Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions

Summary. We propose a goodness-of-fit statistic Q _n based on the Hoeffding maximum correlation for testing uniformity and we show its relationship to Gini's mean difference. We compute exact and asymptotic critical values and study the power of the test proposed against a representative set of alternatives.     

EMPIRICALLY ACCELERATED CONVERGENCE OF THE SPACINGS STATISTICS TO NORMALITY

If the asymptotic normality of a statistic is inadequate for approximating its distribution in practice, then the statistic may be transformed in order to improve the approximation by accelerating the convergence to normality. We treat a goodness-of-fit statistic, the sum of the logarithms of generalized uniform spacings introduced by Cressie (1976, 1978), in this spirit. Specifically, we apply the method of maximum likelihood to simulations of the statistic in order to estimate a power transformation, as in Box & Cox (1964), and hence develop a small sample normal approximation. This approximation provides a more versatile method of applying the statistic than currently available tables of percentiles.     

Testing Homogeneity in Gamma Mixture Models

This paper characterizes the asymptotic behaviour of the likelihood ratio test statistic (LRTS) for testing homogeneity (i.e. no mixture) against gamma mixture alternatives. Under the null hypothesis, the LRTS is shown to be asymptotically equivalent to the square of Davies's Gaussian process test statistic and diverges at a log n rate to infinity in probability. Based on the asymptotic analysis, we propose and demonstrate a computationally efficient method to simulate the null distributions of the LRTS for small to moderate sample sizes.     

On the Cusum test for parameter changes in garch(1,1) Models

This paper considers the problem of testing parameter constancy in GARCH(1,1) models. A cusum of squares test is propesed in analogy Of Incl´n and Tiao (1394)'s statistic. its limiting distribution is derived via using the invariance principle for mixingaie sequences obtained by McLeish(1975). Simulation results are illustrated to demonstrate the validity of the cusum test.     

On a geometrical derivation of the upper tail of the null distribution of w-exponential

Shapiro and Wilk (1972) proposed a goodness of fit test for the exponential distribution. Carrie (1980) obtained an explicit expression of the null distribution of their test statistic W (n) E in a neighbourhood of its upper tail. His derivation uses a certain transformation involving the order statistics from the standard exponential distribution. In this paper we present an alternative derivation of this distribution using an elementary geometrical argument.     

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.     

A Jonckheere-Terpstra-type test for perfect ranking in balanced ranked set sampling

Many methods based on ranked set sampling (RSS) assume perfect ranking of the samples. Here, by using the data measured by a balanced RSS scheme, we propose a nonparametric test for the assumption of perfect ranking. The test statistic that we use formally corresponds to the Jonckheere-Terpstra-type test statistic. We show formal relations of the proposed test for perfect ranking to other methods proposed recently in the literature. Through an empirical power study, we demonstrate that the proposed method performs favorably compared to many of its competitors.     

A predictive leverage statistic for quantile regression with measurement errors

Application of quantile regression models with measurement errors in predictors is becoming increasingly popular. High leverage points in predictors can have substantial impacts on these models. Here, we propose a predictive leverage statistic for these models, assuming that the measurement errors follow a multivariate normal distribution, and derive its exact distribution. We compare its performance versus known predictive leverage statistics using simulation and a real dataset. The proposed statistic is shown to have desirable features. It is also the first predictive leverage statistic having its distribution derived in a closed form.     

Score test of homogeneity for survival data

If follow-up is made for subjects which are grouped into units, such as familial or spatial units then it may be interesting to test whether the groups are homogeneous (or independent for given explanatory variables). The effect of the groups is modelled as random and we consider a frailty proportional hazards model which allows to adjust for explanatory variables. We derive the score test of homogeneity from the marginal partial likelihood and it turns out to be the sum of a pairwise correlation term of martingale residuals and an overdispersion term. In the particular case where the sizes of the groups are equal to one, this statistic can be used for testing overdispersion. The asymptotic variance of this statistic is derived using counting process arguments. An extension to the case of several strata is given. The resulting test is computationally simple; its use is illustrated using both simulated and real data. In addition a decomposition of the score statistic is proposed as a sum of a pairwise correlation term and an overdispersion term. The pairwise correlation term can be used for constructing a statistic more robust to departure from the proportional hazard model, and the overdispesion term for constructing a test of fit of the proportional hazard model.     

Detection of outliers in simple circular regression models using the mean circular error statistic

The investigation on the identification of outliers in linear regression models can be extended to those for circular regression case. In this paper, we propose a new numerical statistic called mean circular error to identify possible outliers in circular regression models by using a row deletion approach. Through intensive simulation studies, the cut-off points of the statistic are obtained and its power of performance investigated. It is found that the performance improves as the concentration parameter of circular residuals becomes larger or the sample size becomes smaller. As an illustration, the statistic is applied to a wind direction data set.     

A note on the completeness of the normal family with constant coefficient of variation

Suppose that Z₁ …Zⁿ are independent and normally distributed with common mean u and variance σ².When σ² = λμ²with given λ> 0, it is well known that the minimal sufficient statistic is not complete for μ if n ≥ 2. The question of completeness seems to be unresolved when n = 1. In this note, we consider this case and, by using the heat equation, establish the second moment completeness of the minimal sufficient statistic, a property which is weaker than completeness but stronger than bounded completeness. Finally, we apply this result to a linear calibration problem.     

