Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

A survey of two-sample location-scale problem, asymptotic relative efficiencies of some rank tests

Here the parametric as well as the non parametric approach to the two-sample location-scale problem are reviewed. Special attention is paid to the quadratic form of the linear rank statistics for the location and the scale suggested by Lepage for testing this problem. Several such quadratic forms are compared through ARE computations.     

Efficient rank tests for semiparametric competing risk models

We consider a semiparametric competing risk model given by k independent survival times. The paper offers an asymptotic treatment of tests for the semiparametric null hypothesis of equality of the underlying risks. It turns out that modified rank tests are asymptotically efficient for certain semiparametric submodels, where the baseline hazard is a nuisance parameter. In addition, the asymptotic relative efficiency of the present tests is derived. A comparison of asymptotic power functions can then be used to classify various tests proposed earlier in the literature. For instance a chi-square type test is efficient for proportional hazards. Data driven tests of likelihood ratio type are proposed for cones of alternatives. We will consider certain stochastically increasing alternatives as a special example. The paper shows how the concept of local asymptotic normality of Le Cam works for hazard oriented models.     

LAD variable selection for linear models with randomly censored data

The least absolute deviations (LAD) variable selection for linear models with randomly censored data is studied through the Lasso. The proposed procedure can select significant variables in the parameters. With appropriate selection of the tuning parameters, we establish the consistency of this procedure and the oracle property of the resulting estimators. Simulation studies are conducted to compare the proposed procedure with an inverse-censoring-probability weighted LAD LASSO-estimator.     

A proof of asymptotic normality for some VARX models

Here we present a proof of the asymptotic normality of least squares estimates for stable multivariate autoregressive models excited by a deterministic second order input signal.     

Linear rank tests for comparing treatments with a control when data are subject to unequal patterns of censorship

A class of asymptotically distribution-free tests is considered for comparing several treatments with a control when the, data are subject to unequal right-censorship. A particular member of this class is proposed for use in practice and an illustrative numerical example is given. A general result for the Pitman efficacy of a test based on an asymptotically normal test statistic is proved for the multiparameter case and using this result the efficacy of the proposed class of tests is obtained under sequences of translation and proportional hazards alternatives. Simulation studies are conducted to compare the performance of the proposed test, in terms of power, with some other tests.     

Estimation of a survival curve with randomly censored data in the presence of a covariate

This paper deals with the estimation of a survival curve in models with random right censoring and dependent censoring mechanism. We consider a specific dependent censorship model in which conditional on a covariate, the survival and censoring times are assumed to be independent. We investigate asymptotic properties of a corrected version of a survival curve estimator introduced by Cheng (1989). In particular we show uniform strong consistency and weak convergence to a Gaussian process. Comparisons of this estimator with the well-known Kaplan-Meier-estimator are included. Finally, some examples illustrate how the estimator performs. Received: February 2000     

Rank tests and random blocking of classified data

From the literature on nonparametric rank tests, limiting distributions of Wilcoxon's test tor symmetry and ot Friedman's test for treatment effect are known for observations that are classified in blocks. It is assumed that there is no interaction between blocks and treatments. In the case of fixed blocks this assumption is quite reasonable, in the case of random blocks it is not, as the presence of a random interaction does not make testing for treatment effect superfluous. For classified, categorical data in random blocks the limiting distribution will be derived in this paper of Wilcoxon's rank test in a model which includes a random interaction between blocks and treatments.
An illustration is given by some data from a judgement comparison experiment for the image quality of Video Long Play discs.     

Exact and asymptotic tests for possibly non-regular hypotheses on stochastic volatility models

We study the problem of testing hypotheses on the parameters of one- and two-factor stochastic volatility models (SV), allowing for the possible presence of non-regularities such as singular moment conditions and unidentified parameters, which can lead to non-standard asymptotic distributions. We focus on the development of simulation-based exact procedures–whose level can be controlled in finite samples–as well as on large-sample procedures which remain valid under non-regular conditions. We consider Wald-type, score-type and likelihood-ratio-type tests based on a simple moment estimator, which can be easily simulated. We also propose a C(α)-type test which is very easy to implement and exhibits relatively good size and power properties. Besides usual linear restrictions on the SV model coefficients, the problems studied include testing homoskedasticity against a SV alternative (which involves singular moment conditions under the null hypothesis) and testing the null hypothesis of one factor driving the dynamics of the volatility process against two factors (which raises identification difficulties). Three ways of implementing the tests based on alternative statistics are compared: asymptotic critical values (when available), a local Monte Carlo (or parametric bootstrap) test procedure, and a maximized Monte Carlo (MMC) procedure. The size and power properties of the proposed tests are examined in a simulation experiment. The results indicate that the C(α)-based tests (built upon the simple moment estimator available in closed form) have good size and power properties for regular hypotheses, while Monte Carlo tests are much more reliable than those based on asymptotic critical values. Further, in cases where the parametric bootstrap appears to fail (for example, in the presence of identification problems), the MMC procedure easily controls the level of the tests. Moreover, MMC-based tests exhibit relatively good power performance despite the conservative feature of the procedure. Finally, we present an application to a time series of returns on the Standard and Poor’s Composite Price Index.     

