Likelihood ratio test against simple stochastic ordering among several multinomial populations

Stochastic ordering between probability distributions has been widely studied in the past 50 years. Because it is often easy to make valuable judgments when such orderings exist, it is desirable to recognize their existence and to model distributional structures under them. Likelihood ratio test is the most commonly used method to test hypotheses involving stochastic orderings. Among the various formally defined notions of stochastic ordering, the least stringent is simple stochastic ordering. In this paper, we consider testing the hypothesis that all multinomial populations are identically distributed against the alternative that they are in simple stochastic ordering. We construct likelihood ratio test statistic for this hypothesis test problem, provide limit form of the objective function corresponding to the test statistic and show that the test statistic is asymptotically distributed as a mixture of chi-squared distributions, i.e., a chi-bar-squared distribution.     

Tests for stochastic ordering under biased sampling

In two-sample comparison problems it is often of interest to examine whether one distribution function majorises the other, that is, for the presence of stochastic ordering. This paper develops a nonparametric test for stochastic ordering from size-biased data, allowing the pattern of the size bias to differ between the two samples. The test is formulated in terms of a maximally selected local empirical likelihood statistic. A Gaussian multiplier bootstrap is devised to calibrate the test. Simulation results show that the proposed test outperforms an analogous Wald-type test, and that it provides substantially greater power over ignoring the size bias. The approach is illustrated using data on blood alcohol concentration of drivers involved in car accidents, where the size bias is due to drunker drivers being more likely to be involved in accidents. Further, younger drivers tend to be more affected by alcohol, so in making comparisons with older drivers the analysis is adjusted for differences in the patterns of size bias.     

Estimating the turning point location in shifted exponential model of time series

We consider the distribution of the turning point location of time series modeled as the sum of deterministic trend plus random noise. If the variables are modeled by shifted exponentials, whose location parameters define the trend, we provide a formula for computing the distribution of the turning point location and consequently to estimate a confidence interval for the location. We test this formula in simulated data series having a trend with asymmetric minimum, investigating the coverage rate as a function of a bandwidth parameter. The method is applied to estimate the confidence interval of the minimum location of two types of real-time series: the RT intervals extracted from the electrocardiogram recorded during the exercise test and an economic indicator, the current account balance. We discuss the connection with stochastic ordering.     

Ordering Properties of Convolutions of Exponential Random Variables

Convolutions of independent random variables are usually compared. In this paper, after a synthetic comparison with respect to hazard rate ordering between sums of independent exponential random variables, we focus on the special case where one sum is identically distributed. So, for a given sum of n independent exponential random variables, we deduce the "best" Erlang-n bounds, with respect to each of the usual orderings: mean ordering, stochastic ordering, hazard rate ordering and likelihood ratio ordering.     

Multiple Comparisons with Control Under Stochastic Ordering: Controlling FDR

Multiple comparisons of the effects of several treatments with a control (MCC) has been a central problem in medicine and other areas. Nearly all of existing papers are devoted to comparing means of the effects. To study medical problems more deeply, one needs more information than mean relationship from the given data. It can be expected to get more useful and deeper conclusion by comparing the probability distributions, i.e., by comparison under stochastic orders. This paper presents a likelihood ratio testing procedure to compare effects under stochastic order for MCC problems, controlling the false discovery rate (FDR). Setting a test controlling FDR under stochastic order faces several non trivial problems. These problems are analyzed and solved in this paper. To facilitate the test more easily, the asymptotic p values for the test are used and their distributions are derived. It is shown that controllability of FDR for this comparison procedure can be guaranteed. A real data example is used to illustrate how to apply this testing procedure and what the test can tell. Simulation results show that this testing procedure works quite well, better than some other tests.     

Permutation Approaches for Stochastic Ordering

In many application problems, when dealing with comparisons between two or more groups, the classical parametric inferential statistical methods are used, although in real problems the quite stringent assumptions required by such methods are rarely satisfied. In particular a parametric approach to the test on ordering of C > 2 populations is very difficult. In order to tackle this problem two alternative methods are proposed in the present paper. Both the methods consist in permutation combination based tests: the first is supposed to be more powerful and it is suitable when the main goal of the study is related to the global ordering of the populations; the second is useful when the interest is in the pairwise comparisons between the populations.     

