Finite Sample Distribution Research Articles

Change-point estimators with the weighted objective function when sequentially estimating breaks

Change-point estimators based on a recently proposed weighted objective function are investigated, focusing on models with two structural breaks estimated sequentially. The asymptotic distributions of the first- and second-step estimators are examined under the long-span asymptotic scheme, which applies under large break sizes. It is shown that these distributions are generally unimodal and asymmetric. However, under small break sizes, the limiting distribution under the aforementioned scheme fails to accurately approximate the finite sample distribution. The first-step estimator’s finite sample distribution tends to exhibit two peaks, while the corresponding asymptotic distribution remains unimodal. This finite sample characteristic is better approximated under the in-fill asymptotic scheme.

DECISION THEORETIC BOOTSTRAPPING

The design and testing of supervised machine learning models combine two fundamental distributions: (1) the training data distribution and (2) the testing data distribution. Although these two distributions are identical and identifiable when the data set is infinite, they are imperfectly known when the data are finite (and possibly corrupted), and this uncertainty must be taken into account for robust uncertainty quantification (UQ). An important case is when the test distribution is coming from a modal or localized area of the finite sample distribution. We present a general decision theoretic bootstrapping solution to this problem: (1) partition the available data into a training subset and a UQ subset; (2) take m subsampled subsets of the training set and train m models; (3) partition the UQ set into n sorted subsets and take a random fraction of them to define <i>n</i> corresponding empirical distributions μ<sub>j</sub>; (4) consider the adversarial game where Player I selects a model i ∈ {1,.....,m}, Player II selects the UQ distribution μ<sub>j</sub>, and Player I receives a loss defined by evaluating the model <i>i</i> against data points sampled from μ<sub>j</sub>; (5) identify optimal mixed strategies (probability distributions over models and UQ distributions) for both players. These randomized optimal mixed strategies provide optimal model mixtures, and UQ estimates given the adversarial uncertainty of the training and testing distributions represented by the game. The proposed approach provides (1) some degree of robustness to in-sample distribution localization/concentration and (2) conditional probability distributions on the output space forming aleatory representations of the uncertainty on the output as a function of the input variable.

Assumption-light and computationally cheap inference on inequality measures by sample splitting: the Student t approach

Inference on inequality indices remains challenging, even in large samples. Heavy right tails in income and wealth distributions hinder the quality and threaten the validity of asymptotic approximations to finite sample distributions. Attempts to improve on asymptotic approximations by bootstrap techniques or permutation tests are only partial successes. We evaluate a different approach to robust inference, relying on Student t statistics obtained from split samples. This relatively simple ‘t-based’ approach requires no consistent variance estimators, no random sampling of populations, and only mild distributional assumptions. We compare its performance with that of refined bootstrap and permutation techniques. We find that the more complex bootstrap methods still have the edge in one-sample tests, where the t-approach suffers from a negative skew. In two-sample comparisons though, the t-approach offers advantages: it is undersized while bootstrap tests and permutation tests are often oversized. In certain circumstances it is less powerful than permutation tests and bootstrap tests, but for large samples, this difference dissipates. It is also more generally applicable than permutation tests and easily generates confidence intervals. These differences are illustrated with an empirical application using two different sources of household data from the Russian Federation.

Finite sample and asymptotic distributions of a statistic for sufficient follow‐up in cure models

AbstractThe existence of immune or cured individuals in a population and whether there is sufficient follow‐up in a sample of censored observations on their lifetimes to be confident of their presence are questions of major importance in medical survival analysis. Here we give a detailed analysis of a statistic designed to test for sufficient follow‐up in a sample. Assuming an i.i.d. censoring model, we obtain exact finite‐sample and asymptotic distributions for the statistic, and use these to calculate the power of a test based on it. A particularly useful finding is that the asymptotic distribution of the test statistic is parameter‐free in the null case when follow‐up is insufficient. The methods are illustrated with application to a glioma cancer dataset.

