A minimum variance adaptive technique for parameter estimation and hypothesis testing

This report presents numerical results of an approach for parameter estimation and hypothesis testing that does not rely on specific assumptions about the underlying distribution of errors in the measured data. This approach combines robust estimation procedures, the bootstrap method for estimation of parameter uncertainties, permutation techniques for hypothesis testing, and adaptive approaches to estimation in order to obtain the minimum variance estimator or test statistic (within a predefined class) for the data under consideration. The technique produces efficient estimators of central tendency and powerful test statistics, even for small sample sizes. (Portions of this work have been presented in preliminary form (Turkheimer et al., 1996)).     

A diagnostic statistic for functional-coefficient autoregressive models

In this paper, functional coefficient autoregressive (FAR) models proposed by Chen and Tsay (1993) are considered. We propose a diagnostic statistic for FAR models constructed by comparing between parametric and nonparametric estimators of the functional form of the FAR models. We show asymptotic properties of our statistic mathematically and it can be applied to the estimation of the delay parameter and the specification of the functional form of FAR models.     

响应变量缺失下半参数变系数EV模型的约束统计推断

文章研究了半参数变系数EV模型在线性约束条件下的估计和检验问题,当响应变量缺失、非参数部分协变量带有测量误差时,利用局部纠偏的Profile最小二乘估计、Lagrange乘子方法和借补技术构造了回归模型参数分量两类纠偏约束估计量。此外,为了检验线性约束条件,构造了借补的Profile Lagrange乘子检验统计量,并通过蒙特卡洛数值模拟验证估计量和检验统计量的有效性。     

Simultaneous closeness among order statistics to population quantiles

We derive expressions for the probability that an individual order statistic is closest to the target parameter among the order statistics from a complete random sample. Results are given for random variables with bounded and complete support. We then apply these general results to location-scale parameter families of distributions with specific applications to estimation of percentiles. In this case, simultaneous-closeness probabilities depend upon the parameters through the value of p in the percentile and the sample size, n. Results are finally illustrated with the estimation of percentiles for normal and exponential distributions.     

The first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood

Abstract

This paper investigates the first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood. The limiting distribution of log empirical likelihood ratio statistic is constructed. Asymptotic convergence and confidence region results of empirical likelihood ratio are given. Hypothesis testing is considering, and maximum empirical likelihood estimation for parameter is acquired. Simulations are given to show that the maximum empirical likelihood estimation is more efficient than the conditional least squares estimation.     

Applying skovgaard's modified directed likelihood statistic to mixed linear models

The introduction of software to calculate maximum likelihood estimates for mixed linear models has made likelihood estimation a practical alternative to methods based on sums of squares. Likelihood based tests and confidence intervals, however, may be misleading in problems with small sample sizes. This paper discusses an adjusted version of the directed log-likelihood statistic for mixed models that is highly accurate for testing one parameter hypotheses. Indroduced by Skovgaard (1996, Journal of the Bernoulli Society,2,145-165), we show in mixed models that the statistic has a simple conpact from that may be obtained from standard software. Simulation studies indicate that this statistic is more accurate than many of the specialized procedure that have been advocated.     

Bootstrapping a consistent nonparametric goodness-of-fit test

In this paper, we employ the parametric bootstrap to approximate the finite sample distribution of a goodness-of-fit test statistic in Fan (1994). We show that the proposed bootstrap procedure works in that the bootstrap distribution conditional on the random sample tends to the asymptotic distribution of the test statistic in probability. A simulation study demonstrates that the bootstrap approximation works extremely well in small samples with only 25 observations and is very robust to the value of the smoothing parameter in the kernel density estimation.     

Improved testing for the binomial distribution using chi-squared components with data-dependent cells

A power study suggests that a good test of fit analysis for the binomial distribution is provided by a data-dependent Chernoff–Lehmann X ² test with class expectations greater than unity, and its components. These data-dependent statistics involve arithmetically simple parameter estimation, convenient approximate distributions and provide a comprehensive assessment of how well the data agree with a binomial distribution. We suggest that a well-performed single test of fit statistic is the Anderson–Darling statistic.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

ON SOME OPTIMALITY PROPERTIES OF FISHER-RAO SCORE FUNCTION IN TESTING AND ESTIMATION

The score function is associated with some optimality features in statistical inference. This review article looks on the central role of the score in testing and estimation. The maximization of the power in testing and the quest for efficiency in estimation lead to score as a guiding principle. In hypothesis testing, the locally most powerful test statistic is the score test or a transformation of it. In estimation, the optimal estimating function is the score. The same link can be made in the case of nuisance parameters: the optimal test function should have maximum correlation with the score of the parameter of primary interest. We complement this result by showing that the same criterion should be satisfied in the estimation problem as well.     

Inference for random coefficient INAR(k) with the occasional level shift random noise based on dual empirical likelihood

The INAR(k) model has been widely used in various kinds of fields. However, there are little discussions about the INAR(k) model with the occasional level shift random noise. In this paper, the maximum likelihood estimation of parameter based on martingale difference sequence is given, the log empirical likelihood ratio test statistic is obtained and the test statistic converges to chi-square distribution, we prove that the confidence region of the parameter is convex. Furthermore, the numerical simulation of the proposed INAR(k) model is given, which illustrates the effectiveness of the model. Then, the proofs of asymptotic results are given in the Appendix.     

