Large Sample Distribution Theory Research Articles

Inference for Support Vector Regression under ℓ1 Regularization

We provide large-sample distribution theory for support vector regression (SVR) with l1-norm along with error bars for the SVR regression coefficients. Although a classical Wald confidence interval obtains from our theory, its implementation inherently depends on the choice of a tuning parameter that scales the variance estimate and thus the width of the error bars. We address this shortcoming by further proposing an alternative large-sample inference method based on the inversion of a novel test statistic that displays competitive power properties and does not depend on the choice of a tuning parameter.

Additive Nonparametric Models with Time Variable and Both Stationary and Nonstationary Regressors

This paper considers nonparametric additive models that have deterministic time trend and both stationary and integrated variables as components. The diverse nature of the regressors caters for applications in a variety of settings. In addition, we extend the analysis to allow the stationary regressor to be instead locally stationary, and we allow the models to include a linear form of the integrated variable. Heteroscedasticity is allowed for in all models. We propose an estimation strategy based on orthogonal series expansion that takes account of the different type of stationarity/nonstationarity possessed by each covariate. We establish the large sample distribution theory for all the estimated functions we consider. In spite of the entanglement of different kinds of regressors, we can separate out the distribution theory for each estimator. We provide Monte Carlo simulations that establish the favourable properties of our procedures in moderate sized samples. Finally, we apply our techniques to the study of a pairs trading strategy.

Large-sample theory for the Bergsma-Dassios sign covariance

The Bergsma-Dassios sign covariance is a recently proposed extension of Kendall’s tau. In contrast to tau or also Spearman’s rho, the new sign covariance $\tau^{*}$ vanishes if and only if the two considered random variables are independent. Specifically, this result has been shown for continuous as well as discrete variables. We develop large-sample distribution theory for the empirical version of $\tau^{*}$. In particular, we use theory for degenerate U-statistics to derive asymptotic null distributions under independence and demonstrate in simulations that the limiting distributions give useful approximations.

Two Useful Reformulations of the Hazard Ratio

The hazard ratio is a standard summary for comparing survival curves yet hazard ratios are often difficult for scientists and clinicians to interpret. Insight into the interpretation of hazard ratios is obtained by relating hazard ratios to the maximum difference and an average difference between survival probabilities. These reformulations of the hazard ratio are useful in classroom discussions of survival analysis and when discussing analyses with scientists and clinicians. Large-sample distribution theory is provided for these reformulations of the hazard ratio. Two examples are used to illustrate the ideas.

Asymptotic properties of simple linear measurement error models

The main ob jective of this paper is to study estimators of regression models on the independent variable X which is not directly observed for some reasons. In such a situation, a substitute variable W is observed instead. This substitution complicates the statistical analysis of the observed data when the purpose of the analysis is inference about a model defined in terms of X. The substitution causes a inconsistent estimator; this is defined as a measurement error problem. To correct this problem, the conditional score and corrected score methods are proposed by Stefanski&Carroll (1985) and Nakamura (1990), respectively. In this study, large sample distribution theory for both the conditional score and corrected score estimators are derived and the performance of the estimators and the adequacy of the large sample distribution theory are obtained via Monte Carlo simulation.

A Note on a Path Counting Formula

ABSTRACT A path counting formula pivotal in the development of certain ladder determinantal ideals is revisited and recast in a probabilistic framework. Results from large-sample distribution theory are exploited to provide an approximate enumeration formula. The results in this note reside on the interface of statistics and combinatorics and hopefully stimulate further research in this direction.

Effective directed tests for models with ordered categorical data

SummaryThis paper offers a new method for testing one‐sided hypotheses in discrete multivariate data models. One‐sided alternatives mean that there are restrictions on the multidimensional parameter space. The focus is on models dealing with ordered categorical data. In particular, applications are concerned with R×C contingency tables. The method has advantages over other general approaches. All tests are exact in the sense that no large sample theory or large sample distribution theory is required. Testing is unconditional although its execution is done conditionally, section by section, where a section is determined by marginal totals. This eliminates any potential nuisance parameter issues. The power of the tests is more robust than the power of the typical linear tests often recommended. Furthermore, computer programs are available to carry out the tests efficiently regardless of the sample sizes or the order of the contingency tables. Both censored data and uncensored data models are discussed.