Asymptotic Comparison of Two Discriminants Used in Normal Covariate Classification

In this paper, we examine the performance of Anderson's classification statistic with covariate adjustment in comparison with the usual Anderson's classification statistic without covariate adjustment in a two-population normal covariate classification problem. The same problem has been investigated using different methods of comparison by some authors. See the bibliography. The aim of this paper is to give a direct comparison based upon the asymptotic probabilities of misclassification. It is shown that for large equal sample size of a training sample from each population, Anderson's classification statistic with covariate adjustment and cut-off point equal to zero, has better performance.     

A two-sample distribution-free scale test of the smirnov type

The two-sample, distribution-free statistics of Smirnov (1939) are used to define a new statistic. While the Smirnov statistics are used as a general goodness-of-fit test, a distribution-free scale test based on this new statistic is developed. It is shown that this new test has higher power than the two-sided Smirnov statistic in detecting differences in scale for some symmetric distributions with equal means/medians. The critical values of the proposed test statistic and its limiting distribution are given     

A Robust Test for Non-nested Hypotheses

We propose a robust version of Cox-type test statistics for the choice between two non-nested hypotheses. We first show that the influence of small amounts of contamination in the data on the test decision can be very large. Secondly, we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable. Finally, we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution, unlike for the classical test statistic. We apply our results to the choice between a Pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept.     

A comparison of tests of homogeneity for sparse contingency tables

Five tests of homogeneity for a 2x(k+l) contingency table are compared using Monte Carlo techniques. For these studiesit is assumed that k becomes large in such a way that thecontingency table is sparse for 2xk of the cells, but the sample size in two of the cells remains large. The test statistics studied are: the chi-square approximation to the Pearson test statistic, the chi-square approximation to the likelihood ratio statistic, the normal approximation to Zelterman's (1984)the normal approximation to Pearson's chi-square, and the normal approximation to the likelihood ratio statistic. For the range of parameters studied the chi-square approximation to Pearson's statistic performs consistently well with regard to its size and power.     

Likelihood Ratio Tests Under Local and Fixed Alternatives in Monotone Function Problems

Abstract.  We focus on a class of non-standard problems involving non-parametric estimation of a monotone function that is characterized by n ^1/3 rate of convergence of the maximum likelihood estimator, non-Gaussian limit distributions and the non-existence of     -regular estimators. We have shown elsewhere that under a null hypothesis of the type ψ ( z ₀) =  θ ₀ ( ψ being the monotone function of interest) in non-standard problems of the above kind, the likelihood ratio statistic has a 'universal' limit distribution that is free of the underlying parameters in the model. In this paper, we illustrate its limiting behaviour under local alternatives of the form ψ _n ( z ), where ψ _n (·) and ψ (·) vary in O ( n ^−1/3) neighbourhoods around z ₀ and ψ _n converges to ψ at rate n ^1/3 in an appropriate metric. Apart from local alternatives, we also consider the behaviour of the likelihood ratio statistic under fixed alternatives and establish the convergence in probability of an appropriately scaled version of the same to a constant involving a Kullback–Leibler distance.     

Estimation of risk ratio in a noncompliance randomized clinical trial with trichotomous dose levels

In randomized clinical trials (RCTs), we may come across the situation in which some patients do not fully comply with their assigned treatment. For an experimental treatment with trichotomous levels, we derive the maximum likelihood estimator (MLE) of the risk ratio (RR) per level of dose increase in a RCT with noncompliance. We further develop three asymptotic interval estimators for the RR. To evaluate and compare the finite sample performance of these interval estimators, we employ Monte Carlo simulation. When the number of patients per treatment is large, we find that all interval estimators derived in this paper can perform well. When the number of patients is not large, we find that the interval estimator using Wald’s statistic can be liberal, while the interval estimator using the logarithmic transformation of the MLE can lose precision. We note that use of a bootstrap variance estimate in this case may alleviate these concerns. We further note that an interval estimator combining interval estimators using Wald’s statistic and the logarithmic transformation can generally perform well with respect to the coverage probability, and be generally more efficient than interval estimators using bootstrap variance estimates when RR>1. Finally, we use the data taken from a study of vitamin A supplementation to reduce mortality in preschool children to illustrate the use of these estimators.     

Exploring Cook's Statistic Graphically

An index plot of Cook's statistic is frequently used to highlight influential observations. In this article we illustrate how enhanced higher dimensional plots of Cook's statistic can provide further useful information about sets of influential observations. We provide examples using normal and generalized linear models.     

Generalized likelihood-ratio test of the number of components in finite mixture models

The number of components is an important feature in finite mixture models. Because of the irregularity of the parameter space, the log-likelihood-ratio statistic does not have a chi-square limit distribution. It is very difficult to find a test with a specified significance level, and this is especially true for testing k — 1 versus k components. Most of the existing work has concentrated on finding a comparable approximation to the limit distribution of the log-likelihood-ratio statistic. In this paper, we use a statistic similar to the usual log likelihood ratio, but its null distribution is asymptotically normal. A simulation study indicates that the method has good power at detecting extra components. We also discuss how to improve the power of the test, and some simulations are performed.     

Adaptive parametric test in a semiparametric regression model

Consider the semiparametric regression model Y_i = x′_iβ +g(t_i)+e_i for i=1,2, …,n. Here the design points (x_i,t_i) are known and nonrandom and the e_i are iid random errors with Ee₁ = 0 and Ee² ₁ = α²<∞. Based on g(.) approximated by a B-spline function, we consider using atest statistic for testing H₀ : β = 0. Meanwhile, an adaptive parametric test statistic is constructed and a large sample study for this adaptive parametric test statistic is presented.