Detection of similar successive groups in a model with diverging number of variable groups

Abstract

In this article, a linear model with grouped explanatory variables is considered. The idea is to perform an automatic detection of different successive groups of the unknown coefficients under the assumption that the number of groups is of the same order as the sample size. The standard least squares loss function and the quantile loss function are both used together with the fused and adaptive fused penalty to simultaneously estimate and group the unknown parameters. The proper convergence rate is given for the obtained estimators and the upper bound for the number of different successive group is derived. A simulation study is used to compare the empirical performance of the proposed fused and adaptive fused estimators, and a real application on the air quality data demonstrates the practical applicability of the proposed methods.     

Second-Order Efficiency for Two-Stage Estimation of a Linear Function of Normal Mean Vectors when Covariance Matrices Have Some Structures

Abstract

We consider two-stage estimation for a fixed-span confidence region about a linear function of mean vectors from π _i : N _p (μ _i , Σ _i ), i = 1,…,k (≥2), when Σ _i 's have some structures. The purpose of this article is to investigate asymptotic efficiency of the estimation up to the second order in terms of the sample size. An adjustment of the design constant and a proper choice of the initial sample size appearing in the two-stage estimation are proposed to have asymptotic second-order efficiency. Some simulations are carried out to see moderate sample size performances of the proposed two-stage estimation. An example is given for a demonstration.     

Estimation of the size of a finite population in sequatial sampling:truncatedcombinatiorial numbers

This note considers the problem of unbiased estimation of the size of a finite population, or the number of equiprobable classes in a population, when sequential sampling plans are aplied. Combinatorial numbers, e.g., Stirling numbers of the second kind or Lah numbers, occurring in sampling distributions when the sample size is fixed, have to be adjusted to take into account the stopping rule. This requires the concept of a truncated combinatorial number, and linearly truncated Stirling numbers and Lah numbers are instances discussed here. UMVU estimators of population size are expressible in terms of ratios of such truncated combiantorial numbers.     

Comparison of Stopping Rules in Sequential Estimation of the Number of Classes in a Population

Abstract

In sequential analysis, investigation of stopping rules is important, as they govern the sampling cost and derivation and accuracy of frequentist inference. We study stopping rules in sampling from a population comprised of an unknown number of classes where all classes are equally likely to occur in each selection. We adopt Blackwell's criterion for a “more informative experiment” to compare stopping rules in our context and derive certain complete class results, which provide some guidance for selecting a stopping rule. We show that it suffices to let the stopping probability, at any time, depend only on the number of selections and the number of discovered classes up to that time. A more informative stopping rule costs a higher expected sample size, and conversely, any given stopping rule can be improved with an increment in expected sample size. Admissibility within all stopping rules with a uniform upper bound on average sample size is also discussed. Any fixed-sample-size rule is shown to be admissible within an appropriate class. Finally, we show that for the minimal sufficient statistic to be complete, which is useful for unbiased estimation, the stopping rule must be nonrandomized.     

Sequential Change-Point Detection and Estimation

Abstract

Two groups of sequential testing procedures are proposed to detect an abrupt change in the distribution of a sequence of observations: truncated and open ended. They are based on large sample strong approximations of the efficient score vector under the null hypothesis of no change and under the alternative hypothesis. An estimator of the time of change is proposed and its approximate bias is analyzed. The estimation of the new parameters that describe the changed distribution naturally follows.     

Applications of Sequentially Estimating the Mean in a Normal Distribution Having Equal Mean and Variance

Abstract

We suppose that we have at our disposal a sequence of independent observations from a N(θ, θ) distribution where θ(>0) is an unknown parameter. Such a model will make good sense to approximate a Poisson(θ) distribution, especially when θ(>0) is moderately large. The Fisher-information contained in the sample mean, the sample variance, and the MLE are included (Appendix A). The derivation of the UMVUE of θ and some remarks regarding computational complexities in numerically evaluating its expression, even for small fixed-sample-size n, are also included (Appendix B). Assuming the availability of a lower bound θ _L (>0) for θ, both purely sequential and two-stage bounded risk methodologies are developed for estimating θ. We have considered the analogs of fixed-sample-size MLE, sample mean, sample median, sample variance, and the UMVUE. First-order asymptotic properties of both purely sequential as well as two-stage sample sizes and the associated risk-bound for an analog of the MLE have been found. Extensive investigations based on computer-simulations have been carried out and the previously stated estimators of θ are compared with one another. The effect of the ratio θ _L /θ on the pilot sample size is critically examined. Overall, we have found that an analog of the fixed-sample-size MLE performs most satisfactorily. In the end, both proposed methodologies are successfully implemented to investigate how these work in two practical situations with the help of real datasets. They show that these methodologies stay fairly robust under mild departures from normality.     

