The X2-goodness-of-fit tests with moment type estimators

In the x²-goodness-of-fit test the underlying null hypothesis usually involves unknown parameters. In this article we study the asymptotic distribution of the Pearson statistic when the unknown parameters are estimated by a moment type estimator based on the ungrouped data. As is expected the usual Pearson statistic is no longer asymptotically x²-distributed in this situation. We propose a statistic [Qcirc] which under certain regularity conditions is asymptotically x²-distributed. We also propose a statistic Q? for the goodness-of-fit test when the class boundaries are random. The asymptotic powers of [Qcirc] and [Qcirc]? tests are discussed.     

TEST OF FIT FOR THE INVERSE GAUSSIAN AND GAMMA DISTRIBUTIONS UNDER CENSORING

The problem of testing the fit of the inverse Gaussian and the gamma distribution when the sample is censored and some of the parameters are unknown, is studied. Empirical Distribution Function (EDF) statistics, namely Cramér-von Mises' W ² and the Anderson-Darling's A ², are used. The limiting covariance functions of the corresponding empirical processes are derived. Asymptotic percentage points are given for some parameter values and censoring proportions. Moreover, a numerical routine is made available upon request, to obtain p-values for both test statistics, thus eliminating the need of tables and interpolation. Finally, a simple Monte Carlo study is presented to evaluate first, the approximation when using the asymptotic distributions in finite samples and second, to support the use of estimated parameter values instead of the unknown parameters needed in the limiting covariance function.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.     

The Effectiveness of Multistage Ranked Set Sampling in Stratifying the Population

The spectral analysis of Gaussian linear time-series processes is usually based on uni-frequential tools because the spectral density functions of degree 2 and higher are identically zero and there is no polyspectrum in this case. In finite samples, such an approach does not allow the resolution of closely adjacent spectral lines, except by using autoregressive models of excessively high order in the method of maximum entropy. In this article, multi-frequential periodograms designed for the analysis of discrete and mixed spectra are defined and studied for their properties in finite samples. For a given vector of frequencies ω, the sum of squares of the corresponding trigonometric regression model fitted to a time series by unweighted least squares defines the multi-frequential periodogram statistic IM(ω). When ω is unknown, it follows from the properties of nonlinear models whose parameters separate (i.e., the frequencies and the cosine and sine coefficients here) that the least-squares estimator of frequencies is obtained by maximizing I ^M(ω). The first-order, second-order and distribution properties of I ^M(ω) are established theoretically in finite samples, and are compared with those of Schuster's uni-frequential periodogram statistic. In the multi-frequential periodogram analysis, the least-squares estimator of frequencies is proved to be theoretically unbiased in finite samples if the number of periodic components of the time series is correctly estimated. Here, this number is estimated at the end of a stepwise procedure based on pseudo-Flikelihood ratio tests. Simulations are used to compare the stepwise procedure involving I ^M(ω) with a stepwise procedure using Schuster's periodogram, to study an approximation of the asymptotic theory for the frequency estimators in finite samples in relation to the proximity and signal-to-noise ratio of the periodic components, and to assess the robustness of I ^M(ω) against autocorrelation in the analysis of mixed spectra. Overall, the results show an improvement of the new method over the classical approach when spectral lines are adjacent. Finally, three examples with real data illustrate specific aspects of the method, and extensions (i.e., unequally spaced observations, trend modeling, replicated time series, periodogram matrices) are outlined.     

Identifying rational distributed lag models using a joint test applied to the corner table

Often a distributed lag response pattern can be usefully represented in rational polynomial form. When the impulse response function decays, the corner table may be useful for model identification if appropriate statistical tests may be done. One or more joint tests are called for since use of the corner table involves studying groups of its elements. We consider an asymptotic x2 statistic that permits joint tests. We report simulation results showing that the distribution of this statistic follows the x ² distribution, for certain sample sizes and degrees of freedom, well enough to be useful in practice. With two data sets we illustrate how this statistic can be a useful aid when using the corner table.     