The Asymptotic Distribution of the S–Gini Index

Several generalizations of the classical Gini index, placing smaller or greater weights on various portions of income distribution, have been proposed by a number of authors. For purposes of statistical inference, the large sample distribution theory of the estimators of those measures of economic inequality is required. The present paper was stimulated by the use of bootstrap by Xu (2000) to estimate the variance of the estimator of the S–Gini index. It shows that the theory of L–statistics (Chernoff, Gastwirth & Johns, 1967; Shorack & Wellner, 1986) makes possible the construction of a consistent estimator for the S–Gini index and proof of its asymptotic normality. The paper also presents an explicit formula for the asymptotic variance. The formula should be helpful in planning the size of samples from which the S–Gini index can be estimated with a prescribed margin of error.

A comparison of unconditional and conditional solutions to the maximum likelihood estimation of a change-point

In this paper, we compare the performance of unconditional and conditional solutions to the problem of estimating an unknown change-point by the method of maximum likelihood estimation. We begin by presenting some asymptotic results for the first two moments of the unconditional solution. Then, considering the cases of normal and exponential distributions, we evaluate through simulations the accuracy of large sample distribution theory results for the above two solutions through bias, mean square error, and graphical displays. In both normal and exponential cases, the unconditional solution shows better performance whenever the amount of change is small and when the true parameters are unknown. The differences are particular sharp when the true change-point is towards the tail of the data. When the amount of change is large, both methods perform at similar levels of accuracy. Finally, we illustrate the two solutions through the data on British coal mine explosions.

On the generalization of the receiver operating characteristic analysis to the population of readers and cases with the jackknife method: An assessment

A new methodology that analyzes receiver operating characteristic (ROC) data sets based on jackknifing and that considers both case and reader variability has been proposed. The purpose of this investigation was to compare results using this method to those using commonly reported methodology. ROC data sets using discrete and continuous rating scales were analyzed using the proposed jackknifing method, and results were compared to analysis of the same data sets using the paired t test. The two methodologies did not result in the same significance levels, and in some cases, the difference was sufficient to affect conclusions regarding comparisons of diagnostic modalities. The probability value for the jackknifing procedure is based on large sample distribution theory, and its appropriateness is unknown for sample sizes used in practice. Also, the jackknifing technique was found to be sensitive to outliers resulting when data from the computer programs used to estimate area under the ROC curve failed to converge. Although the proposed methodology yields reasonable results, several fundamental and practical issues must be addressed before it can be used widely as the analytic method of choice in ROC studies comparing different imaging techniques or reading environments.

Inference for Shift Functions in the Two-Sample Problem With Right-Censored Data: With Applications

Abstract For two distribution functions, F and G, the shift function is defined by Δ(t) = G −1 · F(t) - t. The shift function is the distance from the 45° line and the quantity plotted in Q-Q plots. In the analysis of lifetime data, A represents the difference between two treatments. The shift function can also be used to find crossing points of two distribution functions. The large-sample distribution theory for estimates of Δ is studied for right-censored data. It turns out that the asymptotic covariance function depends on the unknown distribution functions F and G; hence simultaneous confidence bands cannot be directly constructed. A construction of simultaneous confidence bands for Δ is developed via the bootstrap. Construction and application of such bands are explored for the Q-Q plot.

Using Surrogate Failure Time Data to Increase Cost Effectiveness in Clinical Trials

In clinical trials with failure time, outcomes, the cost of follow-up for the endpoint of primary interest is sometimes expensive. In many of these settings, one or more secondary or surrogate failure time outcomes are also available which either occur much earlier or are much less expensive to follow than the primary outcome. This paper presents a method for using these secondary outcomes to increase the power of detecting a treatment effect on the primary outcome only, without introducing bias. The proposed method requires that patients be randomized to either extended or limited follow-up. A test statistic is constructed by forming a minimum variance linear combination of a linear rank statistic based on the primary outcome and a mean zero weighted sum of linear rank statistics based on secondary outcomes from both follow-up arms. Large-sample distribution theory for this statistic is developed while moderate sample behaviour is explored through a simulation study.

Estimation of Two Ordered Mean Residual Lifetime Functions

In many statistical studies involving failure data, biometric mortality data, and actuarial data, mean residual lifetime (MRL) function is of prime importance. In this paper we introduce the problem of nonparametric estimation of a MRL function on an interval when this function is bounded from below by another such function (known or unknown) on that interval, and derive the corresponding two functional estimators. The first is to be used when there is a known bound, and the second when the bound is another MRL function to be estimated independently. Both estimators are obtained by truncating the empirical estimator discussed by Yang (1978, Annals of Statistics 6, 112-117). In the first case, it is truncated at a known bound; in the second, at a point somewhere between the two empirical estimates. Consistency of both estimators is proved, and a pointwise large-sample distribution theory of the first estimator is derived.

