Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. The structural forms are mainly used in econometrics to introduce instantaneous relationships between economic variables. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) portmanteau statistics in this framework. It is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte Carlo experiments illustrate the finite sample performance of the modified portmanteau test.     

Estimating conditional tail expectation with actuarial applications in view

We develop statistical inferential tools for estimating and comparing conditional tail expectation (CTE) functions, which are of considerable interest in actuarial science. In particular, we construct estimators for the CTE functions, develop the necessary asymptotic theory for the estimators, and then use the theory for constructing confidence intervals and bands for the functions. Both parametric and non-parametric approaches are explored. Simulation studies illustrate the performance of estimators in various situations. Results are obtained under minimal assumptions, and the general Vervaat process plays a crucial role in achieving these goals.     

Unbiasedness of fourier integral estimator under weaker assumptions

The Fourier integral estimator (FIE) of the hazard rate function have been introduced by Singpurwalla and Wong (1983) in order to improve the rate of convergence of mean square error. However, the asymptotic properties are derived under several assumptions. These assumptions are too strong to be satisfied in most applications. The situation becomes worse when some of these assumptions appear contradictory to each other. This greatly limits the usage of FIE. This paper endeavors to release or weaken some of these assumptions and therefore expand the applicable coverage of FIE.     

A decision theoretic approach to a distribution–free compound classification problem

A compound decision problem with component decision problem being the classification of a random sample as having come from one of the finite number of univariate populations is investigated. The Bayesian approach is discussed. A distribution–free decision rule is presented which has asymptotic risk equal to zero. The asymptotic efficiencies of these rules are discussed.

The results of a compter simulation are presented which compares the Bayes rule to the distribution–free rule under the assumption of normality. It is found that the distribution–free rule can be recommended in situations where certain key lo cation parameters are not known precisely and/or when certain distributional assumptions are not satisfied.     

Weighted Dickey–Fuller processes for detecting stationarity

Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey–Fuller (DF) processes, and related stopping times, which may be called weighted DF control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For general dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance parameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation.     

Non Asymptotic Minimax Estimation of Functionals with Noisy Observations

The minimax estimation of functionals by a finite number of noisy observations is considered. A new way to formalize the problem that enables one to calculate non asymptotic optimal estimates is proposed. The calculations can be turned into and executed as a computer algorithm or carried out analytically under week assumptions on random variables. Some examples are considered.     

Pattern–mixture and selection models for analysing longitudinal data with monotone missing patterns

Summary. We examine three pattern–mixture models for making inference about parameters of the distribution of an outcome of interest Y that is to be measured at the end of a longitudinal study when this outcome is missing in some subjects. We show that these pattern–mixture models also have an interpretation as selection models. Because these models make unverifiable assumptions, we recommend that inference about the distribution of Y be repeated under a range of plausible assumptions. We argue that, of the three models considered, only one admits a parameterization that facilitates the examination of departures from the assumption of sequential ignorability. The three models are nonparametric in the sense that they do not impose restrictions on the class of observed data distributions. Owing to the curse of dimensionality, the assumptions that are encoded in these models are sufficient for identification but not for inference. We describe additional flexible and easily interpretable assumptions under which it is possible to construct estimators that are well behaved with moderate sample sizes. These assumptions define semiparametric models for the distribution of the observed data. We describe a class of estimators which, up to asymptotic equivalence, comprise all the consistent and asymptotically normal estimators of the parameters of interest under the postulated semiparametric models. We illustrate our methods with the analysis of data from a randomized clinical trial of contracepting women.     

On the Estimation of Integrated Covariance Functions of Stationary Random Fields

Abstract.  For stationary vector-valued random fields on     the asymptotic covariance matrix for estimators of the mean vector can be given by integrated covariance functions. To construct asymptotic confidence intervals and significance tests for the mean vector, non-parametric estimators of these integrated covariance functions are required. Integrability conditions are derived under which the estimators of the covariance matrix are mean-square consistent. For random fields induced by stationary Boolean models with convex grains, these conditions are expressed by sufficient assumptions on the grain distribution. Performance issues are discussed by means of numerical examples for Gaussian random fields and the intrinsic volume densities of planar Boolean models with uniformly bounded grains.     

Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.     

Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Abstract

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.     

Goodness-of-fit Tests for Mixed Models

Abstract.  Mixed linear models have become a very useful tool for modelling experiments with dependent observations within subjects, but to establish their appropriateness several assumptions have to be checked. In this paper, we focus on the normality assumptions, using goodness-of-fit tests that make allowance for possible design imbalance. These tests rely on asymptotic results, which are established via empirical process theory. The power of the tests is explored empirically, and examples illustrate some aspects of the usage of the tests.     

On LSE in regression model for long-range dependent random fields on spheres

We study the asymptotic behaviour of least squares estimators (LSE) in regression models for long-range dependent random fields observed on spheres. The LSE can be given as a weighted functional of long-range dependent random fields. It is known that in this scenario the limits can be non-Gaussian. We derive the limit distribution and the corresponding rate of convergence for the estimators. The results were obtained under rather general assumptions on the random fields. Simulation studies were conducted to support theoretical findings.     

