Asymptotic results for estimation and testing variances in regression models

Usually the variance of independent observations resulting from a linear or a nonlinear relationship is estimated by the Least-Squares residual estimator. In this paper its asymptotic properties are investigated. Further the asymptotic behaviour of tests for one-sided hypotheses on the variance is studied. The paper splits into two parts, the first one concerned with linear and the second one with nonlinear models.     

Some asymptotic results in finite populations

In this paper we investigate limiting properties of predictors of some finite population quantities. Both, the sample size and the population size are considered to become large. Limiting properties like consistency and asymptotic normality of the best linear unbiased predictors of the population total and of the finite population regression coefficient are investigated.     

Extreme value index estimator using maximum likelihood and moment estimation

ABSTRACT

When a distribution function is in the max domain of attraction of an extreme value distribution, its tail can be well approximated by a generalized Pareto distribution. Based on this fact we use a moment estimation idea to propose an adapted maximum likelihood estimator for the extreme value index, which can be understood as a combination of the maximum likelihood estimation and moment estimation. Under certain regularity conditions, we derive the asymptotic normality of the new estimator and investigate its finite sample behavior by comparing with several classical or competitive estimators. A simulation study shows that the new estimator is competitive with other estimators in view of average bias, average MSE, and coefficient of variance of the new device for the optimal selection of the threshold.     

Asymptotic theory for local estimators based on Bregman divergence

This article is concerned with asymptotic theory for local estimators based on Bregman divergence. We consider a localized version of Bregman divergence induced by a kernel weight and minimize it to obtain the local estimator. We provide a rigorous proof for the asymptotic consistency of the local estimator in a situation where both the sample size and the bandwidth involved in the kernel weight increase. Asymptotic normality of the local estimator is also developed under the same asymptotic scenario. Monte Carlo simulations are also performed to confirm the theoretical results. The Canadian Journal of Statistics 47: 628–652; 2019 © 2019 Statistical Society of Canada     

Asymptotic properties for the moment estimators in Rayleigh distribution with two parameters

ABSTRACT

We consider the asymptotic properties for the moment estimators in Rayleigh distribution with two parameters. The law of the iterated logarithm for the estimators can be obtained. Moreover, we can give a simple proof of the asymptotic normality which has been obtained by Li and Li (2012) Li, Y.W., Li, M.H. (2012). Moment estimation of the parameters in Rayleigh distribution with two parameters. Commun. Stat.-Theor. Methods 41:2643–2660.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

Bayesian shrinkage estimation based on Rayleigh type-II censored data

In this paper, we construct a Bayes shrinkage estimator for the Rayleigh scale parameter based on censored data under the squared log error loss function. Risk-unbiased estimator is derived and its risk is computed. A Bayes shrinkage estimator is obtained when a prior point guess value is available for the scale parameter. Risk-bias of the Bayes shrinkage estimator is considered. A comparison between the proposed Bayes shrinkage estimator and the risk-unbiased estimator is provided using calculation of the relative efficiency. A numerical example is presented for illustrative and comparative purposes.     

ROC curve estimation based on local smoothing

ROC curve is a graphical representation of the relationship between sensitivity and specificity of a diagnostic test. It is a popular tool for evaluating and comparing different diagnostic tests in medical sciences. In the literature,the ROC curve is often estimated empirically based on an empirical distribution function estimator and an empirical quantile function estimator. In this paper an alternative nonparametric procedure to estimate the ROC Curve is suggested which is based on local smoothing techniques. Several numerical examples are presented to evaluate the performance of this procedure.     

Asymptotic expansions relating to discrimination based on two-step monotone missing samples

This paper proposes two asymptotic expansions relating to discrimination based on two-step monotone missing samples. These asymptotic expansions have been obtained by Okamoto (1963) and McLachlan (1973) for complete data under multivariate normality. This paper extends the results up to the terms of the first order in the case of two-step monotone missing samples, respectively. Especially, these asymptotic expansions play important roles in obtaining the asymptotic approximations for the probabilities of misclassification in discriminant analysis. The simulation studies have been also conducted in order to evaluate the accuracy of the approximation derived in this paper.     

Small-sample results for maximum-likelihood estimation in branching processes

In this article, small sample properties of the maximum-likelihood estimator (m.l.e.) for the offspring distribution (pk) and its mean m are considered in the context of the simple branching process. A representation theorem is given for the m.l.e. of (Pk) from which the m.l.e. of m is obtained. The case where p0 + p1 + p2 = 1 is studied in detail: numerical results are given for the exact bias of these estimators as a function of the age of the process; a curve fitting analysis expresses the bias of m? as a function of the mean and the variance of the offspring distribution and finally an “approximate m.l.e.” for (pk) is given.     

