Finite Sample Theory Research Articles

Dual-type quantile regression approach for mean estimation incorporating sampled and non-sampled data

Drawing inspiration from recent advancements in robust mean estimation within finite sampling theory, we introduce a novel dual-type class of mean estimators in a design-based framework. The dual-type class is based on quantile regression and is specifically designed to be effective in the presence of extreme observations. Significantly, it integrates the averages of both sampled observations and non-sampled observations of auxiliary variable. In the initial discussion of this class, it is presumed that the target variable is non-sensitive, signifying its relevance to subjects that respondents do not consider embarrassing when queried directly. In this standard setting, we present specific estimators within the class and determine their theoretical properties. The class's scope broadens to include scenarios where the target variable incorporates sensitive topics, giving rise to nonresponse rates and inaccurate reporting. To alleviate these errors, one can promote respondent cooperation by employing scrambled response methods that obscure the actual value of the sensitive variable. Accordingly, the article delves into discussions on additive methods. Subsequently, a numerical study is conducted using asymmetric data to evaluate the effectiveness of the dual-type class by comparing it with several existing estimators, both in the absence and presence of scrambled responses.

Estimation and inference in factor copula models with exogenous covariates

A factor copula model is proposed in which factors are either simulable or estimable from exogenous information. Point estimation and inference are based on a simulated methods of moments (SMM) approach with non-overlapping simulation draws. Consistency and limiting normality of the estimator is established and the validity of bootstrap standard errors is shown. Doing so, previous results from the literature are verified under low-level conditions imposed on the individual components of the factor structure. Monte Carlo evidence confirms the accuracy of the asymptotic theory in finite samples and an empirical application illustrates the usefulness of the model to explain the cross-sectional dependence between stock returns.

Bootstrapping Laplace transforms of volatility

This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed‐span setting using bootstrap methods. Specifically, since standard wild bootstrap procedures deliver inconsistent inference, we propose a local Gaussian (LG) bootstrap, establish its first‐order asymptotic validity, and use Edgeworth expansions to show that the LG bootstrap inference achieves second‐order asymptotic refinements. Moreover, we provide new Laplace transform‐based estimators of the spot variance as well as the covariance, correlation, and beta between two semimartingales, and adapt our bootstrap procedure to the requisite scenario. We establish central limit theory for our estimators and first‐order asymptotic validity of their associated bootstrap methods. Simulations demonstrate that the LG bootstrap outperforms existing feasible inference theory and wild bootstrap procedures in finite samples. Finally, we illustrate the use of the new methods by examining the coherence between stocks and bonds during the global financial crisis of 2008 as well as the COVID‐19 pandemic stock sell‐off during 2020, and by a forecasting exercise.

Joint Hypothesis Testing from Heterogeneous Samples under Cross-dependence

A testing principle is introduced that allows to combine evidence from N potentially correlated samples. It builds on a (weighted) sum of entities from the individual samples, which is fed into a self-normalizing variance ratio type statistic. Due to self-normalization the (autoco)variances within each sample as well as the cross-covariances between the samples melt into one scaling parameter that cancels from the ratios asymptotically. Tests constructed from this principle are hence robust with respect to cross-dependence without having to estimate any nuisance parameters. The weighting and the entities from the individual samples depend on the testing problem at hand. Two cases are discussed in detail. The first one are tests of restrictions on a parameter vector (e. g. testing restrictions on expected values), while the second one focusses on time series: panel integration tests (unit root as well as stationarity tests). The validity of the asymptotic theory in finite samples is established by means of simulation evidence.

