Unit Root Testing in Heteroscedastic Panels Using the Cauchy Estimator

The Cauchy estimator of an autoregressive root uses the sign of the first lag as instrumental variable. The resulting IV t-type statistic follows a standard normal limiting distribution under a unit root case even under unconditional heteroscedasticity, if the series to be tested has no deterministic trends. The standard normality of the Cauchy test is exploited to obtain a standard normal panel unit root test under cross-sectional dependence and time-varying volatility with an orthogonalization procedure. The article’s analysis of the joint N, T asymptotics of the test suggests that (1) N should be smaller than T and (2) its local power is competitive with other popular tests. To render the test applicable when N is comparable with, or larger than, T, shrinkage estimators of the involved covariance matrix are used. The finite-sample performance of the discussed procedures is found to be satisfactory.     

Permutation Tests for Comparing Inequality Measures

ABSTRACT

Asymptotic and bootstrap tests for inequality measures are known to perform poorly in finite samples when the underlying distribution is heavy-tailed. We propose Monte Carlo permutation and bootstrap methods for the problem of testing the equality of inequality measures between two samples. Results cover the Generalized Entropy class, which includes Theil’s index, the Atkinson class of indices, and the Gini index. We analyze finite-sample and asymptotic conditions for the validity of the proposed methods, and we introduce a convenient rescaling to improve finite-sample performance. Simulation results show that size correct inference can be obtained with our proposed methods despite heavy tails if the underlying distributions are sufficiently close in the upper tails. Substantial reduction in size distortion is achieved more generally. Studentized rescaled Monte Carlo permutation tests outperform the competing methods we consider in terms of power.     

Goodness-of-fit testing of a weak memoryless property of life distributions

A life distribution is said to have a weak memoryless property if its conditional probability of survival beyond a fixed time point is equal to its (unconditional) survival probability at that point. Goodness‐of‐fit testing of this notion is proposed in the current investigation, both when the fixed time point is known and when it is unknown but estimable from the data. The limiting behaviour of the proposed test statistic is obtained and the null variance is explicitly given. The empirical power of the test is evaluated for a commonly known alternative using Monte Carlo methods, showing that the test performs well. The case when the fixed time point t₀ equals a quantile of the distribution F gives a distribution‐free test procedure. The procedure works even if t₀ is unknown but is estimable.     

The Consistency and Robustness of Modified Cramér–Von Mises and Kolmogorov–Cramér Estimators

This article focuses on the minimum distance estimators under two newly introduced modifications of Cramér–von Mises distance. The generalized power form of Cramér–von Mises distance is defined together with the so-called Kolmogorov–Cramér distance which includes both standard Kolmogorov and Cramér–von Mises distances as limiting special cases. We prove the consistency of Kolmogorov-Cramér estimators in the (expected) L₁-norm by direct technique employing domination relations between statistical distances. In our numerical simulation we illustrate the quality of consistency property for sample sizes of the most practical range from n = 10 to n = 500. We study dependence of consistency in L₁-norm on ?-contamination neighborhood of the true model and further the robustness of these two newly defined estimators for normal families and contaminated samples. Numerical simulations are used to compare statistical properties of the minimum Kolmogorov–Cramér, generalized Cramér–von Mises, standard Kolmogorov, and Cramér–von Mises distance estimators of the normal family scale parameter. We deal with the corresponding order of consistency and robustness. The resulting graphs are presented and discussed for the cases of the contaminated and uncontaminated pseudo-random samples.     

Statistical inference of partially linear panel data regression models with fixed individual and time effects

This article considers a partially linear panel data model with fixed individual and time effects in a setting where both N and T are large. Based on the within transformation and profile likelihood method, we propose an approach to estimating the parametric and non parametric components of the partially linear model. The resultant estimators are shown to be consistent and asymptotically normal. Monte Carlo simulations are also conducted to illustrate the finite-sample performance of the proposed estimators.     

A Nonrandomized,Nonconservative Version of the Fisher Exact Test

The Fisher exact test has been unjustly dismissed by some as ‘only conditional,’ whereas it is unconditionally the uniform most powerful test among all unbiased tests, tests of size α and with power greater than its nominal level of significance α. The problem with this truly optimal test is that it requires randomization at the critical value(s) to be of size α. Obviously, in practice, one does not want to conclude that ‘with probability x the we have a statistical significant result.’ Usually, the hypothesis is rejected only if the test statistic's outcome is more extreme than the critical value, reducing the actual size considerably.