Estimating changes in a multi-parameter exponential family

A two-stage procedure is studied for estimating changes in the parameters of the multi-parameter exponential family, given a sample X ₁,…,X _n. The first step is a likelihood ratio test of the hypothesis H_oof no change. Upon rejection of this hypothesis, the change point index and pre- and post-change parameters are estimated by maximum likelihood. The asymptotic (n → ∞) distribution of the log-likelihood ratio statistic is obtained under both H_oand local alternatives. The m.l.e.fs o of the pre- and post-change parameters are shown to be asymptotically jointly normal. The distribution of the change point estimate is obtained under local alternatives. Performance of the procedure for moderate samples is studied by Monte Carlo methods.     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Monitoring Change in Persistence Against the Null of Difference-Stationarity in Infinite Variance Observations

In this article, we propose a moving kernel-weighted variance ratio statistic to monitor persistence change in infinite variance observations. We focus on I(1) to I(0) persistence change for sequences in the domain of attraction of a stable law and local-to-finite variance sequences. The null distribution of the monitoring statistic and its consistency are proved. In particular, a bootstrap procedure is proposed to determine the critical values for the derived asymptotic distribution depends on unknown tail index. The small sample performances of proposed monitoring procedure are illustrated by both simulation and application to a high frequency financial data.     

The likelihood ratio test for homogeneity in finite mixture models

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re‐sampling procedure is recommended to obtain the asymptotic p‐value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.     

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ ² reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ ² reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795–800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.     

Empirical likelihood for nonlinear models with missing responses

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.     

A novel approach based on multiple correspondence analysis for monitoring social networks with categorical attributed data

In in most cases, the distribution of communications is unknown and one may summarize social network communications with categorical attributes in a contingency table. Due to the categorical nature of the data and a large number of features, there are many parameters to be considered and estimated in the model. Hence, the accuracy of estimators decreases. To overcome the problem of high dimensionality and unknown communications distribution, multiple correspondence analysis is used to reduce the number of parameters. Then the rescaled data are studied in a Dirichlet model in which the parameters should be estimated. Moreover, two control charts, Hotelling’s T² and multivariate exponentially weighted moving average (MEWMA), are developed to monitor the parameters of the Dirichlet distribution. The performance of the proposed method is evaluated through simulation studies in terms of average run length criterion. Finally, the proposed method is applied to a real case.     

A simple test for comparing regression curves versus one-sided alternatives

In this article we present a simple procedure to test for the null hypothesis of equality of two regression curves versus one-sided alternatives in a general nonparametric and heteroscedastic setup. The test is based on the comparison of the sample averages of the estimated residuals in each regression model under the null hypothesis. The test statistic has asymptotic normal distribution and can detect any local alternative of rate n^-1/2. Some simulations and an application to a data set are included.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

Testing for linearity in Markov switching models: a bootstrap approach

Testing for linearity in the context of Markov switching models is complicated because standard regularity conditions for likelihood based inference are violated. In particular, under the null hypothesis of linearity, some parameters are not identified and scores are identically zero. Thus, the asymptotic distribution of the relevant test statistic does not possess the standard χ ²-distribution. A bootstrap resampling scheme to approximate the distribution of the relevant test statistic under the null of linearity is proposed. The procedure is relatively easy to program and computation requirements are reasonable. The performance of the bootstrap-based test is investigated by means of Monte Carlo simulations. Results show that this test works well and outperforms the Hansen test and the Carrasco et al. test.     

Partitioning Anderson's statistic for tied data

Anderson (Biometrics 15 (1959) 582) proposed a χ²-type statistic for the nonparametric analysis of a randomized blocks design with no ties in the data. In this paper, we propose an Anderson statistic that allows for ties in the data. We show that the asymptotic distribution of the statistic under the null hypothesis of no treatment effect is a χ² distribution. Under weak assumptions on the tie structure it is shown that the degrees of freedom for the asymptotic distribution is unchanged compared to the untied case. An extended analysis based on a partition of the statistic into independent components is suggested. The first component is shown to equal the Friedman rank statistic corrected for ties. The subsequent components allow for the detection of dispersion effects, higher order effects and differences in distribution. A simulation study is given and the new analysis is applied to a sensory evaluation data set.     

Omnibus tests for the error distribution in the linear regression model

Test procedures are constructed for testing the goodness-of-fit of the error distribution in the regression context. The test statistic is based on an L ²-type distance between the characteristic function of the (assumed) error distribution and the empirical characteristic function of the residuals. The asymptotic null distribution as well as the behavior of the test statistic under contiguous alternatives is investigated, while the issue of the choice of suitable estimators has been particularly emphasized. Theoretical results are accompanied by a simulation study.