Sharper Sub-Weibull Concentrations

Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations from the mean. These bounds are new and improve existing bounds with sharper constants. In addition, a new sub-Weibull parameter is also proposed, which enables recovering the tight concentration inequality for a random variable (vector). For statistical applications, we give an ℓ2-error of estimated coefficients in negative binomial regressions when the heavy-tailed covariates are sub-Weibull distributed with sparse structures, which is a new result for negative binomial regressions. In applying random matrices, we derive non-asymptotic versions of Bai-Yin’s theorem for sub-Weibull entries with exponential tail bounds. Finally, by demonstrating a sub-Weibull confidence region for a log-truncated Z-estimator without the second-moment condition, we discuss and define the sub-Weibull type robust estimator for independent observations {Xi}i=1n without exponential-moment conditions.

Estimation of coefficient of variation using linear moments and calibration approach for nonsensitive and sensitive variables

AbstractAssessment of coefficient of variation (CV) is of major importance in numerous examinations. However, the appearance of extreme observations raises concerns about the outcomes of CV estimates based on conventional moments. So, motivated by some recent developments in finite sampling theory, we propose some new estimators of CV based on the properties of linear moments (L‐moments and Trimmed L‐moments), which are highly robust whenever outliers or extreme observations appear in a dataset. The proposed estimators are initially established on the premise that the variable of interest is nonsensitive which deals with the subjects that do not embarrass respondents when asked about them explicitly. These estimators are also applied to situations where the variable of interest is associated with sensitive issues that cause measurement errors resulting from nonresponse or unreliable reporting where such issues can be mitigated by increasing respondent participation by scrambled response models which obscure the true value of the sensitive variable. Four models are considered for this article: additive, multiplicative, mixed, and combined additive‐multiplicative models. Finally, in both nonsensitive and sensitive settings, real‐life data sets are employed to undertake simulation‐based analysis. In all cases, the proposed estimators considerably increased efficiency.

Multi-scale Fisher's independence test for multivariate dependence.

Identifying dependency in multivariate data is a common inference task that arises in numerous applications. However, existing nonparametric independence tests typically require computation that scales at least quadratically with the sample size, making it difficult to apply them in the presence of massive sample sizes. Moreover, resampling is usually necessary to evaluate the statistical significance of the resulting test statistics at finite sample sizes, further worsening the computational burden. We introduce a scalable, resampling-free approach to testing the independence between two random vectors by breaking down the task into simple univariate tests of independence on a collection of 2 × 2 contingency tables constructed through sequential coarse-to-fine discretization of the sample space, transforming the inference task into a multiple testing problem that can be completed with almost linear complexity with respect to the sample size. To address increasing dimensionality, we introduce a coarse-to-fine sequential adaptive procedure that exploits the spatial features of dependency structures. We derive a finite-sample theory that guarantees the inferential validity of our adaptive procedure at any given sample size. We show that our approach can achieve strong control of the level of the testing procedure at any sample size without resampling or asymptotic approximation and establish its large-sample consistency. We demonstrate through an extensive simulation study its substantial computational advantage in comparison to existing approaches while achieving robust statistical power under various dependency scenarios, and illustrate how its divide-and-conquer nature can be exploited to not just test independence, but to learn the nature of the underlying dependency. Finally, we demonstrate the use of our method through analysing a dataset from a flow cytometry experiment.

Finite sample theory for high-dimensional functional/scalar time series with applications

Statistical analysis of high-dimensional functional times series arises in various applications. Under this scenario, in addition to the intrinsic infinite-dimensionality of functional data, the number of functional variables can grow with the number of serially dependent observations. In this paper, we focus on the theoretical analysis of relevant estimated cross-(auto)covariance terms between two multivariate functional time series or a mixture of multivariate functional and scalar time series beyond the Gaussianity assumption. We introduce a new perspective on dependence by proposing functional cross-spectral stability measure to characterize the effect of dependence on these estimated cross terms, which are essential in the estimates for additive functional linear regressions. With the proposed functional cross-spectral stability measure, we develop useful concentration inequalities for estimated cross-(auto)covariance matrix functions to accommodate more general sub-Gaussian functional linear processes and, furthermore, establish finite sample theory for relevant estimated terms under a commonly adopted functional principal component analysis framework. Using our derived non-asymptotic results, we investigate the convergence properties of the regularized estimates for two additive functional linear regression applications under sparsity assumptions including functional linear lagged regression and partially functional linear regression in the context of high-dimensional functional/scalar time series.

