TESTING FOR CONTINUOUS LOCAL MARTINGALES USING THE CROSSING TREE

We present statistical tests for the continuous martingale hypothesis; that is, for whether an observed process is a continuous local martingale, or equivalently a continuous time‐changed Brownian motion. Our technique is based on the concept of the crossing tree. Simulation experiments are used to assess the power of the tests, which is generally higher than that of recently proposed tests using the estimated quadratic variation (i.e. realized volatility). In particular, the crossing tree shows significantly higher power with shorter data sets. We then show results from applying the methodology to five high‐frequency currency exchange rate data sets from 2003. For four of them we show that at small time‐scales (less than 15 minutes or so) the continuous martingale hypothesis is rejected, but not so at larger time‐scales. For the fifth, the hypothesis is rejected at small time‐scales and at some moderate time‐scales, but not all.     

Central Limit Theorems of Local Polynomial Threshold Estimator for Diffusion Processes with Jumps

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the non‐parametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the σ ‐field generated by diffusion processes is destroyed, so the classical central limit theorem for martingale difference sequences cannot work. In high‐frequency data, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps by Jacod's stable convergence theorem. We believe that our proof procedure for local polynomial threshold estimators provides a new method in this field, especially in the local linear case.     

A monte carlo analysis of two spectral tests of the martingale hypothesis

Summary The size, power, and robustness properties of the Kolmogorov-Smirnov and Cramér-von Mises spectral tests of the martingale (difference) hypothesis are investigated by Monte Carlo methods. The results highlight a marked superiority of the Cramér-von Mises with respect to the Kolmogorov-Smirnov test. The paper also shows that the Cramér-von Mises test is simple to compute, more general and more powerful than other converntionally used tests.     

Parameter estimation for partially observed hypoelliptic diffusions

Summary.  Hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis. We study parameter estimation for such processes in situations where we observe some components of the solution at discrete times. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small intersample times Δ t and large total observation times N  Δ t . Hypoellipticity together with partial observation leads to ill conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. We combine these in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters. Numerical experiments illustrate asymptotic consistency of the method when applied to simulated data. The paper concludes with an application of the Gibbs sampler to molecular dynamics data.     

Checking the censored two-sample accelerated life model using integrated cumulative hazard difference

In this paper, new statistical tests for the censored two-sample accelerated life model are discussed. From the estimating functions using integrated cumulative hazard difference, stochastic processes are constructed. They can be described by martingale residuals, and, given the data, conditional distributions can be approximated by zero mean Gaussian processes. The new methods, based on these processes, provide asymptotically consistent tests against a general departure from the model. A graphical method is also discussed. In various numerical studies, the new tests performed better than the existing method, especially when the hazard curves cross. The proposed procedures are illustrated with two real data sets.     

The local fractional bootstrap

We introduce a bootstrap procedure for high‐frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high‐frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first‐order validity of the bootstrap method, and in simulations, we observe that the bootstrap‐based hypothesis test provides considerable finite‐sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data. We illustrate this by applying the bootstrap method to two empirical data sets: We assess the roughness of a time series of high‐frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.     

Full Bayesian Analysis for a Class of Jump-Diffusion Models

The Full Bayesian Significance Test (FBST) is adjusted for jump detection in a diffusion process. Under a natural parameterization, pure diffusion can be seen as a precise hypothesis. The evidence measure defined by FBST deals with absolutely continuous posterior distributions, when posterior rates for precise hypotheses are not appropriate. Applications to simulated and real data are shown.     

Testing the Martingale Hypothesis

We propose new tests of the martingale hypothesis based on generalized versions of the Kolmogorov–Smirnov and Cramér–von Mises tests. The tests are distribution-free and allow for a weak drift in the null model. The methods do not require either smoothing parameters or bootstrap resampling for their implementation and so are well suited to practical work. The article develops limit theory for the tests under the null and shows that the tests are consistent against a wide class of nonlinear, nonmartingale processes. Simulations show that the tests have good finite sample properties in comparison with other tests particularly under conditional heteroscedasticity and mildly explosive alternatives. An empirical application to major exchange rate data finds strong evidence in favor of the martingale hypothesis, confirming much earlier research.     

