Numerical computation of exact moments of the least squares estimator in a first-order stationary autoregressive model

This paper examines the least squares estimator of the autoregressive coefficient in a first-order sta¬tionary autoregressive model. Exact lower-order moments are computed by numerical integrations. From the study of moment values, it is found that the exact distribution of the least squares estimator may be well approximated by a beta distribution.     

Dependency and truncated forms of combinations in multivariate combination-based permutation tests and ordered categorical variables

ABSTRACT

Quite an important problem usually occurs in several multi-dimensional hypotheses testing problems when variables are correlated. In this framework the non-parametric combination (NPC) of a finite number of dependent permutation tests is suitable to cover almost all real situations of practical interest since the dependence relations among partial tests are implicitly captured by the combining procedure itself without the need to specify them [Pesarin F, Salmaso L. Permutation tests for complex data: theory, applications and software. Chichester: Wiley; 2010a]. An open problem related to NPC-based tests is the impact of the dependency structure on combined tests, especially in the presence of categorical variables. This paper’s goal is firstly to investigate the impact of the dependency structure on the possible significance of combined tests in cases of ordered categorical responses using Monte Carlo simulations, then to propose some specific procedures aimed at improving the power of multivariate combination-based permutation tests. The results show that an increasing level of correlation/association among responses negatively affects the power of combination-based multivariate permutation tests. The application of special forms of combination functions based on the truncated product method [Zaykin DV, Zhivotovsky LA, Westfall PH, Weir BS. Truncated product method for combining p-values. Genet Epidemiol. 2002;22:170–185; Dudbridge F, Koeleman BPC. Rank truncated product of p-values, with application to genomewide association scans. Genet Epidemiol. 2003;25:360–366] or on Liptak combination allowed us, using Monte Carlo simulations, to demonstrate the possibility of mitigating the negative effect on power of combination-based multivariate permutation tests produced by an increasing level of correlation/association among responses.     

Children’s narrative story stem responses: Contributions of executive functioning and language proficiency to relationship representations

Story stem narrative tasks provide insight into young children's inner experiences. Little research has investigated how developmental capacities impact narrative content and process. This study evaluates the influence of executive functioning (EF) and language ability on children's narratives. Data were gathered from 210 low‐income children. EF and language ability were assessed at age 4 and EF was assessed at age 6 via direct examination, cognitive‐motor tasks, and examiner ratings. Children's responses to eight story stems were gathered at age 6; three latent narrative variables were constructed (Prosocial, Aggressive/Conflict, and Avoidance/Danger), and Narrative Coherence was rated. Results of longitudinal path analyses in structural equation modeling demonstrated age 4 EF significantly predicted all narrative outcomes. Language ability independently predicted Prosocial and Narrative Coherence, and had an influence on Aggression/Conflict when modeled with EF. Age 6 EF mediated the relationship between age 4 EF and narrative outcomes. Findings provide insight into developmental abilities that influence on children's narrative responses to challenging interpersonal scenarios. Future research should study parent–child relations, EF, and language abilities simultaneously.     

Exact distributions of two non-central generalized Wilks's statistics

We derive the exact expressions of the probability density function (pdf) and the cumulative distribution function (cdf) of Wilks's likelihood ratio criterion Λ and Wilks-Lawley's statistic U in the non-central linear and the non-central planar cases. Those expressions are given in rapidly converging infinite series and can be used for numerical computation. For applications, we compute the exact power of these statistics in a multivariate analysis of variance exercise, and show by simulation the precision of our analytic formulae.     

Estimating ordered quantiles of two exponential populations with a common minimum guarantee time

The problem of estimating ordered quantiles of two exponential populations is considered, assuming equality of location parameters (minimum guarantee times), using the quadratic loss function. Under order restrictions, we propose new estimators which are the isotonized version of the MLEs, call it, restricted MLE. A sufficient condition for improving equivariant estimators is derived under order restrictions on the quantiles. Consequently, estimators improving upon the old estimators have been derived. A detailed numerical study has been done to evaluate the performance of proposed estimators using the Monte-Carlo simulation method and recommendations have been made for the use of the estimators.     

A note on the mean residual life of a coherent system with a cold standby component

In this paper, we investigate the effect of a cold standby component on the mean residual life (MRL) of a system. When the system fails, a cold standby component is immediately put in operation. We particularly focus on the coherent systems in which, after putting the standby component into operation, the failure of the system is due to the next component failure. For these systems, we define MRL functions and obtain their explicit expressions. Also some stochastic ordering results are provided. Such systems include k-out-of-n systems. Hence, our results extend some results in literature.     

Effect of tuberculosis on immune restoration among HIV-infected patients receiving antiretroviral therapy

In this article, time to immune recovery during antiretroviral therapy was estimated and compared between HIV-infected children with and without tuberculosis (TB). CD4?T-cell restoration was used as a criterion for determining immune recovery. The median residual lifetime function, which is more intuitive and robust compared to the frequently used measures of lifetime data, was used to estimate time to CD4?T-cell restoration. The median residual lifetime is not influenced by extreme observations and heavy-tailed distributions which are commonly encountered in clinical studies. Permutation-based methods were used to compare the CD4?T-cell restoration times between the two groups of patients. Our results indicate that children with TB had uniformly higher median residual lifetimes to immune recovery compared to those without TB. Although TB was associated with slower CD4?T-cell restoration, the differences between the restoration times of the two groups were not statistically significant.     

A robust automatic clustering algorithm for probability density functions with application to categorizing color images

This study develops a robust automatic algorithm for clustering probability density functions based on the previous research. Unlike other existing methods that often pre-determine the number of clusters, this method can self-organize data groups based on the original data structure. The proposed clustering method is also robust in regards to noise. Three examples of synthetic data and a real-world COREL dataset are utilized to illustrate the accurateness and effectiveness of the proposed approach.     

An algorithmic approach to construct D-optimal saturated designs for logistic model

In this paper, locally D-optimal saturated designs for a logistic model with one and two continuous input variables have been constructed by modifying the famous Fedorov exchange algorithm. A saturated design not only ensures the minimum number of runs in the design but also simplifies the row exchange computation. The basic idea is to exchange a design point with a point from the design space. The algorithm performs the best row exchange between design points and points form a candidate set representing the design space. Naturally, the resultant designs depend on the candidate set. For gain in precision, intuitively a candidate set with a larger number of points and the low discrepancy is desirable, but it increases the computational cost. Apart from the modification in row exchange computation, we propose implementing the algorithm in two stages. Initially, construct a design with a candidate set of affordable size and then later generate a new candidate set around the points of design searched in the former stage. In order to validate the optimality of constructed designs, we have used the general equivalence theorem. Algorithms for the construction of optimal designs have been implemented by developing suitable codes in R.