Nonparametric Partial Sequential Test for Location Shift at an Unknown Time Point

Abstract

A sufficient condition for the uniqueness of multinomial sequential unbiased estimators is provided generalizing a classical result for binomial samples. Unbiased estimators are applied to infer the parameters of multidimensional or multinomial random walks that are observed until they reach a boundary. Clinical trials are shown to be representable within this scheme and an application to the estimation of the multinomial probabilities following multinomial clinical trials is presented.     

Estimation in Functional Lagged Regression

The paper introduces a functional time series (lagged) regression model. The impulse‐response coefficients in such a model are operators acting on a separable Hilbert space, which is the function space L² in applications. A spectral approach to the estimation of these coefficients is proposed and asymptotically justified under a general nonparametric condition on the temporal dependence of the input series. Since the data are infinite‐dimensional, the estimation involves a spectral‐domain dimension‐reduction technique. Consistency of the estimators is established under general data‐dependent assumptions on the rate of the dimension‐reduction parameter. Their finite‐sample performance is evaluated by a simulation study that compares two ad hoc approaches to dimension reduction with an alternative, asymptotically justified method.     

Sequential Fixed Width Confidence Intervals for the Offset Between Two Network Clocks

Abstract

Estimation of the offset between two network clocks has received a lot of attention in the literature, with the motivating force being data networking applications that require synchronous communication protocols. Statistical modeling techniques have been used to develop improved estimation algorithms, with a recent development being the construction of a confidence interval based on a fixed sample size. Lacking in the fixed sample size confidence interval procedures is a useable relationship between sample size and the width of the resulting confidence interval. Were that available, an optimum sample size could be determined to achieve a specified level of precision in the estimator and thereby improve the efficiency of the estimation procedure by reducing unnecessary overhead in the network that is associated with collecting the data used by the estimation schemes. A fixed sample size confidence interval that has a prescribed width is not available for this problem. However, in this paper we develop and compare alternative sequential intervals with fixed width and demonstrate that an effective solution is available.     

On a Type of Probability Stopping Rule for Toxicity Study

Abstract

In early phase cancer clinical trials where toxicity events follow independent and identical Bernoulli distributions indexed by patients, the Bayesian stopping rule has been used for continuous monitoring of toxicity along with an affordable maximum sample size (N). This article studies some properties of an heuristic procedure where the trial will stop at the first time that the posterior probability that the toxicity rate (p) is greater than a threshold (η) is greater than certain probability threshold (τ). Specifically, we study the pattern formed by stopping times and regions, recursive stopping probability computation, and toxicity rate estimation. Some relevant theoretical results are given. The presented results are potentially useful for guiding toxicity clinical trial designs.     

An Approximate Innovation Method For The Estimation Of Diffusion Processes From Discrete Data

Abstract. In this paper, an approximate innovation method is introduced for the estimation of diffusion processes, given a set of discrete and noisy observations of some of their components. The method is based on a recent extension of local linearization filters to the general case of continuous–discrete state–space models with multiplicative noise. This filtering method provides adequate approximations for the prediction and filter estimates that are required by the innovation method in the estimation of the unknown parameters and the unobserved component of the diffusion process. The performance of approximate innovation estimators is illustrated by means of numerical simulations.     

Parameter Estimation Under Failure-Censored Constant-Stress Life Testing Model: A Bayesian Approach

Abstract

This article compares likelihood and Bayesian estimations for partially accelerated constant-stress life test model under type II censoring assuming Pareto distribution of the second kind. Both maximum likelihood and Bayesian estimators of the model parameters are derived. The posterior means and posterior variances are obtained under the squared error loss function using Lindley's approximation procedure. The advantages of this proposed procedure are shown. Monte Carlo simulations are conducted under different samples sizes and different parameter values to assess and compare the proposed methods of estimation. A noninformative prior on the model parameters is used to make the comparison more meaningful. It has been observed that Lindley's method usually provides posterior variances and mean squared errors smaller than those of the maximum likelihood estimators. That is, Lindley's method produces improved estimates, which is an advantage of this method.     

Robust and powerful serial correlation tests with new robust estimates in ARX models

Abstract. We consider robust serial correlation tests in autoregressive models with exogenous variables (ARX). Since the least squares estimators are not robust when outliers are present, a new family of estimators is introduced, called residual autocovariances for ARX (RA‐ARX). They provide resistant estimators that are less sensible to abnormal observations in the output variable of the dynamic model. Such ‘bad’ observations could be due to unexpected phenomena such as economic crisis or equipment failure in engineering, among others. We show that the new robust estimators are consistent and we can consider robust and powerful tests of serial correlation in ARX models based on these estimators. The new one‐sided tests of serial correlation are obtained in extending Hong's (1996) approach in a framework resistant to outliers. They are based on a weighted sum of robust squared residual autocorrelations and on any robust and n^1/2‐consistent estimators. Our approach generalizes Li's (1988) test statistic, that can be interpreted as a test using the truncated uniform kernel. However, many kernels deliver a higher power. This is confirmed in a simulation study, where we investigate the finite sample properties of the new robust serial correlation tests in comparison to some commonly used robust and non‐robust tests.     

