Positive quadrant dependence testing and constrained copula estimation

Positive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution of the test statistics under the null hypothesis of positive quadrant dependence is to resample from a constrained copula. This requires constrained estimation of a copula function. We show that this use of resampling under a constrained copula improves considerably the power performance of existing testing procedures. We propose two resampling procedures, one based on a parametric constrained copula estimation and one relying on nonparametric estimation of a positive quadrant dependence copula, and discuss their properties. The finite‐sample performances of the resulting testing procedures are evaluated via a simulation study that also includes comparisons with existing tests. Finally, a data set of Danish fire insurance claims is tested for positive quadrant dependence. The Canadian Journal of Statistics 41: 36–64; 2013 © 2012 Statistical Society of Canada     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

A note on admissibility of some tests of independence against “Positive Dependence” in R×C contingeney table

Admissibility of some asymptotically optimal tests of independence against “positive dependence” in a R × C contingency table is deduced from earlier results of the author. “Positive dependence” is specified to be positive regression dependence and positive quadrant dependence.     

Conditional tests in a competing risks model

Testing for equality of competing risks based on their cumulative incidence functions (CIFs) or their cause specific hazard rates (CSHRs) has been considered by many authors. The finite sample distributions of the existing test statistics are in general complicated and the use of their asymptotic distributions can lead to conservative tests. In this paper we show how to perform some of these tests using the conditional distributions of their corresponding test statistics instead (conditional on the observed data). The resulting conditional tests are initially developed for the case of k = 2 and are then extended to k > 2 by performing a sequence of two sample tests and by combining several risks into one. A simulation study to compare the powers of several tests based on their conditional and asymptotic distributions shows that using conditional tests leads to a gain in power. A real life example is also discussed to show how to implement such conditional tests.     

Average Collapsibility of Distribution Dependence and Quantile Regression Coefficients

Abstract. The Yule–Simpson paradox notes that an association between random variables can be reversed when averaged over a background variable. Cox and Wermuth introduced the concept of distribution dependence between two random variables X and Y, and gave two dependence conditions, each of which guarantees that reversal of qualitatively similar conditional dependences cannot occur after marginalizing over the background variable. Ma, Xie and Geng studied the uniform collapsibility of distribution dependence over a background variable W, under stronger homogeneity condition. Collapsibility ensures that associations are the same for conditional and marginal models. In this article, we use the notion of average collapsibility, which requires only the conditional effects average over the background variable to the corresponding marginal effect and investigate its conditions for distribution dependence and for quantile regression coefficients.     

A class of exact UMP unbiased tests for conditional symmetry in small-sample square contingency tables

Testing conditional symmetry against various alternative diagonals-parameter symmetry models often provides a point of departure in studies of square contingency tables with ordered categories. Typically, chi-square or likelihood-ratio tests are used for such purposes. Since these tests depend on the validity of asymptotic approximation, they may be inappropriate in small-sample situations where exact tests are required. In this paper, we apply the theory of UMP unbiased tests to develop a class of exact tests for conditional symmetry in small samples. Oesophageal cancer and longitudinal income data are used to illustrate the approach.     

A resampling procedure for nonparametric combination of several dependent tests

Summary This paper deals with nonparametric methods for combining dependent permutation or randomization tests. Particularly, they are nonparametric with respect to the underlying dependence structure. The methods are based on a without replacement resampling procedure (WRRP) conditional on the observed data, also called conditional simulation, which provide suitable estimates, as good as computing time permits, of the permutational distribution of any statistic. A class C of combining functions is characterized in such a way that all its members, under suitable and reasonable conditions, are found to be consistent and unbiased. Moreover, for some of its members, their almost sure asymptotic equivalence with respect to best tests, in particular cases, is shown. An applicational example to a multivariate permutationalt-paired test is also discussed.     

Estimation of a Conditional Copula and Association Measures

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y ₁ and Y ₂ conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.     

Accurately sized test statistics with misspecified conditional homoskedasticity

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.     

DISTRIBUTION-FREE TESTS BASED ON SUB-SAMPLE EXTREMA FOR TESTING AGAINST POSITIVE DEPENDENCE

A class of distribution-free tests based on U-statistics has been proposed for testing the null hypothesis of independence against positive quadrant dependence. The tests are based on U-statistics and the Kendall's-tau test belongs to this class.     

Conditional Heteroscedasticity in Qualitative Response Models of Time Series: A Gibbs-Sampling Approach to the Bank Prime Rate

Previous time series applications of qualitative response models have ignored features of the data, such as conditional heteroscedasticity, that are routinely addressed in time series econometrics of financial data. This article addresses this issue by adding Markov-switching heteroscedasticity to a dynamic ordered probit model of discrete changes in the bank prime lending rate and estimating via the Gibbs sampler. The dynamic ordered probit model of Eichengreen, Watson, and Grossman allows for serial autocorrelation in probit analysis of a time series, and this article demonstrates the relative simplicity of estimating a dynamic ordered probit using the Gibbs sampler instead of the Eichengreen et al. maximum likelihood procedure. In addition, the extension to regime-switching parameters and conditional heteroscedasticity is easy to implement under Gibbs sampling. The article compares tests of goodness of fit between dynamic ordered probit models of the prime rate that have constant variance and conditional heteroscedasticity.     

Testing dependence between the failure time and failure modes: An application of enlarged filtration

The model of independent competing risks provides no information for the assessment of competing failure modes if the failure mechanisms underlying these modes are coupled. Models for dependent competing risks in the literature can be distinguished on the basis of the functional behaviour of the conditional probability of failure due to a particular failure mode given that the failure time exceeds a fixed time, as a function of time. There is an interesting link between monotonicity of such conditional probability and dependence between failure time and failure mode, via crude hazard rates. In this paper, we propose tests for testing the dependence between failure time and failure mode using the crude hazards and using the conditional probabilities mentioned above. We establish the equivalence between the two approaches and provide an asymptotically efficient weight function under a sequence of local alternatives. The tests are applied to simulated data and to mortality follow-up data.     

