Moment Estimation of the Parameters in Rayleigh Distribution with Two Parameters

In this article, the moment estimation of the parameters in two-parameter Rayleigh distribution is studied. For given sample values, the necessary and sufficient conditions on the estimating equations which have unique solution are obtained, and it is also proved that these conditions are satisfied on asymptotic probability 1. Furthermore, the strong consistency and asymptotic normality are obtained. Finally, some simulation experiments are made to show the conclusions.     

Estimation in linear mixed models for longitudinal data under linear restricted conditions

In the paper, we consider a linear mixed model (LMM) for longitudinal data under linear restriction and find the estimators for the parameters of interest. The strong consistency and asymptotic normality of the estimators are obtained under some regularity conditions. Besides, we derive the strong consistent estimator of the fourth moment for the error which is useful for statistical inference for random effects and error variance. Simulations and an example are reported for illustration.     

On Simultaneous Estimation of the Number of Signals and Frequencies of Multiple Sinusoids When Some Observations Are Missing

In an earlier article (Bai et al., 1999 Bai , Z. D. , Rao , C. R. , Wu , Y. H. , Zen , M. M. , Zhao , L. C. ( 1999 ). The simultaneous estimation of the number of signals and frequencies of multiple sinusods when some observations are missing: I. Asymptotics . Proc. Natl. Acad. Sci. 96 : 11,106 – 11,110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), the problem of simultaneous estimation of the number of signals and frequencies of multiple sinusoids is considered in the case that some observations are missing. The number of signals is estimated with an information theoretic criterion and the frequencies are estimated with eigenvariation linear prediction. Asymptotic properties of the procedure are investigated but the Monte Carlo simulation is not performed. In this article, a slightly different but scale invariant criterion for detection is proposed and the estimation of frequencies remains the same. Asymptotic properties of this new procedure are provided. Monte Carlo Simulation for both procedures is carried out. Furthermore, comparison on the real signals is also given.     

Two-stage generalized moment method approach for bidimensional random coefficient autoregressive models

ABSTRACT

A two-dimensionally indexed random coefficients autoregressive models (2D ? RCAR) and the corresponding statistical inference are important tools for the analysis of spatial lattice data. The study of such models is motivated by their second-order properties that are similar to those of 2D ? (G)ARCH which play an important role in spatial econometrics. In this article, we study the asymptotic properties of two-stage generalized moment method (2S ? GMM) under general asymptotic framework for 2D ? RCA models. So, the efficiency, strong consistency, the asymptotic normality, and hypothesis tests of 2S ? GMM estimation are derived. A simulation experiment is presented to highlight the theoretical results.     

Statistical Analysis of Non Linear Least Squares Estimation for Harmonic Signals in Multiplicative and Additive Noise

Abstract

This article concerns the stochastically constrained linear model under a biased assumption. We propose a quasi-stochastically constrained least squares estimator. Furthermore, we provide the expectation of this estimator, demonstrate its consistency and asymptotic normality. In the end of the article, the simulation study of the new estimator shows that it is superior to the least squares estimator, ridge estimator, and the linear constrained estimators under certain conditions by comparing the mean squared errors of these estimators.     

Semiparametric Estimation for Two-Sample Location-Scale Models under Type I Censorship

To compare two samples under Type I censorship, this article proposes a method of semiparametric inference for the two-sample location-scale problem when the model for two samples is characterized by an unknown distribution and two unknown parameters. Simultaneous estimators for both the location shift and scale change parameters are given. It is shown that the two estimators are strongly consistent and asymptotically normal. The approach in this article can also be used for scale-shape models. Monte Carlo studies indicate that the proposed estimation procedure performs well in finite and heavily censored samples, maintains high relative efficiencies for a wide range of censoring proportions and is robust to the model misspecification, and also outperforms other competitive estimators.     

A modified likelihood ratio test for homogeneity in finite mixture models

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in practice. We propose a modified LRT for homogeneity in finite mixture models with a general parametric kernel distribution family. The modified LRT has a χ-type of null limiting distribution and is asymptotically most powerful under local alternatives. Simulations show that it performs better than competing tests. They also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic, even for moderate sample sizes.     

Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.     

Some linear model techniques for analyzing small-circle spherical data

The author investigates least squares as a method for fitting small-circle models to a sample of unit vectors in R³. He highlights a local linear model underlying the estimation of the parameters of a circle. This model is used to construct an estimation algorithm and regression-type inference procedures for the parameters of a circle. It makes it possible to compare the fit of a small circle with that of a spherical ellipse. The limitations of the least-squares approach are emphasized: when the errors are bounded away from 0, the least-squares estimators are not consistent as the sample size goes to infinity. Two examples, concerned with the migration of elephant seals and with the classification of geological folds, are analyzed using the linear model techniques proposed in this work.     