Asymptotic Results of a Nonparametric Conditional Quantile Estimator for Functional Time Series

We consider the estimation of the conditional quantile function when the covariates take values in some abstract function space. The main goal of this article is to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional quantile under the α-mixing assumption and on the concentration properties on small balls of the probability measure of the functional regressors. Some applications and particular cases are studied. This approach can be applied in time series analysis to the prediction and building of confidence bands. We illustrate our methodology with El Niño data.     

Non Parametric Regression Quantile Estimation for Dependent Functional Data under Random Censorship: Asymptotic Normality

In this paper we study a smooth estimator of the regression quantile function in the censorship model when the covariates take values in some abstract function space. The main goal of this paper is to establish the asymptotic normality of the kernel estimator of the regression quantile, under α-mixing condition and, on the concentration properties on small balls probability measure of the functional regressors. Some applications and particular cases are studied. This study can be applied in time series analysis to the prediction and building confidence bands. Some simulations are drawn to lend further support to our theoretical results and to compare the quality of behavior of the estimator for finite samples with different rates of censoring and sizes.     

Semiparametric estimation in copula models

The author recalls the limiting behaviour of the empirical copula process and applies it to prove some asymptotic properties of a minimum distance estimator for a Euclidean parameter in a copula model. The estimator in question is semiparametric in that no knowledge of the marginal distributions is necessary. The author also proposes another semiparametric estimator which he calls “rank approximate Z‐estimator” and whose asymptotic normality he derives. He further presents Monte Carlo simulation results for the comparison of various estimators in four well‐known bivariate copula models.     

Smooth nonparametric quantile estimation under censoring: simulations and bootstrap methods

Based on right-censored data from a lifetime distribution F , a smooth nonparametric estimator of the quantile function Q (p) is given by Qn(p)=h 1jjQn(t)K((t-p)/h)dt, where QR(p) denotes the product-limit quantile function. Extensive Monte Carlo simulations indicate that at fixed p this kernel-type quantile estimator has smaller mean squared error than (L(p) for a range of values of the bandwidth h. A method of selecting an "optimal" bandwidth (in the sense of small estimated mean squared error) based on the bootstrap is investigated yielding results consistent with the simulation study. The bootstrap is also used to obtain interval estimates for Q (p) after determining the optimal value of h.     

Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations

We study asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, while in the critical and supercritical cases, non-standard asymptotic behaviour is described.     

Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACT

In this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.     

Minimum variance unbiased estimation based on bootstrap iterations

Practical computation of the minimum variance unbiased estimator (MVUE) is often a difficult, if not impossible, task, even though general theory assures its existence under regularity conditions. We propose a new approach based on iterative bootstrap bias correction of the maximum likelihood estimator to accurately approximate the MVUE. Viewing bootstrap iteration as a Markov process, we develop a computational algorithm for bias correction based on arbitrarily many bootstrap iterations. The algorithm, when applied parametrically to finite sample spaces, does not involve Monte Carlo simulation. For infinite sample spaces, a nonparametric version of the algorithm is combined with a preliminary round of Monte Carlo simulation to yield an approximate MVUE. Both algorithms are computationally more efficient and stable than conventional simulation-based bootstrap iterations. Examples are given of both finite and infinite sample spaces to illustrate the effectiveness of our new approach. Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7026/97P).     

Some theoretical results concerning non parametric estimation by using a judgment poststratification sample

ABSTRACT

In this paper, some of the properties of non parametric estimation of the expectation of g(X) (any function of X), by using a judgment poststratification sample (JPS), have been discussed. A class of estimators (including the standard JPS estimator and a JPS estimator proposed by Frey and Feeman (2012 Frey, J., Feeman, T.G. (2012). An improved mean estimator for judgment post-stratification. Comput. Stat. Data Anal. 56(2):418–426.[Crossref], [Web of Science ®] , [Google Scholar], Comput. Stat. Data An.) is considered. The paper provides mean and variance of the members of this class, and examines their consistency and asymptotic distribution. Specifically, the results are for the estimation of population mean, population variance, and cumulative distribution function. We show that any estimators of the class may be less efficient than simple random sampling (SRS) estimator for small sample sizes. We prove that the relative efficiency of some estimators in the class with respect to balanced ranked set sampling (BRSS) estimator tends to 1 as the sample size goes to infinity. Furthermore, the standard JPS mean estimator, and Frey–Feeman JPS mean estimator are specifically studied and we show that two estimators have the same asymptotic distribution. For the standard JPS mean estimator, in perfect ranking situations, optimum values of H (the ranking class size), for different sample sizes, are determined non parametrically for populations that are not heavily skewed or thick tailed.