Estimation and inference in adaptive learning models with slowly decreasing gains

An asymptotic theory for estimation and inference in adaptive learning models with strong mixing regressors and martingale difference innovations is developed. The maintained polynomial gain specification provides a unified framework which permits slow convergence of agents' beliefs and contains recursive least squares as a prominent special case. Reminiscent of the classical literature on co‐integration, an asymptotic equivalence between two approaches to estimation of long‐run equilibrium and short‐run dynamics is established. Notwithstanding potential threats to inference arising from non‐standard convergence rates and a singular variance–covariance matrix, hypotheses about single, as well as joint restrictions remain testable. Monte Carlo evidence confirms the accuracy of the asymptotic theory in finite samples.

Unordering of Estimators in Sampling Theory: Revisited

Rao–Blackwellization, a term credited to C. R. Rao based on his 1945 ‘breakthrough’ paper published in the Bulletin of the Calcutta Mathematical Society, besides providing improved estimators in conventional, adaptive, link-tracing, size-biased sampling theories, found applications in post simulation improvement of Monte Carlo methods, cross validation and non parametric boot strapping, particle filtering, stereology, data compression, Rao-Blackwellized Field Goal percentage estimator (RB-FG %), Rao-Blackwellized Gaussian Smoothing, Rao–Blackwellized Parts-Constellation Tracker, Rao-Blackwellized Tempered Sampling (RTS) and a host of others. In this paper, we shall consider applications related to improving of estimators in finite population sampling theory. Taking cue from Basu (1958) wherein he showed that the ‘order statistic ‘(sample units in ascending order of their labels) is a sufficient statistic, Pathak (Sankhya A 23:409–414, 1961) in the context of sampling from finite populations,first noticed that ‘any estimator which is not a function of the order statistic’, can be uniformly improved by the use of Rao–Blackwellization technique. See also Sinha and Sen (Sankhya [B] 51: 65–83, 1989) who go beyond variance comparisons and generalize to convex loss functions. In this paper, we shall revisit some of the estimators in Probability Proportional to Size sampling With Out Replacement (PPSWOR) schemes and show how Rao–Blackwellization provides improved estimators.

All in order: Distribution of serially correlated order statistics with applications to hydrological extremes

Classic extreme value theory provides asymptotic distributions of block maxima (BM) Y or peaks over threshold (POT). However, as BM and POT are relatively small subsets of the values assumed by the parent process Z, alternative approaches have been proposed to make inferences on extremes by using intermediate values and non-asymptotic models. In this study, we investigate the finite sample theory of extremes based on order statistics, and present a set of results enabling the analysis of the properties of non-asymptotic distributions of BM in finite-size blocks of data under the assumption that the parent process Z has stationary temporal dependence. In particular, we suggest the beta-binomial distribution (FβB) as a suitable approximation of the marginal distribution of order statistics and BM under dependence, thus generalizing the theoretical results available under independence. We demonstrate the usefulness of the FβB distribution in three conceptual applications of hydrological interest. Firstly, we review the so-called Complete Time-series Analysis (CTA) framework, showing that the differences between FY and FZ are due to the inherent theoretical nature of the processes Y and Z and the different probabilistic failure scenarios described by FY and FZ. Secondly, we provide a theoretical rule to select the number of the largest maxima (LM) and over-threshold exceedances (OT) required to approximate the desired portion of the upper tail of FZ with specified accuracy. Finally, we discuss how the FβB distribution offers an interpretation and generalization of the so-called metastatistical extreme value (MEV) framework and its simplified version (SMEV), avoiding the use of high-dimensional joint distributions and preliminary data thresholding and declustering. All methodological results are validated by Monte Carlo simulations involving widely used stochastic processes with persistence. Real-world stream flow data are also analyzed as proof of concept.