UMP unbiased tests for multiparameter testing problems with restricted alternatives

Applications of the standard theory of UMP unbiased tests depends on conditions which in general are difficult to verify. In the present paper, however, we suggest more simple rules for applying this theory for regular exponential families of distributions. This approach leads to UMP unbiased tests for various multiparameter testing problems with restricted alternatives, and is shown to give justification for conditional tests for testing symmetry, diagonals-parameter symmetry and independence in two-way contingency tables. The derived tests are shown to possess attractive small sample properties. Received: June 1998     

Locally Most Powerful and Other Rank Tests for Independence – with a Contaminated Weighted Alternative

Several authors in the literature have attempted the quantification of the concept of stochastic dependence for bivariate distribution. Two weighted rank tests for testing independence against a weighted contamination alternative is proposed and their distributional properties are studied. We also derived a locally most powerful rank test for the alternative setting. The rank tests proposed are shown to be asymptotic locally most powerful for specific distributions.     

Exact linear rank tests for two‐sample equivalence problems with continuous data

The present paper introduces a methodology for the semiparametric or non‐parametric two‐sample equivalence problem when the effects are specified by statistical functionals. The mean relative risk functional of two populations is given by the average of the time‐dependent risk. This functional is a meaningful non‐parametric quantity, which is invariant under strictly monotone transformations of the data. In the case of proportional hazard models, the functional determines just the proportional hazard risk factor. It is shown that an equivalence test of the type of the two‐sample Savage rank test is appropriate for this functional. Under proportional hazards, this test can be carried out as an exact level α test. It also works quite well under other semiparametric models. Similar results are presented for a Wilcoxon rank‐sum test for equivalence based on the Mann–Whitney functional given by the relative treatment effect.     

Type I error and test power of different tests for testing interaction effects in factorial experiments

A simulation study was conducted to investigate the effect of non normality and unequal variances on Type I error rates and test power of the classical factorial anova F‐test and different alternatives, namely rank transformation procedure (FR), winsorized mean (FW), modified mean (FM) and permutation test (FP) for testing interaction effects. Simulation results showed that as long as no significant deviation from normality and homogeneity of the variances exists, generally all of the tests displayed similar results. However, if there is significant deviation from the assumptions, the other tests are observed to be affected at considerably high levels except FR and FP tests. As a result, when the assumptions of factorial anova F‐test are not met or, in the case those assumptions are not tested whether met, it can be concluded that using FR and FP tests is more suitable than the classical factorial anova F‐test.     

Forecasting performance of multivariate time series models with full and reduced rank: an empirical examination

Since Quenouille's influential work on multiple time series, much progress has been made towards the goal of parameter reduction and model fit. Relatively less attention has been paid to the systematic evaluation of out-of-sample forecast performance of multivariate time series models. In this paper, we update the hog data set studied by Quenouille (and other researchers who followed him). We re-estimate his model with extended observations (1867–1966), and generate recursive one- to four-steps-ahead forecasts for the period of 1967 through 2000. These forecasts are compared to forecasts from an unrestricted vector autoregression, a reduced rank regression model, an index model and a cointegration-based error correction model. The error correction model that takes into account both nonstationarity of the data and rank reduction performs best at all four forecasting horizons. However, differences among competing models are statistically insignificant in most cases. No model consistently encompasses the others at all four horizons.     

Some rank tests for one–sided many–one comparisons

We compare some nonparametric tests for the (/+ 1)–sample problem with additive effects under the constraint that in every sample the treatment effect is not less than that in the first sample, i.e. of some control. The behavior of the Pitman efficiency of the respective tests (essentially tests of a Kruskal–Wallis–, Wilcoxon–, Fligner–Wolfe–, Steel–, and Nemenyi–type) is discussed which turns out to depend on the level and power of the tests as well as on the directions, from which the alternative tends to the hypothesis. It will be shown that none of the tests under consideration is uniformly superior to the others.     

Tests for balanced incomplete block ranked data with ties

We consider balanced incomplete block data when ties occur and propose new statistics for testing (a) differences in mean ranks, (b) differences in distributions of ranks, (c) differences in nonlinear effects of ranks and (d) linear contrasts. A sensory evaluation example where the data are ranks is given.     

Asymptotically optimal tests for some growth curve models under non-normal error structure

This paper considers growth curve models consisting of two stages. In the first stage we have a regression model with random parameter. The second stage is given by a linear model for the mean of the random parameter of the first stage. In the present paper we prove asymptotic optimality of tests for linear functions of the mean of the random parameter under non-normal error. Received: September 1999     

Rank tests for testing randomness of a regression coefficient in a linear regression model

We propose a class of nonparametric tests for testing non-stochasticity of the regression parameterβ in the regression modely _i =βx _i +ɛ _i ,i=1, ...,n. We prove that the test statistics are asymptotically normally distributed both underH ₀ and under contiguous alternatives. The asymptotic relative efficiencies (in the Pitman sense) with respect to the best parametric test have also been computed and they are quite high. Some simulation studies are carried out to illustrate the results. Research was supported by the University Grants Commission, India.     

Diagnostics analysis for log‐Birnbaum–Saunders regression models with censored data

This article develops influence diagnostics for log‐Birnbaum–Saunders (LBS) regression models with censored data based on case‐deletion model (CDM). The one‐step approximations of the estimates in CDM are given and case‐deletion measures are obtained. Meanwhile, it is shown that CDM is equivalent to mean shift outlier model (MSOM) in LBS regression models and an outlier test is presented based on MSOM. Furthermore, we discuss a score test for homogeneity of shape parameter in LBS regression models. Two numerical examples are given to illustrate our methodology and the properties of score test statistic are investigated through Monte Carlo simulations under different censoring percentages.