U-Statistics based on spacings

In this paper, we investigate the asymptotic theory for U-statistics based on sample spacings, i.e. the gaps between successive observations. The usual asymptotic theory for U-statistics does not apply here because spacings are dependent variables. However, under the null hypothesis, the uniform spacings can be expressed as conditionally independent Exponential random variables. We exploit this idea to derive the relevant asymptotic theory both under the null hypothesis and under a sequence of close alternatives.The generalized Gini mean difference of the sample spacings is a prime example of a U-statistic of this type. We show that such a Gini spacings test is analogous to Rao's spacings test. We find the asymptotically locally most powerful test in this class, and it has the same efficacy as the Greenwood statistic.     

Extreme value analysis of actuarial risks: estimation and model validation

We give an overview of several aspects arising in the statistical analysis of extreme risks with actuarial applications in view. In particular it is demonstrated that empirical process theory is a very powerful tool, both for the asymptotic analysis of extreme value estimators and to devise tools for the validation of the underlying model assumptions. While the focus of the paper is on univariate tail risk analysis, the basic ideas of the analysis of the extremal dependence between different risks are also outlined. Here we emphasize some of the limitations of classical multivariate extreme value theory and sketch how a different model proposed by Ledford and Tawn can help to avoid pitfalls. Finally, these theoretical results are used to analyze a data set of large claim sizes from health insurance.     

Generalised regression estimation via the bootstrap

A generalised regression estimation procedure is proposed that can lead to much improved estimation of population characteristics, such as quantiles, variances and coefficients of variation. The method involves conditioning on the discrepancy between an estimate of an auxiliary parameter and its known population value. The key distributional assumption is joint asymptotic normality of the estimates of the target and auxiliary parameters. This assumption implies that the relationship between the estimated target and the estimated auxiliary parameters is approximately linear with coefficients determined by their asymptotic covariance matrix. The main contribution of this paper is the use of the bootstrap to estimate these coefficients, which avoids the need for parametric distributional assumptions. First‐order correct conditional confidence intervals based on asymptotic normality can be improved upon using quantiles of a conditional double bootstrap approximation to the distribution of the studentised target parameter estimate.     

Consistency of the estimator of binary response models based on AUC maximization

This paper examines the asymptotic properties of a binary response model estimator based on maximization of the Area Under receiver operating characteristic Curve (AUC). Given certain assumptions, AUC maximization is a consistent method of binary response model estimation up to normalizations. As AUC is equivalent to Mann-Whitney U statistics and Wilcoxon test of ranks, maximization of area under ROC curve is equivalent to the maximization of corresponding statistics. Compared to parametric methods, such as logit and probit, AUC maximization relaxes assumptions about error distribution, but imposes some restrictions on the distribution of explanatory variables, which can be easily checked, since this information is observable.     

Estimating the p-values of robust tests for the linear model

In this paper, we study the estimation of p-values for robust tests for the linear regression model. The asymptotic distribution of these tests has only been studied under the restrictive assumption of errors with known scale or symmetric distribution. Since these robust tests are based on robust regression estimates, Efron's bootstrap (1979) presents a number of problems. In particular, it is computationally very expensive, and it is not resistant to outliers in the data. In other words, the tails of the bootstrap distribution estimates obtained by re-sampling the data may be severely affected by outliers.We show how to adapt the Robust Bootstrap (Ann. Statist 30 (2002) 556; Bootstrapping MM-estimators for linear regression with fixed designs, http://mathstat.carleton.ca/~matias/pubs.html) to this problem. This method is very fast to compute, resistant to outliers in the data, and asymptotically correct under weak regularity assumptions. In this paper, we show that the Robust Bootstrap can be used to obtain asymptotically correct, computationally simple p-value estimates. A simulation study indicates that the tests whose p-values are estimated with the Robust Bootstrap have better finite sample significance levels than those obtained from the asymptotic theory based on the symmetry assumption.Although this paper is focussed on robust scores-type tests (in: Directions in Robust Statistics and Diagnostics, Part I, Springer, New York), our approach can be applied to other robust tests (for example, Wald- and dispersion-type also discussed in Markatou et al., 1991).     

On the asymptotic behaviour of selector statistics

Adaptive procedures proposed by Hogg are based on selector statistics for the skewness and the tails. The asymptotic properties of several proposed selector statistics are investigated. Since ail these statistics have under some assumptions asymptoti¬cally a normal distribution, their properties depend on the asymptotic bias and variance. The reasonable concept to compare the different selector statistics is based on the selection probabil¬ities in discriminating the type of the underlying distribution. These values are numerically calculated and analyzed in detail for a number of underlying distributions.     

Properties of maximum likelihood estimators of variance components in the one-way classification,balanced data

We studied properties of maximum likelihood estimators (MLEs) of the variance components obtained from balanced data of the one-way classification. Exact and asymptotic expected values and variances of these MLEs were derived under the usual normality assumptions. Numerical studies illustrate these expected values and variances, and also illustrate the probability of obtaining a negative solution to the maximum likelihood (ML) equation for the between-class variance component. Simulations were used to study the robustness of the ML estimators under non-normal distributions.     

On approximate least squares estimators of parameters of one-dimensional chirp signal

Chirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.