部分函数型线性变系数模型的序列相关检验

部分函数型线性变系数模型(PFLVCM)是近几年出现的一个比较灵活、应用广泛的新模型。在实际应用中,搜集到的经济和金融数据往往存在序列相关性。如果不考虑数据间的相关性直接对其进行建模,会影响模型中参数估计的精度和有效性。本文主要研究了PFLVCM中误差的序列相关性的检验问题,基于经验似然,把标量时间序列数据相关性检验的方法拓展到函数型数据中,提出了经验对数似然比检验统计量,并在零假设下得到了检验统计量的近似分布。通过蒙特卡洛数值模拟说明该统计量在有限样本下有良好的水平和功效。最后,把该方法用于检验美国商业用电消费数据是否有序列相关性,证明该统计量的有效性和实用性。     

Rank-based analysis of repeated measures block designs

A rank-based inference is developed for repeated measures balanced incomplete block and randomized complete block designs using a suitable dispersion function. Asymptotic distributions of rank estimators are developed after establishing approximate linearity of the gradient vector of the dispersion function. Unlike available nonparametric procedures for those designs, estimation and testing are tied together. Three different test statistics are developed for testing the linear hypotheses. Friedman's (1937) statistic and Durbin's (1951) statistic are particular cases of one of the three proposed statistics. An estimate of a scale parameter which appears in the ARE expression as well as as in the variences and covariances of the rank estimators is discussed.     

No-cointegration test based on fractional differencing: Some Monte Carlo results

This paper examines the use of the t-statistic in the Geweke–Porter-Hudak regression for the estimation of the fractional differencing parameter as a test for cointegration. The critical values of the test statistic are estimated using Monte Carlo methods. The results confirm that the test will over-reject the null hypothesis of no cointegration if the standard-normal critical values are used. The estimated critical values are generally robust to the nuisance parameters in the autoregressive or moving average specification of the error process of the component time series. Exceptions occur when the dependent variable in the cointegration regression follows an autoregressive process with a large positive parameter or a moving average process with a large negative parameter.     

Improving the bandwidth-free inference methods by prewhitening

In this paper we consider inference of parameters in time series regression models. In the traditional inference approach, the heteroskedasticity and autocorrelation consistent (HAC) estimation is often involved to consistently estimate the asymptotic covariance matrix of regression parameter estimator. Since the bandwidth parameter in the HAC estimation is difficult to choose in practice, there has been a recent surge of interest in developing bandwidth-free inference methods. However, existing simulation studies show that these new methods suffer from severe size distortion in the presence of strong temporal dependence for a medium sample size. To remedy the problem, we propose to apply the prewhitening to the inconsistent long-run variance estimator in these methods to reduce the size distortion. The asymptotic distribution of the prewhitened Wald statistic is obtained and the general effectiveness of prewhitening is shown through simulations.     

Asymptotic cumulants of the estimator of the canonical parameter in the exponential family

Asymptotic cumulants of the maximum likelihood estimator of the canonical parameter in the exponential family are obtained up to the fourth order with the added higher-order asymptotic variance. In the case of a scalar parameter, the corresponding results with and without studentization are given. These results are also obtained for the estimators by the weighted score, especially for those using the Jeffreys prior. The asymptotic cumulants are used for reducing bias and mean square error to improve a point estimator and for interval estimation to have higher-order accuracy. It is shown that the kurtosis to squared skewness ratio of the sufficient statistic plays a fundamental role.     

The Relationship Between the T2 Statistic and the Influence Function

Hotelling's T² statistic has many applications in multivariate analysis. In particular, it can be used to measure the influence that a particular observation vector has on parameter estimation. For example, in the bivariate case, there exists a direct relationship between the ellipse generated using a T² statistic for individual observations and the hyperbolae generated using Hampel's influence function for the corresponding correlation coefficient. In this paper, we jointly use the components of an orthogonal decomposition of the T² statistic and some influence functions to identify outliers or influential observations. Since the conditional components in the T² statistic are related to the possible changes in the correlation between a variable and a group of other variables, we consider the theoretical influence functions of the correlations and multiple correlation coefficients. Finite-sample versions of these influence functions are used to find the estimated influence function values.     

Linear regression with compositional explanatory variables

Compositional explanatory variables should not be directly used in a linear regression model because any inference statistic can become misleading. While various approaches for this problem were proposed, here an approach based on the isometric logratio (ilr) transformation is used. It turns out that the resulting model is easy to handle, and that parameter estimation can be done in like in usual linear regression. Moreover, it is possible to use the ilr variables for inference statistics in order to obtain an appropriate interpretation of the model.     

Interval estimation for the breakpoint in segmented regression: a smoothed score‐based approach

This paper is concerned with interval estimation for the breakpoint parameter in segmented regression. We present score‐type confidence intervals derived from the score statistic itself and from the recently proposed gradient statistic. Due to lack of regularity conditions of the score, non‐smoothness and non‐monotonicity, naive application of the score‐based statistics is unfeasible and we propose to exploit the smoothed score obtained via induced smoothing. We compare our proposals with the traditional methods based on the Wald and the likelihood ratio statistics via simulations and an analysis of a real dataset: results show that the smoothed score‐like statistics perform in practice somewhat better than competitors, even when the model is not correctly specified.     

Application of Concomitants of Order Statistics of Independent Non Identically Distributed Random Variables in Estimation

In this article, we obtain expressions for the pdf of a single concomitant of order statistic and the joint pdf of a pair of concomitants of order statistics of independent non identically distributed random variables. Using these expressions, we find the means, variances and covariances of order statistics arising from independent non identically distributed bivariate Pareto distributions. A method of estimation of a common parameter involved in several bivariate Pareto distributions using concomitants of order statistics is also discussed.