In-Fill Asymptotic Distribution of the Change Point Estimator when Estimating Breaks One at a Time

Abstract In this study, we investigate the least squares (LS) estimator of a structural change point by the in-fill asymptotic theory, which has been recently used by Jiang, Wang, and Yu (2018. “New Distribution Theory for the Estimation of Structural Break Point in Mean.” Journal of Econometrics 205 (1): 156–76; 2020. “In-Fill Asymptotic Theory for Structural Break Point in Autoregressions.” Econometric Reviews 40 (4): 359–86), when the model with two structural changes is estimated as the model with only a one-time structural change. We, hence, show that the finite sample distribution of the estimator of the first break has four peaks, which is different from the classical long-span asymptotic distribution, which contains only one peak. Conversely, the in-fill asymptotic distribution of the estimator has four peaks and can approximate the finite sample distribution very well. We also demonstrate that the estimator is consistent in the in-fill asymptotic framework with a relatively large magnitude of the break. In the latter case, the finite sample distribution of the estimator has only one peak and is well approximated by both the in-fill and long-span asymptotic theory.

A Monte Carlo permutation procedure for testing variance components using robust estimation methods

Permutation methods offer an acceptable and convenient tool for inferring zero variance components in linear mixed models using the likelihood ratio test. However, when data exhibit heavy-tailed distribution, heavy-skewed distribution or outliers, maximum likelihood estimation may not be the best choice in constructing useful test statistics. In this article, we propose the use of robust rank-based estimation as an alternative. The finite sample distribution of our test statistic is well approximated using suitable permutations of the cluster indices that are exchangeable when the null hypothesis is true. Empirical results, comparing the new test to existing tests, indicate that all tests maintain acceptable Type I error rates when data exhibit heavy-tailed or heavy-skewed distributions. However, only our new test remains robust against the presence of outlier in the response space. Besides, it is only the latter case where other tests could show a competing power to our test. Otherwise, the new test is superior with an outstanding power under the remaining settings.

Estimation of a Structural Break Point in Linear Regression Models

This study proposes a point estimator of the break location for a one-time structural break in linear regression models. If the break magnitude is small, the least-squares estimator of the break date has two modes at the ends of the finite sample period, regardless of the true break location. To solve this problem, I suggest an alternative estimator based on a modification of the least-squares objective function. The modified objective function incorporates estimation uncertainty that varies across potential break dates. The new break point estimator is consistent and has a unimodal finite sample distribution under small break magnitudes. A limit distribution is provided under an in-fill asymptotic framework. Monte Carlo simulation results suggest that the new estimator outperforms the least-squares estimator. I apply the method to estimate the break date in U.S. and U.K. stock return prediction models.

Asymptotic distributions of a new type of design‐based incomplete U‐statistics

The U‐statistic has been an important part of the arsenal of statistical tools. Meanwhile, the computation of it could easily become expensive. As a remedy, the idea of incomplete U‐statistics has been adopted in practice, where only a small fraction of combinations of units are evaluated. Recently, researchers proposed a new type of incomplete U‐statistics called ICUDO, which needs substantially less time of computing than all existing methods. This paper aims to study the asymptotic distributions of ICUDO to facilitate the corresponding statistical inference. This is a non‐trivial task due to the restricted randomization in the sampling scheme of ICUDO. The bootstrap approach for the finite sample distribution of ICUDO is also discussed. Lastly, we observe some intrinsic connections between U‐statistics and computer experiments in the context of integration approximation. This allows us to generalize some existing theoretical results in the latter topic.

Splitting the sample at the largest uncensored observation

We calculate finite sample and asymptotic distributions for the largest censored and uncensored survival times, and some related statistics, from a sample of survival data generated according to an iid censoring model. These statistics are important for assessing whether there is sufficient follow-up in the sample to be confident of the presence of immune or cured individuals in the population. A key structural result obtained is that, conditional on the value of the largest uncensored survival time, and knowing the number of censored observations exceeding this time, the sample partitions into two independent subsamples, each subsample having the distribution of an iid sample of censored survival times, of reduced size, from truncated random variables. This result provides valuable insight into the construction of censored survival data, and facilitates the calculation of explicit finite sample formulae. We illustrate by calculating distributions of statistics useful for testing for sufficient follow-up in a sample, and apply extreme value methods to derive asymptotic distributions for some of those.

Robust inference for change points in high dimension

This paper proposes a new test for a change point in the mean of high-dimensional data based on the spatial sign and self-normalization. The test is easy to implement with no tuning parameters, robust to heavy-tailedness and theoretically justified with both fixed-n and sequential asymptotics under both null and alternatives, where n is the sample size. We demonstrate that the fixed-n asymptotics provide a better approximation to the finite sample distribution and thus should be preferred in both testing and testing-based estimation. To estimate the number and locations when multiple change-points are present, we propose to combine the p-value under the fixed-n asymptotics with the seeded binary segmentation (SBS) algorithm. Through numerical experiments, we show that the spatial sign based procedures are robust with respect to the heavy-tailedness and strong coordinate-wise dependence, whereas their non-robust counterparts proposed in Wang et al. (2022)[28] appear to under-perform. A real data example is also provided to illustrate the robustness and broad applicability of the proposed test and its corresponding estimation algorithm.