Asymptotic properties of the kernel mode estimator under twice censorship model

In this article, we study the asymptotic properties of the kernel estimator of the mode and density function when the data are twice censored. More specifically, we first establish a strong uniform consistency over a compact set with a rate of the kernel density estimator and then we give the consistency with rate and asymptotic normality for the kernel mode estimator. An application to confidence bands is given.     

A Further Study on Reliable Life Estimation under Ranked Set Sampling

This article considers nonparametric estimation of reliable life based on ranked set sampling and its properties. It is proven analytically that the large sample efficiency of the reliable life estimator under the balanced ranked set sampling is higher than that under the simple random sampling of the same size, but the relative efficiency damps away as the reliable life moves away from the median on both directions. To improve the efficiency for the estimation of extreme reliable life, we then propose a reliable life estimator under a modified ranked set sampling protocol, its strong consistency and asymptotic normality are established. The proposed sampling is shown to be superior to the balanced ranked set sampling, and the relative advantage improves as the reliable life moves away from median. Finally, results of simulation studies for small sample as well as an application to a real data set are presented to illustrate some of the theoretical findings.     

Statistical Analysis of Parameter Estimation for 2-D Harmonics in Multiplicative and Additive Noise

This article considers the problem of parameter estimation for two dimensional (2-D) multi-component harmonics in non zero-mean multiplicative and additive noise. The least squares estimators (LSEs) are proposed to estimate the coherent model parameters, and some statistical results of the LSEs are obtained, including strong consistency, strong convergence rate, and asymptotic normality. Furthermore, the LSEs-based estimators are proposed to estimate the noncoherent model parameters, and the strong consistency and the asymptotic normality are also proved. Finally, some numerical experiments are performed to see how the asymptotic results work for finite sample sizes.     

The credibility premiums based on estimated moment-generating function

Traditionally, experience ratemaking is in principle based on the idea of Bühlmann’s credibility theory that, except for net premiums, was rarely applied to other premium calculation principles. This article uses Bühlmann’s credibility procedure to estimate moment-generating functions (MGFs) of risks and then deduces estimates of moments of those risks. For the premium calculation principles that can be expressed as functions of certain moments or more directly of the MGFs, this article develops a new type of experience ratemaking methods by means of the estimated MGFs and discusses their consistency and asymptotic normality. Numerical simulation shows that, under the Esscher and exponential premium principles, the new credibility estimates are better than existing credibilityestimates in the literature.     

ON ASYMPTOTIC PROPERTIES OF A TWO DIMENSIONAL FREQUENCY ESTIMATOR

Estimating parameters of a two dimensional frequency model is an important problem in statistical signal processing. In this paper, we consider the two-dimensional frequency model in presence of an additive stationary noise. We consider two different estimators and obtain their asymptotic properties. The asymptotic properties can be used to construct confidence intervals of the unknown parameters and for testing purposes also. The small sample performances of these estimators are observed using numerical simulations.     

Maximum log-likelihood ratio test for a change in three parameter Weibull distribution

Asymptotic behavior of a log-likelihood ratio statistic for testing a change in a three parameter Weibull distribution is studied. It is shown that if a shape parameter α>2$\alpha >2$ the law of iterated logarithm for maximum-likelihood estimators is still valid and the log-likelihood testing statistic is asymptotically distributed (after an appropriate normalization) according to a Gumbel distribution.     

A goodness-of-fit test for the stratified proportional hazards model for survival data

Abstract

Goodness-of-fit testing is addressed in the stratified proportional hazards model for survival data. A test statistic based on within-strata cumulative sums of martingale residuals over covariates is proposed and its asymptotic distribution is derived under the null hypothesis of model adequacy. A Monte Carlo procedure is proposed to approximate the critical value of the test. Simulation studies are conducted to examine finite-sample performance of the proposed statistic.     

Estimation in inverse Gaussian regression: comparison of asymptotic and bootstrap distributions

This paper presents a brief review of the asymptotic properties of the pseudo-maximum likelihood estimator in the regression model where the reciprocal of the mean of the dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo-maximum likelihood estimator presented in Babu and Chaubey (1996) is highlighted and a simulation study is carried out to compare the approximation yielded by the bootstrap distribution to that of the asymptotic distribution.     

Asymptotic Properties of Quasi-Maximum Likelihood Estimates in Generalized Linear Models

This article introduces a novel method, named JC ₁, for obtaining G-efficient mixture design to fit quadratic models. The advantage of JC ₁ method over existing algorithms is that it gives G-efficient designs without need of generating all the extreme vertices, edge centroids and constraint plane centroids of the mixture experimental region. The performance of the new method is illustrated and its comparison is given with popularly used algorithms—Snee (1975) algorithm and Welch (1985 Welch , W. J. ( 1985 ). ACED: Algorithms for the construction of experimental designs . Amer. Statistician 39 : 146 .[Crossref] , [Google Scholar]) ACED algorithm for second-order (quadratic model) designs and it is observed that JC ₁ method performs as well as the existing methods or sometimes better than those with additional advantage of large savings in computational efforts.