The randomized unconditional Fisher exact is constructed (using Neyman–structure arguments) by deriving a conditional randomized test randomizing at critical values c(t) by probabilities γ(t), that both depend on the total number of successes T (the complete-sufficient statistic for the nuisance parameter—the common success probability) conditioned upon.

In this paper, the Fisher exact is approximated by deriving nonrandomized conditional tests with critical region including the critical value only if γ (t) > γ₀, for a fixed threshold value γ₀, such that the size of the unconditional modified test is for all value of the nuisance parameter—the common success probability—smaller, but as close as possible to α. It will be seen that this greatly improves the size of the test as compared with the conservative nonrandomized Fisher exact test.

Size, power, and p value comparison with the (virtual) randomized Fisher exact test, and the conservative nonrandomized Fisher exact, Pearson's chi-square test, with the more competitive mid-p value, the McDonald's modification, and Boschloo's modifications are performed under the assumption of two binomial samples.     

Cross-Sectional Dependence in Panel Data Analysis

This article provides an overview of the existing literature on panel data models with error cross-sectional dependence (CSD). We distinguish between weak and strong CSD and link these concepts to the spatial and factor structure approaches. We consider estimation under strong and weak exogeneity of the regressors for both T fixed and T large cases. Available tests for CSD and methods for determining the number of factors are discussed in detail. The finite-sample properties of some estimators and statistics are investigated using Monte Carlo experiments.     

The exact unconditional z-pooled test for equality of two binomial probabilities: optimal choice of the Berger and Boos confidence coefficient

Exact unconditional tests for comparing two binomial probabilities are generally more powerful than conditional tests like Fisher's exact test. Their power can be further increased by the Berger and Boos confidence interval method, where a p-value is found by restricting the common binomial probability under H ₀ to a 1?γ confidence interval. We studied the average test power for the exact unconditional z-pooled test for a wide range of cases with balanced and unbalanced sample sizes, and significance levels 0.05 and 0.01. The detailed results are available online on the web. Among the values 10^?3, 10^?4, …, 10^?10, the value γ=10^?4 gave the highest power, or close to the highest power, in all the cases we looked at, and can be given as a general recommendation as an optimal γ.     

A more powerful unconditional exact test of homogeneity for 2 × c contingency table analysis

The classical unconditional exact p-value test can be used to compare two multinomial distributions with small samples. This general hypothesis requires parameter estimation under the null which makes the test severely conservative. Similar property has been observed for Fisher's exact test with Barnard and Boschloo providing distinct adjustments that produce more powerful testing approaches. In this study, we develop a novel adjustment for the conservativeness of the unconditional multinomial exact p-value test that produces nominal type I error rate and increased power in comparison to all alternative approaches. We used a large simulation study to empirically estimate the 5th percentiles of the distributions of the p-values of the exact test over a range of scenarios and implemented a regression model to predict the values for two-sample multinomial settings. Our results show that the new test is uniformly more powerful than Fisher's, Barnard's, and Boschloo's tests with gains in power as large as several hundred percent in certain scenarios. Lastly, we provide a real-life data example where the unadjusted unconditional exact test wrongly fails to reject the null hypothesis and the corrected unconditional exact test rejects the null appropriately.     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Box-Cox transformed linear models: A parameter-based asymptotic approach

A Box-Cox transformed linear model usually has the form y(λ) = μ + β₁x₁ +… + β_px_p + oe, where y(λ) is the power transform of y. Although widely used in practice, the Fisher information matrix for the unknown parameters and, in particular, its inverse have not been studied seriously in the literature. We obtain those two important matrices to put the Box-Cox transformed linear model on a firmer ground. The question of how to make inference on β = (β₁,…,β_p)^T when λ; is estimated from the data is then discussed for large but finite sample size by studying some parameter-based asymptotics. Both unconditional and conditional inference are studied from the frequentist point of view.     