Normal but skewed?

SummaryWe propose a multivariate normality test against skew normal distributions using higher‐order log‐likelihood derivatives, which is asymptotically equivalent to the likelihood ratio but only requires estimation under the null. Numerically, it is the supremum of the univariate skewness coefficient test over all linear combinations of the variables. We can simulate its exact finite sample distribution for any multivariate dimension and sample size. Our Monte Carlo exercises confirm its power advantages over alternative approaches. Finally, we apply it to the joint distribution of US city sizes in two consecutive censuses finding that non‐normality is very clearly seen in their growth rates.

Asymptotic properties of the maximum likelihood estimator in regime-switching models with time-varying transition probabilities

Summary Time-varying transition probability (TVTP) regime-switching models extend the constant regime transition probability in Markov-switching models to include information from observations. We show consistency and asymptotic normality of the maximum likelihood estimator (MLE) in general TVTP regime-switching models where the conditional distribution of $Y_t$ depends on lagged regimes. Consistency of the MLE is also shown under misspecification. The assumptions are verified in regime-switching autoregressive models with widely applied TVTP specifications. A simulation study examines the finite-sample distributions of the MLE and compares the asymptotic variance estimates constructed from the Hessian matrix and the outer product of the score. The simulation results favour the latter. As an empirical example, we compare three leading economic indicators in terms of describing U.S. industrial production.

Confidence Intervals for Seroprevalence

This paper concerns the construction of confidence intervals in standard seroprevalence surveys. In particular, we discuss methods for constructing confidence intervals for the proportion of individuals in a population infected with a disease using a sample of antibody test results and measurements of the test’s false positive and false negative rates. We begin by documenting erratic behavior in the coverage probabilities of standard Wald and percentile bootstrap intervals when applied to this problem. We then consider two alternative sets of intervals constructed with test inversion. The first set of intervals are approximate, using either asymptotic or bootstrap approximation to the finite-sample distribution of a chosen test statistic. We consider several choices of test statistic, including maximum likelihood estimators and generalized likelihood ratio statistics. We show with simulation that, at empirically relevant parameter values and sample sizes, the coverage probabilities for these intervals are close to their nominal level and are approximately equi-tailed. The second set of intervals are shown to contain the true parameter value with probability at least equal to the nominal level, but can be conservative in finite samples.

A Quasi-Bayesian change point detection with exchangeable weights

A Quasi-Bayesian change point test statistic is derived, under the fixed and random exchangeable priors, which is asymptotically close to an important subclass of Poisson–Dirichlet weights. While detecting change at each point, the random prior can be updated for future change points in a sequential sampling manner. Asymptotic behaviors of quasi-Bayesian test statistics, under the null and alternative hypothesis are represented in terms of stochastic integrals. Also, the M-estimate approaches for change in mean for both cases of finite and infinite variance observations are discussed. Aside to the application in change point detections, the asymptotic analysis reveals some interesting probabilistic properties. Multiple change point detection and simulation of finite sample distribution of test statistics are studied Finally, a conclusion section is also given.

Monitoring Value-at-Risk and Expected Shortfall Forecasts

This paper introduces formal monitoring procedures as a risk-management tool. Continuously monitoring risk forecasts allows practitioners to swiftly review and update their forecasting procedures as soon as forecasts turn inadequate. Similarly, regulators may take timely action in case reported risk forecasts become poor. Extant (one-shot) backtests require, however, that all data are available prior to testing and are not informative of when inadequacies might have occurred. To monitor value-at-risk and expected shortfall forecasts “online”—that is, as new observations become available—we construct sequential testing procedures. We derive the exact finite-sample distributions of the proposed procedures and discuss the suitability of asymptotic approximations. Simulations demonstrate good behavior of our exact procedures in finite samples. An empirical application to major stock indices during the COVID-19 pandemic illustrates the economic benefits of our monitoring approach. This paper was accepted by Agostino Capponi, finance. Funding: Y. Hoga gratefully acknowledges support of the German Research Foundation [Grant HO 6305/1-1]. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2022.4460 .

