Truncation invariant copulas and a testing procedure

ABSTRACT

The class of bivariate copulas that are invariant under truncation with respect to one variable is considered. A simulation algorithm for the members of the class and a novel construction method are presented. Moreover, inspired by a stochastic interpretation of the members of such a class, a procedure is suggested to check whether the dependence structure of a given data set is truncation invariant. The overall performance of the procedure has been illustrated on both simulated and real data.     

A stochastic procedure to solve linear ill-posed problems

In this work, we propose a stochastic procedure of Robbins–Monro type to resolve linear inverse problems in Hilbert space. We study the probability of large deviation between the exact solution and the approximated one and build a confidence domain for the approximated solution while precising the rate of convergence. To check the validity of our work, we give a simulation application into a deconvolution problem.     

Assessing respondent-driven sampling: A simulation study across different networks

The purpose was to assess RDS estimators in populations simulated with diverse connectivity characteristics, incorporating the putative influence of misreported degrees and transmission processes. Four populations were simulated using different random graph models. Each population was “infected” using four different transmission processes. From each combination of population x transmission, one thousand samples were obtained using a RDS-like sampling strategy. Three estimators were used to predict the population-level prevalence of the “infection”. Several types of misreported degrees were simulated. Also, samples were generated using the standard random sampling method and the respective prevalence estimates, using the classical frequentist estimator. Estimation biases in relation to population parameters were assessed, as well as the variance. Variability was associated with the connectivity characteristics of each simulated population. Clustered populations yield greater variability and no RDS-based strategy could address the estimation biases. Misreporting degrees had modest effects, especially when RDS estimators were used. The best results for RDS-based samples were observed when the “infection” was randomly attributed, without any relation with the underlying network structure.     

A non-iterative posterior sampling algorithm for linear quantile regression model

In this article, a non-iterative posterior sampling algorithm for linear quantile regression model based on the asymmetric Laplace distribution is proposed. The algorithm combines the inverse Bayes formulae, sampling/importance resampling, and the expectation maximization algorithm to obtain independently and identically distributed samples approximately from the observed posterior distribution, which eliminates the convergence problems in the iterative Gibbs sampling and overcomes the difficulty in evaluating the standard deviance in the EM algorithm. The numeric results in simulations and application to the classical Engel data show that the non-iterative sampling algorithm is more effective than the Gibbs sampling and EM algorithm.     

Sampling schemes for price index construction: a performance comparison across the classification of individual consumption by purpose food groups

Five sampling schemes (SS) for price index construction – one cut-off sampling technique and four probability-proportional-to-size (pps) methods – are evaluated by comparing their performance on a homescan market research data set across 21 months for each of the 13 classification of individual consumption by purpose (COICOP) food groups. Classifications are derived for each of the food groups and the population index value is used as a reference to derive performance error measures, such as root mean squared error, bias and standard deviation for each food type. Repeated samples are taken for each of the pps schemes and the resulting performance error measures analysed using regression of three of the pps schemes to assess the overall effect of SS and COICOP group whilst controlling for sample size, month and population index value. Cut-off sampling appears to perform less well than pps methods and multistage pps seems to have no advantage over its single-stage counterpart. The jackknife resampling technique is also explored as a means of estimating the standard error of the index and compared with the actual results from repeated sampling.     

Asymptotic expansions for sequential confidence levels in inverse linear regression

Non-linear renewal theory is used to derive second order asymptotic expansions for the coverage probability of a fixed-width sequential confidence interval for an unknown parameter xin the inverse linear regression model. These expansions are obtained for a two-stage sequential procedure, proposed by Perng and Tong (1974) for the construction of a confidence interval for x.     

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.     

Effect of a new selection procedure on adaptive estimation

Adaptive estimation of parameters of some failure time distributionsis considered. A new procedure named the F-procedure has beendeveloped for selecting an appropriate model out of two possible models by Pandey et.al. (1991). Applying this F-procedure adaptive estimatorsof parameters of exponential, Wei bull, inverse Gaussian (IG) and Wald failure time distributions have been proposed in this paper. Comparison of these estimators has been undertaken with MLE's of the respective parameters and with some previous adaptiveestimators by simulation of samples using the Monte Carlo method.Adaptive estimation of parameters of some failure time distributions is considered. A new procedure named the F-procedure has been developedfor selecting an appropriate model out of two possible models by Pandey et.al. (1991). Applying this F-procedure adaptive estimators of parameters of exponential, Wei bull, inverse Gaussian (IG) and Wald failure time distributions have been proposed in this paper. Comparison of these estimators has been undertaken with MLE's of the respective parameters and with some previous adaptive estimators by simulation of samples using the Monte Carlo method.     

Application of the Bootstrap Approach to the Choice of Dimension and the α Parameter in the SIRα Method

To reduce the dimensionality of regression problems, sliced inverse regression approaches make it possible to determine linear combinations of a set of explanatory variables X related to the response variable Y in general semiparametric regression context. From a practical point of view, the determination of a suitable dimension (number of the linear combination of X) is important. In the literature, statistical tests based on the nullity of some eigenvalues have been proposed. Another approach is to consider the quality of the estimation of the effective dimension reduction (EDR) space. The square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. In this article, we focus on the SIR_α method and propose a naïve bootstrap estimation of the square trace correlation criterion. Moreover, this criterion could also select the α parameter in the SIR_α method. We indicate how it can be used in practice. A simulation study is performed to illustrate the behavior of this approach.     

A note on approximate confidence limits for the negative binomial model

Four methods of approximating confidence limits for the single negative binomial parameter, P, are outlined and an empirical study is presented. Some remarks on prediction intervals are also included.