On the spectrum of [v,k, t] trades

A [v, k, t] trade of volume m consists of two disjoint collections T₁ and T₂, each of m k-subsets of a v-set V, such that each t-subset of V is contained in the same number of blocks of T₁ and T₂, and each element of V is contained in at least one block of T₁. We study [v, k, t] trades, and investigate their spectrum (i.e., the collections of allowable volumes), using both theoretical techniques and computer-based searches.     

A multivariate control median test

In this paper, we extend the univariate control median test to the multivariate case. We apply the permutation principle for the null distribution function of the test statistic and obtain a conditionally nonparametric test procedure. Because of the amount of computational work involved in implementing the test, we consider the normal approximation. We prove the consistency and derive the asymptotic efficiency of our control median test relative to Puri and Sen's median test. Finally, we compare the power of our control median test with those of Hotelling's T² test and Puri and Sen's median test through the simulations.     

Sur un test d'égalité des autocovariances de deux séries chronologiques

We propose a test for the equality of the autocovariance functions of two independent and stationary time series. The test statistic is a quadratic form in the vector of differences of the first J + 1 autocovariances. Its asymptotic distribution is derived under the null hypothesis, and the finite-sample properties of the test, namely the bias and the power, are investigated by Monte Carlo methods. A by-product of this study is a new estimator of the covariance between two sample autocovariances which provides a positive definite covariance matrix. We establish the convergence of this estimator in the L₁ norm.     

Uniqueness of permuted treatment estimator distributions in the proportional hazards model

We discuss findings regarding the permutation distributions of treatment effect estimators in the proportional hazards model. For fixed sample size n, we will prove that all uncensored and untied event times yield the same permutation distribution of treatment effect estimators in the proportional hazards model. In other words this distribution is irrelevant with respect to the actual event times. We will show several uniqueness properties under different conditions. These properties are useful for small sample permutation tests and also helpful to large sample cases.     

A robust permutation test for the concordance correlation coefficient

In this work, we developed a robust permutation test for the concordance correlation coefficient (ρ_c) for testing the general hypothesis H₀ : ρ_c = ρ_c(0). The proposed test is based on an appropriately studentized statistic. Theoretically, the test is proven to be asymptotically valid in the general setting when two paired variables are uncorrelated but dependent. This desired property was demonstrated across a range of distributional assumptions and sample sizes in simulation studies, where the test exhibits robust type I error control in all settings tested, even when the sample size is small. We demonstrated the application of this test in two real world examples across cardiac output measurements and endocardiographic imaging.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.     

ON TESTING THE EQUALITY OF LOCATION PARAMETERS IN k CENSORED EXPONENTIAL DISTRIBUTIONS

Kumar and Patel (1971) have considered the problem of testing the equality of location parameters of two exponential distributions on the basis of samples censored from above, when the scale parameters are the same and unknown. The test proposed by them is shown to be biased for n₁≠n₂, while for n₁=n₂ the test possesses the property of monotonicity and is equivalent to the likelihood ratio test, which is considered by Epstein and Tsao (1953) and Dubey (1963a, 1963b). Epstein and Tsao state that the test is unbiased. We may note that when the scale parameters of k exponential distributions are unknown the problem of testing the equality of location parameters is reducible to that of testing the equality of parameters in k rectangular populations for which a test and its power function were given by Khatri (1960, 1965); Jaiswal (1969) considered similar problems in his thesis. Here we extend the problem of testing the equality of k exponential distributions on the basis of samples censored from above when the scale parameters are equal and unknown, and we establish the likelihood ratio test (LET) and the union-intersection test (UIT) procedures. Using the results previously derived by Jaiswal (1969), we obtain the power function for the LET and for k= 2 show that the test possesses the property of monotonicity. The power function of the UIT is also given.     

Large sample results for permutation tests of association

This paper deals with the asymptotics of permutation tests based on a certain rather general class of measures of association for R by C contingency tables, given marginal totals. This class includes the classical chi-square test, the T _b and γ indices of Goodman and Kruskall (1954) and the popular Rand (1971) index. The asymptotic distribution of this class of permutation tests for association is a weighted sum of non-central (gen-erally speaking) chi-squares. The formulae for the asymptotic moments of such tests are also given. If non-centrality holds under the null hypothe-sis of independence, the distribution in question converges to the normal distribution. The efficacies for such measures of association are obtained. Several applications are analysed in detail, including the above mentioned indices. Approximations to the permutation distribution are also discussed.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.