Exogeneity tests, incomplete models, weak identification and non-Gaussian distributions: Invariance and finite-sample distributional theory

We study the distribution of Durbin–Wu–Hausman (DWH) and Revankar–Hartley (RH) tests for exogeneity from a finite-sample viewpoint, under the null and alternative hypotheses. We consider linear structural models with possibly non-Gaussian errors, where structural parameters may not be identified and where reduced forms can be incompletely specified (or nonparametric). On level control, we characterize the null distributions of all the test statistics. Through conditioning and invariance arguments, we show that these distributions do not involve nuisance parameters. In particular, this applies to several test statistics for which no finite-sample distributional theory is yet available, such as the standard statistic proposed by Hausman (1978). The distributions of the test statistics may be non-standard – so corrections to usual asymptotic critical values are needed – but the characterizations are sufficiently explicit to yield finite-sample (Monte-Carlo) tests of the exogeneity hypothesis. The procedures so obtained are robust to weak identification, missing instruments or misspecified reduced forms, and can easily be adapted to allow for parametric non-Gaussian error distributions. We give a general invariance result (block triangular invariance) for exogeneity test statistics. This property yields a convenient exogeneity canonical form and a parsimonious reduction of the parameters on which power depends. In the extreme case where no structural parameter is identified, the distributions under the alternative hypothesis and the null hypothesis are identical, so the power function is flat, for all the exogeneity statistics. However, as soon as identification does not fail completely, this phenomenon typically disappears. We present simulation experiments which confirm the finite-sample theory. The theoretical results are illustrated with an empirical example : the relation between trade and economic growth.

Multivariate Normal Inference based on Singly Imputed Synthetic Data under Plug-in Sampling

In this paper we consider singly imputed synthetic data generated via plug-in sampling under the multivariate normal model. Based on the observed synthetic dataset, we derive a statistical test for the generalized variance, the sphericity test, a test for independence between two subsets of variables, and a test for the regression of one set of variables on the other. The procedures are based on finite sample theory.

Bootstrapping Laplace Transforms of Volatility

This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed-span setting using bootstrap methods. Specifically, since standard wild bootstraps provide inconsistent inference, we propose local Gaussian (LG) and modified wild (MW) bootstrap procedures, and establish their first-order asymptotic validity. Moreover, motivated by its superior finite sample performance in simulations, we use Edgeworth expansions to show that the LG inference achieves second-order asymptotic refinements. To further broaden the scope of the bootstraps, we provide new Laplace transform-based estimators of the spot variance as well as the covariance, correlation and beta between two semimartingales, and adapt our inference procedures to the requisite scenario. We establish central limit theory for our estimators and show first-order asymptotic validity of their associated bootstraps. Not surprisingly, a Monte Carlo study shows that the LG bootstrap outperforms the MW bootstrap and existing (first-order) feasible inference theory in finite samples. Moreover, it demonstrates that our new spot measure estimators and inference procedures are very accurate. Finally, we illustrate the use of the new methods by examining the volatility of, and the coherence between, stocks and bonds during the large equity sell-off in December 2018.

Volatility estimation and jump detection for drift–diffusion processes

The logarithmic prices of financial assets are conventionally assumed to follow a drift–diffusion process. While the drift term is typically ignored in the infill asymptotic theory and applications, the presence of temporary nonzero drifts is an undeniable fact. The finite sample theory for integrated variance estimators and extensive simulations provided in this paper reveal that the drift component has a nonnegligible impact on the estimation accuracy of volatility, which leads to a dramatic power loss for a class of jump identification procedures. We propose an alternative construction of volatility estimators and observe significant improvement in the estimation accuracy in the presence of nonnegligible drift. The analytical formulas of the finite sample bias of the realized variance, bipower variation, and their modified versions take simple and intuitive forms. The new jump tests, which are constructed from the modified volatility estimators, show satisfactory performance. As an illustration, we apply the new volatility estimators and jump tests, along with their original versions, to 21 years of 5-minute log returns of the NASDAQ stock price index.