Modelling Marginal Hazards in Multivariate Failure Time Data

SUMMARY We consider the problem of estimating relative risk parameters in the marginal distribution of failure times from a sample in which individuals are grouped into clusters and failure times for individuals within clusters may be dependent. We assume that the marginal hazard of failure for individuals within a cluster is specified by a relative risk regression model and the structure of dependence between individuals within a cluster is left unspecified. Estimating equations for the regression parameters and an estimator for the cumulative base-line marginal hazard are described. Large sample distribution theory based on a representation of the estimating functions as U-statistics is developed and results from a small simulation study are described. Two examples which originally motivated this work are presented.

Using surrogate failure time data to increase cost effectiveness in clinical trials

SUMMARY In clinical trials with failure time outcomes, the cost of follow-up for the endpoint of primary interest is sometimes expensive. In many of these settings, one or more secondary or surrogate failure time outcomes are also available which either occur much earlier or are much less expensive to follow than the primary outcome. This paper presents a method for using these secondary outcomes to increase the power of detecting a treatment effect on the primary outcome only, without introducing bias. The proposed method requires that patients be randomized to either extended or limited follow-up. A test statistic is constructed by forming a minimum variance linear combination of a linear rank statistic based on the primary outcome and a mean zero weighted sum of linear rank statistics based on secondary outcomes from both follow-up arms. Large-sample distribution theory for this statistic is developed while moderate sample behaviour is explored through a simulation study.

Inference using surrogate outcome data and a validation sample

SUMMARY In the context of estimating ,3 from the regression model P ( YI X), relating response Y to covariates X, suppose that only a surrogate response S is available for most study subjects. Suppose that for a random subsample of the study cohort, termed the validation sample, the true outcome Y is available in addition to S. We consider maximum likelihood estimation of ,3 from such data and show that it is nonrobust to misspecification of the distribution relating the surrogate to the true outcome, P(S I Y, X). An alternative semiparametric method is also considered, which is nonparametric with respect to P(S I Y, X). Large-sample distribution theory for maximum estimated likelihood estimates is developed. An illustrative example is presented.

Monitoring statistics for detecting interventions for autoregressive models

AbstractMonitoring statistics for detecting interventions in autoregressive time series are considered. These statistics are defined for stochastic regressor variables analogous to the Bayesian‐type statistics derived for non‐stochastic regressors. Large sample distribution theory is derived for the changedetection statistic for the first‐order autoregressive model.

Tables and Large-Sample Distribution Theory for Censored-Data Correlation Statistics for Testing Normality

Abstract Plotting order statistics versus some variant of the normal scores is a standard graphical technique for assessing the assumption of normality. To obtain an objective evaluation of the normal assumption, it is customary to calculate the correlation coefficient associated with this plot. The Shapiro—Francia statistic is the square of the correlation between the observed order statistics and the expected values of standard normal order statistics, whereas the Shapiro—Wilk statistic also involves the covariances of the standard normal order statistics. In a wide variety of applications, an investigation of the plausibility of the normal (or lognormal) model is needed when the observations on strength or life length are right-censored. The plotting procedure still applies if the observations are censored at a fixed order statistic or a fixed time. Here, the corresponding distribution theory for some modified versions of the Shapiro—Wilk correlation statistic is investigated. Because the asymptotic theory used in this article shows a surprisingly slow rate of convergence even for complete samples, a table of critical values based on a Monte Carlo study is provided. Results from an empirical power study are also presented. Finally, large-sample critical values are obtained and compared with the Monte Carlo values.

Tables and Large-Sample Distribution Theory for Censored-Data Correlation Statistics for Testing Normality

Abstract Plotting order statistics versus some variant of the normal scores is a standard graphical technique for assessing the assumption of normality. To obtain an objective evaluation of the normal assumption, it is customary to calculate the correlation coefficient associated with this plot. The Shapiro—Francia statistic is the square of the correlation between the observed order statistics and the expected values of standard normal order statistics, whereas the Shapiro—Wilk statistic also involves the covariances of the standard normal order statistics. In a wide variety of applications, an investigation of the plausibility of the normal (or lognormal) model is needed when the observations on strength or life length are right-censored. The plotting procedure still applies if the observations are censored at a fixed order statistic or a fixed time. Here, the corresponding distribution theory for some modified versions of the Shapiro—Wilk correlation statistic is investigated. Because the asymptotic th...

A note on testing for constant hazard against a change-point alternative

The problem of testing for constant hazard against a change-point alternative is considered. It is shown that this problem is related to another one in quality control. Based on this relationship, a test is proposed. The main advantages of this test are its computational simplicity and the ready availability of small and large sample distribution theory.