, in addition, the test requires a noteworthy computational effort, the estimation of the critical values may need unacceptably long runtimes.

To overcome the problem, the paper proposes a regression-based refinement of an initial MC estimate of the critical values, also allowing an approximation of the achieved improvement. Moreover, the paper presents an application of the method to two tests: SNHT (standard normal homogeneity test, widely used in climatology), and SNH2T (a version of SNHT showing a squared numerical complexity). For both, the paper reports the critical values for α ranging between 0.1 and 0.0001 (useful for the p-value estimation), and the series length ranging from 10 (widely adopted size in climatological change-point detection literature) to 70,000 elements (nearly the length of a daily data time series 200 years long), estimated with coefficients of variation within 0.22%. For SNHT, a comparison of our results with approximated, theoretically derived, critical values is also performed; we suggest adopting those values for the series exceeding 70,000 elements.     

Discriminant analysis under the common principal components model

For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015 Pepler, P. T., Uys, D. W. and Nel, D. G. (2015). Regularised covariance matrix estimation under the common principal components model. Communications in Statistics: Simulation and Computation. (In press). [Google Scholar]) proposed a regularized CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations, where the CPC model is applicable. This article extends their work to the context of discriminant analysis for two groups, by plugging the regularized CPC estimator into the ordinary quadratic discriminant function. Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations, where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures.     

A hierarchical Bayesian model for binary data incorporating selection bias

We consider a Bayesian nonignorable model to accommodate a nonignorable selection mechanism for predicting small area proportions. Our main objective is to extend a model on selection bias in a previously published paper, coauthored by four authors, to accommodate small areas. These authors assume that the survey weights (or their reciprocals that we also call selection probabilities) are available, but there is no simple relation between the binary responses and the selection probabilities. To capture the nonignorable selection bias within each area, they assume that the binary responses and the selection probabilities are correlated. To accommodate the small areas, we extend their model to a hierarchical Bayesian nonignorable model and we use Markov chain Monte Carlo methods to fit it. We illustrate our methodology using a numerical example obtained from data on activity limitation in the U.S. National Health Interview Survey. We also perform a simulation study to assess the effect of the correlation between the binary responses and the selection probabilities.     

Testing the validity of the logistic model based on the empirical distribution function

The logistic distribution is one of the fundamental distribution and is widely used for describing model growth curves in survival analysis and biological studies. Applications of this distribution are presented in statistical literature. In this article, goodness of fit tests for the logistic distribution based on the empirical distribution function (EDF) are considered. In order to compute the test statistics, because the MLEs cannot be obtained explicitly, we use the approximate maximum likelihood estimates (AMLEs) suggested by Balakrishnan and Cohen (1990 Balakrishnan, N., Cohen, A. C. (1990). Order Statistics and Inference: Estimation Methods. Boston: Academic Press. [Google Scholar]), which are simple explicit estimators. Power comparisons of the considered tests are carried out via simulations. Finally, two illustrative examples are presented and analyzed.     

Robust generalized confidence intervals

Most interval estimates are derived from computable conditional distributions conditional on the data. In this article, we call the random variables having such conditional distributions confidence distribution variables and define their finite-sample breakdown values. Based on this, the definition of breakdown value of confidence intervals is introduced, which covers the breakdowns in both the coverage probability and interval length. High-breakdown confidence intervals are constructed by the structural method in location-scale families. Simulation results are presented to compare the traditional confidence intervals and their robust analogues.     

Extended zero-one inflated beta and adjusted three-part regression models for proportional data analysis

Extended zero-one inflated beta and adjusted three-part regression models are introduced to analyze proportional response data where there are nonzero probabilities that the response variable takes the values zero and one. The proposed models adapt skewness and heteroscedasticity of the fractional response data, and are constructed to estimate the unknown parameters. Extensive Monte Carlo simulation studies are used to compare the performance of the two approaches with respect to bias and root mean square error. A real data example is presented to illustrate the application of both regression models.     

Bayesian analysis of multivariate threshold autoregressive models with missing data

In some fields, we are forced to work with missing data in multivariate time series. Unfortunately, the data analysis in this context cannot be carried out in the same way as in the case of complete data. To deal with this problem, a Bayesian analysis of multivariate threshold autoregressive models with exogenous inputs and missing data is carried out. In this paper, Markov chain Monte Carlo methods are used to obtain samples from the involved posterior distributions, including threshold values and missing data. In order to identify autoregressive orders, we adapt the Bayesian variable selection method in this class of multivariate process. The number of regimes is estimated using marginal likelihood or product parameter-space strategies.