Local Information Theoretic Methods for smooth Coefficients Dynamic Panel Data Models

This paper considers estimation and inference in semiparametric smooth coefficients dynamic panel data models. It proposes a class of local estimators that can be given an interesting information‐theoretic interpretation and a number of test statistics that can be used to test for the (local) correct specification of the model and for the constancy of the smooth coefficients. The results of the paper are rather general as they allow for the three cases of ‘large N, small T’, ‘small N, large T’ and ‘large N, large T’, for the possibility that some of the regressors might be correlated with the unobservable errors and for the possibility that some of the variables used in the estimation might not be directly observable. Simulations show that the proposed method have competitive finite sample properties.     

Volatility Estimation and Jump Testing via Realized Information Variation

We put forward a new method to construct jump‐robust estimators of integrated volatility, namely realized information variation (RIV) and realized information power variation (RIPV). The ‘information’ here refers to the difference between two‐grid of ranges in high‐frequency intervals, which preserves continuous variation and eliminates jump variation asymptotically. We show that such kind of estimators have several superior statistical properties, i.e., the estimators are generally more efficient with sufficiently using the opening, high, low, closing (OHLC) data in high‐frequency intervals, and have faster jump convergence rate due to a new type of construction. For example, the RIV is much more efficient than the estimators that only use closing prices or ranges, and the RIPV has faster jump convergence rate at O_p(1/n), while the other (multi)power‐based estimators are usually . We also extend our results to integrated quarticity and higher‐order variation estimation, and then propose the corresponding jump testing method. Simulation studies provide extensive evidence on the finite sample properties of our estimators and tests, comparing with alternative prevalent methods. Empirical results further demonstrate the practical relevance and advantages of our method.     

Sequential model selection method for nonparametric autoregression

Abstract

In this article, the nonparametric autoregression estimation problem for quadratic risks is considered. To this end, we develop a new adaptive sequential model selection method based on the efficient sequential kernel estimators proposed by Arkoun and Pergamenshchikov (2016). Moreover, we develop a new analytical tool for general regression models to obtain the non-asymptotic sharp oracle inequalities for both usual quadratic and robust quadratic risks. Then, we show that the constructed sequential model selection procedure is optimal in the sense of oracle inequalities.     

An Estimating Method for Parametric Spectral Densities of Gaussian Time Series

An estimating method for spectral densities of Gaussian time series that belong to a parametric model is proposed. Spectral density estimators are evaluated by using average Kullback–Leibler divergence from the true spectral density to estimated spectral densities. In the classical approach, unknown spectral densities are estimated by replacing the unknown parameters by asymptotically efficient estimates. In the alternative method introduced in the present paper, spectral density estimates usually do not belong to the model. The alternative spectral density estimators asymptotically dominate the classical ones. The difference in average Kullback–Leibler divergence between them can be regarded as the mixture mean curvature of the model in the space of all spectral densities. The explicit expression for the proposed estimators of spectral densities of autoregressive processes is obtained. The accuracy of prediction can be improved by using predictors that correspond to the alternative spectral density estimators.     

BIAS-CORRECTED NONPARAMETRIC SPECTRAL ESTIMATION

Abstract. The theory of nonparametric spectral density estimation based on an observed stretch X₁,…, X_N from a stationary time series has been studied extensively in recent years. However, the most popular spectral estimators, such as the ones proposed by Bartlett, Daniell, Parzen, Priestley and Tukey, are plagued by the problem of bias, which effectively prohibits ?N-convergence of the estimator. This is true even in the case where the data are known to be m-dependent, in which case ?N-consistent estimation is possible by a simple plug-in method. In this report, an intuitive method for the reduction in the bias of a nonparametric spectral estimator is presented. In fact, applying the proposed methodology to Bartlett's estimator results in bias-corrected estimators that are related to kernel estimators with lag-windows of trapezoidal shape. The asymptotic performance (bias, variance, rate of convergence) of the proposed estimators is investigated; in particular, it is found that the trapezoidal lag-window spectral estimator is ?N-consistent in the case of moving-average processes, and ?(N/log/N)-consistent in the case of autoregressive moving-average processes. The finite-sample performance of the trapezoidal lag-window estimator is also assessed by means of a numerical simulation.