Statistical power to detect violation of the proportional hazards assumption when using the Cox regression model

The use of the Cox proportional hazards regression model is wide-spread. A key assumption of the model is that of proportional hazards. Analysts frequently test the validity of this assumption using statistical significance testing. However, the statistical power of such assessments is frequently unknown. We used Monte Carlo simulations to estimate the statistical power of two different methods for detecting violations of this assumption. When the covariate was binary, we found that a model-based method had greater power than a method based on cumulative sums of martingale residuals. Furthermore, the parametric nature of the distribution of event times had an impact on power when the covariate was binary. Statistical power to detect a strong violation of the proportional hazards assumption was low to moderate even when the number of observed events was high. In many data sets, power to detect a violation of this assumption is likely to be low to modest.     

A Simulation-Based Specification Test for Diffusion Processes

This article makes two contributions. First, we outline a simple simulation-based framework for constructing conditional distributions for multifactor and multidimensional diffusion processes, for the case where the functional form of the conditional density is unknown. The distributions can be used, for example, to form predictive confidence intervals for time period t + τ, given information up to period t. Second, we use the simulation-based approach to construct a test for the correct specification of a diffusion process. The suggested test is in the spirit of the conditional Kolmogorov test of Andrews. However, in the present context the null conditional distribution is unknown and is replaced by its simulated counterpart. The limiting distribution of the test statistic is not nuisance parameter-free. In light of this, asymptotically valid critical values are obtained via appropriate use of the block bootstrap. The suggested test has power against a larger class of alternatives than tests that are constructed using marginal distributions/densities. The findings of a small Monte Carlo experiment underscore the good finite sample properties of the proposed test, and an empirical illustration underscores the ease with which the proposed simulation and testing methodology can be applied.     

On Threshold Choice in Hypothesis Testing for Dynamical Systems with Small Noise

We considered the problem of testing a simple hypothesis against composite one-sided alternative by the continuous time observations of diffusion process with small noise. Moreover, we propose a test which is asymptotically equivalent to the Neyman-Pearson test for local alternatives. The special choice of the threshold allows us to improve the rate of convergence of the first type error to the given value. The calculation of this threshold is based on the stochastic expansion of the test statistics and on the Edgeworth expansion of its distribution function.     

Finding local departures from a parametric model using nonparametric regression

Goodness-of-fit evaluation of a parametric regression model is often done through hypothesis testing, where the fit of the model of interest is compared statistically to that obtained under a broader class of models. Nonparametric regression models are frequently used as the latter type of model, because of their flexibility and wide applicability. To date, this type of tests has generally been performed globally, by comparing the parametric and nonparametric fits over the whole range of the data. However, in some instances it might be of interest to test for deviations from the parametric model that are localized to a subset of the data. In this case, a global test will have low power and hence can miss important local deviations. Alternatively, a naive testing approach that discards all observations outside the local interval will suffer from reduced sample size and potential overfitting. We therefore propose a new local goodness-of-fit test for parametric regression models that can be applied to a subset of the data but relies on global model fits, and propose a bootstrap-based approach for obtaining the distribution of the test statistic. We compare the new approach with the global and the naive tests, both theoretically and through simulations, and illustrate its practical behavior in an application. We find that the local test has a better ability to detect local deviations than the other two tests.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

Testing for Threshold Diffusion

The threshold diffusion model assumes a piecewise linear drift term and a piecewise smooth diffusion term, which constitutes a rich model for analyzing nonlinear continuous-time processes. We consider the problem of testing for threshold nonlinearity in the drift term. We do this by developing a quasi-likelihood test derived under the working assumption of a constant diffusion term, which circumvents the problem of generally unknown functional form for the diffusion term. The test is first developed for testing for one threshold at which the drift term breaks into two linear functions. We show that under some mild regularity conditions, the asymptotic null distribution of the proposed test statistic is given by the distribution of certain functional of some centered Gaussian process. We develop a computationally efficient method for calibrating the p-value of the test statistic by bootstrapping its asymptotic null distribution. The local power function is also derived, which establishes the consistency of the proposed test. The test is then extended to testing for multiple thresholds. We demonstrate the efficacy of the proposed test by simulations. Using the proposed test, we examine the evidence of nonlinearity in the term structure of a long time series of U.S. interest rates.     