Fully observed INAR(1) processes

The innovations of an INAR(1) process (integer-valued autoregressive) are usually assumed to be unobservable. There are, however, situations in practice, where also the innovations can be uncovered, i.e. where we are concerned with a fully observed INAR(1) process. We analyze stochastic properties of such a fully observed INAR(1) process and explore the relation between the INAR(1) model and certain metapopulation models. We show how the additional knowledge about the innovations can be used for parameter estimation, for model diagnostics, and for forecasting. Our findings are illustrated with two real-data examples.     

A robust modification of the ordered-heterogeneity test

An ordered heterogeneity (OH) test is a test for a trend that combines a non-directional heterogeneity test with the rank-order information specified under the alternative. We propose two modifications of the OH test procedure: (1) to use the mean ranks of the groups rather than the sample means to determine the observed ordering of the groups, and (2) to use the maximum correlation out of the 2^k???1 – 1 possibilities under the alternative rather than the single ordering (1, 2, … , k), where k is the number of independent groups. A simulation study indicates that these two changes increase the power of the ordered heterogeneity test when, as common in practice, the underlying distribution may deviate from a normal distribution and the trend pattern is a priori unknown. In contrast to the original OH test, the modified OH test can detect all possible patterns under the alternative with a relatively high power.     

Do maternal health problems influence child's worrying status? Evidence from the British Cohort Study

Conventional methods apply symmetric prior distributions such as a normal distribution or a Laplace distribution for regression coefficients, which may be suitable for median regression and exhibit no robustness to outliers. This work develops a quantile regression on linear panel data model without heterogeneity from a Bayesian point of view, i.e. upon a location-scale mixture representation of the asymmetric Laplace error distribution, and provides how the posterior distribution is summarized using Markov chain Monte Carlo methods. Applying this approach to the 1970 British Cohort Study (BCS) data, it finds that a different maternal health problem has different influence on child's worrying status at different quantiles. In addition, applying stochastic search variable selection for maternal health problems to the 1970 BCS data, it finds that maternal nervous breakdown, among the 25 maternal health problems, contributes most to influence the child's worrying status.     

On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.     

Conditional inference under simultaneous stochastic ordering constraints

Testing for stochastic ordering is of considerable importance when increasing does of a treatment are being compared, but in applications involving multivariate responses has received much less attention. We propose a permutation test for testing against multivariate stochastic ordering. This test is distribution-free and no assumption is made about the dependence relations among variables. A comparative simulation study shows that the proposed solution exhibits a good overall performance when compared with existing tests that can be used for the same problem.     

Testing for order among K populations: theory and examples

Testing for stochastic order among K populations is a common and important problem in statistical practice. It arises in the analysis of both planned experiments and observational studies. The authors develop a new nonparametric test for order among K populations that can accommodate any stochastic ordering. The test is based on a maximally selected chi‐bar‐square statistic. The authors find its limiting distribution and use simulations to derive critical values. Three important examples are used to illustrate the applicability of the general method. The authors find that the new tests outperform the existing methods in many practical cases. The Canadian Journal of Statistics 38: 97–115; 2010 © 2009 Statistical Society of Canada     

On- and offline detection of structural breaks in thermal spraying processes

We investigate and develop methods for structural break detection, considering time series from thermal spraying process monitoring. Since engineers induce technical malfunctions during the processes, the time series exhibit structural breaks at known time points, giving us valuable information to conduct the investigations. First, we consider a recently developed robust online (also real-time) filtering (i.e. smoothing) procedure that comprises a test for local linearity. This test rejects when jumps and trend changes are present, so that it can also be useful to detect such structural breaks online. Second, based on the filtering procedure we develop a robust method for the online detection of ongoing trends. We investigate these two methods as to the online detection of structural breaks by simulations and applications to the time series from the manipulated spraying processes. Third, we consider a recently developed fluctuation test for constant variances that can be applied offline, i.e. after the whole time series has been observed, to control the spraying results. Since this test is not reliable when jumps are present in the time series, we suggest data transformation based on filtering and demonstrate that this transformation makes the test applicable.     

On the Comparison of Two Ordinal Responses

Responses of two groups, measured on the same ordinal scale, are compared through the column effect association model, applied on the corresponding 2 × J contingency table. Monotonic or umbrella shaped ordering for the scores of the model are related to stochastic or umbrella ordering of the underlying response distributions, respectively. An algorithm for testing all possible hypotheses of stochastic ordering and deciding on an appropriate one is proposed.     