Finite-sample properties of estimators for first and second order autoregressive processes

The class of autoregressive (AR) processes is extensively used to model temporal dependence in observed time series. Such models are easily available and routinely fitted using freely available statistical software like R. A potential problem is that commonly applied estimators for the coefficients of AR processes are severely biased when the time series are short. This paper studies the finite-sample properties of well-known estimators for the coefficients of stationary AR(1) and AR(2) processes and provides bias-corrected versions of these estimators which are quick and easy to apply. The new estimators are constructed by modeling the relationship between the true and originally estimated AR coefficients using weighted orthogonal polynomial regression, taking the sampling distribution of the original estimators into account. The finite-sample distributions of the new bias-corrected estimators are approximated using transformations of skew-normal densities, combined with a Gaussian copula approximation in the AR(2) case. The properties of the new estimators are demonstrated by simulations and in the analysis of a real ecological data set. The estimators are easily available in our accompanying R-package for AR(1) and AR(2) processes of length 10–50, both giving bias-corrected coefficient estimates and corresponding confidence intervals.

BOOTSTRAP INFERENCE FOR MULTIPLE CHANGE-POINTS IN TIME SERIES

In this paper, we propose two bootstrap procedures, namely parametric and block bootstrap, to approximate the finite sample distribution of change-point estimators for piecewise stationary time series. The bootstrap procedures are then used to develop a generalized likelihood ratio scan method (GLRSM) for multiple change-point inference in piecewise stationary time series, which estimates the number and locations of change-points and provides a confidence interval for each change-point. The computational complexity of using GLRSM for multiple change-point detection is as low as $O(n(\log n)^{3})$ for a series of length n. Extensive simulation studies are provided to demonstrate the effectiveness of the proposed methodology under different scenarios. Applications to financial time series are also illustrated.

Exact distribution of the F-statistic under heteroskedasticity of unknown form for improved inference

In this paper, we derive the exact finite sample distribution of the F () statistic for a single linear restriction on the regression parameters. We show that the F statistic can be expressed as a ratio of quadratic forms, and therefore its exact cumulative distribution under the null hypothesis can be derived from the result of Imhof [Computing the distribution of quadratic forms in normal variables. Biometrika. 1961;48(3/4):419–426]. A numerical calculation is carried out for the exact distribution of the F statistic using various HC covariance matrix estimators, and the rejection probability under the null hypothesis (size) based on the exact distribution is examined. The results show the exact finite sample distribution is remarkably reliable, while, in comparison, the use of the F-table leads to a serious over-rejection when the sample is not large or leveraged/unbalanced.

Empirical likelihood ratio test on quantiles under a density ratio model

Population quantiles are important parameters in many applications. Enthusiasm for the development of effective statistical inference procedures for quantiles and their functions has been high for the past decade. In this article, we study inference methods for quantiles when multiple samples from linked populations are available. The research problems we consider have a wide range of applications. For example, to study the evolution of the economic status of a country, economists monitor changes in the quantiles of annual household incomes, based on multiple survey datasets collected annually. Even with multiple samples, a routine approach would estimate the quantiles of different populations separately. Such approaches ignore the fact that these populations are linked and share some intrinsic latent structure. Recently, many researchers have advocated the use of the density ratio model (DRM) to account for this latent structure and have developed more efficient procedures based on pooled data. The nonparametric empirical likelihood (EL) is subsequently employed. Interestingly, there has been no discussion in this context of the EL-based likelihood ratio test (ELRT) for population quantiles. We explore the use of the ELRT for hypotheses concerning quantiles and confidence regions under the DRM. We show that the ELRT statistic has a chi-square limiting distribution under the null hypothesis. Simulation experiments show that the chi-square distributions approximate the finite-sample distributions well and lead to accurate tests and confidence regions. The DRM helps to improve statistical efficiency. We also give a real-data example to illustrate the efficiency of the proposed method.

Mildly Explosive Autoregression with Anti‐persistent Errors*

AbstractAn asymptotic distribution is derived for the least squares (LS) estimate of a first‐order autoregression with a mildly explosive root and anti‐persistent errors. While the sample moments depend on the Hurst parameter asymptotically, the Cauchy limiting distribution theory remains valid for the LS estimates in the model without intercept and a model with an asymptotically negligible intercept. Monte Carlo studies are designed to check the precision of the Cauchy distribution in finite samples. An empirical study based on the monthly NASDAQ index highlights the usefulness of the model and the new limiting distribution.