Normality testing: two new tests using L-moments

Establishing that there is no compelling evidence that some population is not normally distributed is fundamental to many statistical inferences, and numerous approaches to testing the null hypothesis of normality have been proposed. Fundamentally, the power of a test depends on which specific deviation from normality may be presented in a distribution. Knowledge of the potential nature of deviation from normality should reasonably guide the researcher's selection of testing for non-normality. In most settings, little is known aside from the data available for analysis, so that selection of a test based on general applicability is typically necessary. This research proposes and reports the power of two new tests of normality. One of the new tests is a version of the R-test that uses the L-moments, respectively, L-skewness and L-kurtosis and the other test is based on normalizing transformations of L-skewness and L-kurtosis. Both tests have high power relative to alternatives. The test based on normalized transformations, in particular, shows consistently high power and outperforms other normality tests against a variety of distributions.     

A distribution free test to detect general dependence between a response variable and a covariate in the presence of heteroscedastic treatment effects

In this paper, we present a test of independence between the response variable, which can be discrete or continuous, and a continuous covariate after adjusting for heteroscedastic treatment effects. The method involves first augmenting each pair of the data for all treatments with a fixed number of nearest neighbours as pseudo‐replicates. Then a test statistic is constructed by taking the difference of two quadratic forms. The statistic is equivalent to the average lagged correlations between the response and nearest neighbour local estimates of the conditional mean of response given the covariate for each treatment group. This approach effectively eliminates the need to estimate the nonlinear regression function. The asymptotic distribution of the proposed test statistic is obtained under the null and local alternatives. Although using a fixed number of nearest neighbours pose significant difficulty in the inference compared to that allowing the number of nearest neighbours to go to infinity, the parametric standardizing rate for our test statistics is obtained. Numerical studies show that the new test procedure has robust power to detect nonlinear dependency in the presence of outliers that might result from highly skewed distributions. The Canadian Journal of Statistics 38: 408–433; 2010 © 2010 Statistical Society of Canada     

Martingales without tears

This pedagogical paper presents a casual introduction to martingales, or fair gambling processes. Our objective is to describe the concept of a martingale and its application to common statistical tests used in the analysis of survival data, but without the mathematical rigor required for formal proofs.We use heuristic arguements to demonstrate that the logrank statistic evaluated over followup time is a fair gambling process, and introduce some mathematical notation and terminology along the way. We then employ the counting process approach to show that the logrank statistic computed over followup time can be expressed as the difference of two martingale transforms, and thus is a martingale. These ideas are first introduced in the context of a discrete time process, and are then generalized to a continuous time process. With slight modifications, the same idea extends from the logrank to other weighted Mantel-Haenszel statistics computed over time.     

On heteroscedastic hazards regression models: theory and application

A class of non-proportional hazards regression models is considered to have hazard specifications consisting of a power form of cross-effects on the base-line hazard function. The primary goal of these models is to deal with settings in which heterogeneous distribution shapes of survival times may be present in populations characterized by some observable covariates. Although effects of such heterogeneity can be explicitly seen through crossing cumulative hazards phenomena in k -sample problems, they are barely visible in a one-sample regression setting. Hence, heterogeneity of this kind may not be noticed and, more importantly, may result in severely misleading inference. This is because the partial likelihood approach cannot eliminate the unknown cumulative base-line hazard functions in this setting. For coherent statistical inferences, a system of martingale processes is taken as a basis with which, together with the method of sieves, an overidentified estimating equation approach is proposed. A Pearson's χ² type of goodness-of-fit testing statistic is derived as a by-product. An example with data on gastric cancer patients' survival times is analysed.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

Simple Robust Tests for Autocorrelated Errors in Time Series Design Intervention Models

A simple, robust test for the autocorrelation parameter in an intervention time-series model (AB design) is proposed. It is analogous to the traditional tests and can easily be computed by using the freeware R. In the same way as traditional tests of autocorrelation are based on least squares (LS) fits of a linear model, our robust test is based on the highly efficient Wilcoxon fit of the linear model. We present the results of a Monte Carlo study which show that our robust test inherits the good efficiency properties of this Wilcoxon fit. Its empirical power is only slightly less than the empirical power of the least squares test over situations with normally distributed errors while it exhibited much more power over situations with error distributions having tails heavier than those of a normal distribution. It also showed robustness of validity over all null situations simulated. We also present the results of the application of our test to a real data set which illustrates the robustness of our test.