On nonparametric test for the residual probability order

Residual probability order is a partial ordering analog of the total order stochastic precedence which can be fruitfully applied in many reliability problems. We introduce a U-statistic test for the null hypothesis that two random variables are equally distributed versus the alternative that one strictly dominates the other in residual probability order. The critical value is obtained by normal approximation and bootstrap procedures. The performance of the test statistic is evaluated using a simulation study. Finally, a numerical example illustrating the theory is also given.     

The analysis of multicentre clinical trials when there is heterogeneity between centres

Two-treatment multicentre clinical trials are very common in practice. In cases where a non-parametric analysis is appropriate, a rank-sum test for grouped data called the van Elteren test can be applied. As an alternative approach, one may apply a combination test such as Fisher's combination test or the inverse normal combination test (also called Liptak's method) in order to combine centre-specific P-values. If there are no ties and no differences between centres with regard to the groups’ sample sizes, the inverse normal combination test using centre-specific Wilcoxon rank-sum tests is equivalent to the van Elteren test. In this paper, the van Elteren test is compared with Fisher's combination test based on Wilcoxon rank-sum tests. Data from two multicentre trials as well as simulated data indicate that Fisher's combination of P-values is more powerful than the van Elteren test in realistic scenarios, i.e. when there are large differences between the centres’ P-values, some quantitative interaction between treatment and centre, and/or heterogeneity in variability. The combination approach opens the possibility of using statistics other than the rank sum, and it is also a suitable method for more complicated designs, e.g. when covariates such as age or gender are included in the analysis.     

我国区域经济增长收敛的计量分析——基于面板数据分析

本文综合运用回归分析和时间序列分析法,从不同角度较全面地考察了我国区域经济增长收敛问题。计量结果表明:改革开放以来,我国各省区的经济增长不存在收敛趋势,但东、中、西部三大经济带内基本上可以认为存在具有不同稳定性的"收敛俱乐部"现象,即东部经济带存在稳定的收敛趋势,任何外部冲击都应该是暂时的,而中、西部经济带却经常会由于外部冲击而随时可能呈现发散。     

Band-limited component estimation in time-limited economic series

This paper tackles the issue of economic time-series modeling from a joint time and frequency-domain standpoint, with the objective of estimating the latent trend-cycle component. Since time-series records are data strings over a finite time span, they read as samples of contiguous data drawn from realizations of stochastic processes aligned with the time arrow. This accounts for the interpretation of time series as time-limited signals. Economic time series (up to a disturbance term) result from latent components known as trend, cycle, and seasonality, whose generating stochastic processes are harmonizable on a finite average-power argument. In addition, since trend is associated with long-run regular movements, and cycle with medium-term economic fluctuation, both of these turn out to be band-limited components. Recognizing such a frequency-domain location permits a filter-based approach to component estimation. This is accomplished through a Toeplitz matrix operator with sinc functions as entries, mirroring the ideal low-pass filter impulse response. The notion of virtual transfer function is developed and its closed-form expression derived in order to evaluate the filter features. The paper is completed by applying this filter to quarterly data from Italian industrial production, thus shedding light on the performance of the estimation procedure.     

Hierarchical Variable Selection in Polynomial Regression Models

Significance tests on coefficients of lower-order terms in polynomial regression models are affected by linear transformations. For this reason, a polynomial regression model that excludes hierarchically inferior predictors (i.e., lower-order terms) is considered to be not well formulated. Existing variable-selection algorithms do not take into account the hierarchy of predictors and often select as “best” a model that is not hierarchically well formulated. This article proposes a theory of the hierarchical ordering of the predictors of an arbitrary polynomial regression model in m variables, where m is any arbitrary positive integer. Ways of modifying existing algorithms to restrict their search to well-formulated models are suggested. An algorithm that generates all possible well-formulated models is presented.     

Inference on measurements of agreement using marginal association

A statistical test concerning the comparison of two agreements for dependent observations is studied. The concept of stochastic ordering plays an important role in defining order of agreements. A one-sided likelihood ratio test for equality of two agreements is proposed. This test is closely related to the test of marginal homogeneity against marginal stochastic ordering. A real example is analyzed for illustration purposes.