A Nonparametric Test for Autoregressive Conditional Heteroscedasticity: A Markov-Chain Approach

In this article we propose a nonparametric test for autoregressive conditional heteroscedasticity based on finite-state Markov chains. A simple Monte Carlo experiment suggests that in finite samples it performs comparably to the Lagrange multiplier test under conditional normality and is superior for the t, lognormal, and exponential distributions. As an illustration, we apply both tests to Canadian/U.S. forward foreign exchange data.     

Measure of departure from conditional marginal homogeneity for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Tomizawa et al. (S. Tomizawa, N. Miyamoto, and N. Ashihara, Measure of departure from marginal homogeneity for square contingency tables having ordered categories, Behaviormetrika 30 (2003), pp. 173–193.) and Tahata et al. (K. Tahata, T. Iwashita, and S. Tomizawa, Measure of departure from symmetry of cumulative marginal probabilities for square contingency tables with ordered categories, SUT J. Math., 42 (2006), pp. 7–29.) considered the measures which represent the degree of departure from the marginal homogeneity (MH) model. The present paper proposes a measure that represents the degree of departure from the conditional MH, given that an observation will fall in one of the off-diagonal cells of the table. The measure proposed is expressed by using the Cressie–Read power-divergence or the Patil–Taillie diversity index, which is applied for the conditional cumulative marginal probabilities given that an observation will fall in one of the off-diagonal cells of the table. When the MH model does not hold, the measure is useful for seeing how far the conditional cumulative marginal probabilities are from those with an MH structure and for comparing the degree of departure from MH in several tables. Examples are given.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

Maximum Likelihood Estimation and Inference in Multivariate Conditionally Heteroscedastic Dynamic Regression Models With Student t Innovations

We provide numerically reliable analytical expressions for the score, Hessian, and information matrix of conditionally heteroscedastic dynamic regression models when the conditional distribution is multivariatet. We also derive one-sided and two-sided Lagrange multiplier tests for multivariate normality versus multivariate t based on the first two moments of the squared norm of the standardized innovations evaluated at the Gaussian pseudo-maximum likelihood estimators of the conditional mean and variance parameters. Finally, we illustrate our techniques through both Monte Carlo simulations and an empirical application to 26 U.K. sectorial stock returns that confirms that their conditional distribution has fat tails.     

Evaluating Multivariate GARCH Models in the Nordic Electricity Markets

ABSTRACT

This article considers a variety of specification tests for multivariate GARCH models that are used for dynamic hedging in electricity markets. The test statistics include the robust conditional moments tests for sign-size bias along with the recently introduced copula tests for an appropriate dependence structure. We consider this effort worthwhile, since quite often the tests of multivariate GARCH models are omitted and the models become selected ad hoc depending on the results they generate. Hedging performance comparisons, in terms of unconditional and conditional ex-post variance portfolio reduction, are conducted.     

Shift in re‐randomization distribution with conditional randomization test

Proschan, Brittain, and Kammerman made a very interesting observation that for some examples of the unequal allocation minimization, the mean of the unconditional randomization distribution is shifted away from 0. Kuznetsova and Tymofyeyev linked this phenomenon to the variations in the allocation ratio from allocation to allocation in the examples considered in the paper by Proschan et al. and advocated the use of unequal allocation procedures that preserve the allocation ratio at every step. In this paper, we show that the shift phenomenon extends to very common settings: using conditional randomization test in a study with equal allocation. This phenomenon has the same cause: variations in the allocation ratio among the allocation sequences in the conditional reference set, not previously noted. We consider two kinds of conditional randomization tests. The first kind is the often used randomization test that conditions on the treatment group totals; we describe the variations in the conditional allocation ratio with this test on examples of permuted block randomization and biased coin randomization. The second kind is the randomization test proposed by Zheng and Zelen for a multicenter trial with permuted block central allocation that conditions on the within‐center treatment totals. On the basis of the sequence of conditional allocation ratios, we derive the value of the shift in the conditional randomization distribution for specific vector of responses and the expected value of the shift when responses are independent identically distributed random variables. We discuss the asymptotic behavior of the shift for the two types of tests. Copyright © 2013 John Wiley & Sons, Ltd.     

A dynamic double asymmetric copula generalized autoregressive conditional heteroskedasticity model: application to China's and US stock market

Modeling the relationship between multiple financial markets has had a great deal of attention in both literature and real-life applications. One state-of-the-art technique is that the individual financial market is modeled by generalized autoregressive conditional heteroskedasticity (GARCH) process, while market dependence is modeled by copula, e.g. dynamic asymmetric copula-GARCH. As an extension, we propose a dynamic double asymmetric copula (DDAC)-GARCH model to allow for the joint asymmetry caused by the negative shocks as well as by the copula model. Furthermore, our model adopts a more intuitive way of constructing the sample correlation matrix. Our new model yet satisfies the positive-definite condition as found in dynamic conditional correlation-GARCH and constant conditional correlation-GARCH models. The simulation study shows the performance of the maximum likelihood estimate for DDAC-GARCH model. As a case study, we apply this model to examine the dependence between China and US stock markets since 1990s. We conduct a series of likelihood ratio test tests that demonstrate our extension (dynamic double joint asymmetry) is adequate in dynamic dependence modeling. Also, we propose a simulation method involving the DDAC-GARCH model to estimate value at risk (VaR) of a portfolio. Our study shows that the proposed method depicts VaR much better than well-established variance–covariance method.