Asymptotics for the conditional self-weighted M-estimator of GRCA(1) models with possibly heavy-tailed errors

Consider a generalized random coefficient AR(1) model, $$y_t=\Phi _t y_{t-1}+u_t$$ , where $$\{(\Phi _t, u_t)^\prime , t\ge 1\}$$ is a sequences of i.i.d. random vectors, and a conditional self-weighted M-estimator of $$\textsf {E}\Phi _t$$ is proposed. The asymptotically normality of this new estimator is established with $$\textsf {E}u_t^2$$ being possibly infinite. Simulation experiments are carried out to assess the performance of the theory and method in finite samples and a real data example is given.

Knit Product of Finite Groups and Sampling

A finite sampling theory associated with a unitary representation of a finite non-abelian group \({\mathbf {G}}\) on a Hilbert space is established. The non-abelian group \({\mathbf {G}}\) is a knit product \({\mathbf {N}}\bowtie {\mathbf {H}}\) of two finite subgroups \({\mathbf {N}}\) and \({\mathbf {H}}\) where at least \({\mathbf {N}}\) or \({\mathbf {H}}\) is abelian. Sampling formulas where the samples are indexed by either \({\mathbf {N}}\) or \({\mathbf {H}}\) are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space \(\ell ^2({\mathbf {G}})\) having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results.

Optimal two-sided tests for instrumental variables regression with heteroskedastic and autocorrelated errors

This paper considers two-sided tests for the parameter of an endogenous variable in an instrumental variable (IV) model with heteroskedastic and autocorrelated errors. We develop the finite-sample theory of weighted-average power (WAP) tests with normal errors and a known long-run variance. A typical weight choice yields tests with power near zero for parts of the parameter space.We propose two boundary conditions that yield two-sided tests with HAC errors. The locally unbiased (LU) condition is related to the power around the null hypothesis and is a weaker requirement than unbiasedness. The strongly unbiased (SU) condition is more restrictive than LU, but the associated WAP tests are easier to implement. The WAP-SU tests can outperform several tests under the weak IV asymptotics, as they make use of the data beyond the Anderson–Rubin and score statistics.

Estimation and Inference in Adaptive Learning Models with Slowly Decreasing Gains

An asymptotic theory for estimation and inference in adaptive learning models with strong mixing regressors and martingale difference innovations is developed. The maintained polynomial gain specification provides a unified framework which permits slow convergence of agents' beliefs and contains recursive least squares as a prominent special case. Reminiscent of the classical literature on co-integration, an asymptotic equivalence between two approaches to estimation of long-run equilibrium and short-run dynamics is established. Notwithstanding potential threats to inference arising from non-standard convergence rates and a singular variance-covariance matrix, hypotheses about single, as well as joint restrictions remain testable. Among tests of joint hypotheses, superiority of an Anderson-Rubin inspired test in terms of local power is established. Monte Carlo evidence confirms the accuracy of the asymptotic theory in finite samples.

The second-order bias of quantile estimators

The finite sample theory using higher-order asymptotics provides better approximations of the bias for a class of estimators. Phillips (1991) demonstrated the higher-order asymptotic expansions for LAD estimators. Rilstone et al. (1996) provided the second-order bias results of conditional mean regression estimators. This paper develops new analytical results on the second-order bias of the conditional quantile regression estimators, which enables an improved bias correction and thus to obtain improved quantile estimation. In particular, we show that the second-order bias is larger towards the tails of the conditional density than near the median, and therefore the benefit of the second-order bias correction is greater when we are interested in the deeper tail quantiles, e.g., for the study of income distribution and financial risk management. The Monte Carlo simulation confirms the theory that the bias is larger at the tail quantiles, and the second-order bias correction improves the behavior of the quantile estimators.