Estimation in proportional hazard and log-linear models

Proportional hazard models and models where the dependent variables follow a linear model are shown to be equivalent if and only if the hazard rate is the product of a non-negative periodic function and a Weibull factor. Estimates based on rank statistics are proposed for the parameters in the proportional hazard model.     

Estimation of a linear transformation and an associated distributional problem

A multivariate “errors in variables” regression model is proposed which generalizes a model previously considered by Gleser and Watson (1973). Maximum likelihood estimators [MLE's] for the parameters of this model are obtained, and the consistency properties of these estimators are investigated. Distribution of the MLE of the “error” variance is obtained in a simple case while the mean and the variance of the estimator are obtained in this case without appealing to the exact distribution.     

Maximum Likelihood Estimation of Parameters in a Mixture Model

The estimation of parameters of the log normal distribution based on complete and censored samples are considered in the literature. In this article, the problem of estimating the parameters of log normal mixture model is considered. The Expectation Maximization algorithm is used to obtain maximum likelihood estimators for the parameters, as the likelihood equation does not yield closed form expression. The standard errors of the estimates are obtained. The methodology developed here is then illustrated through simulation studies. The confidence interval based on large-sample theory is obtained.     

A construction of repeated measurements designs with balance for residual effects

Families of Repeated Measurements Designs balanced for residual effects are constructed (whenever the divisibility conditions allow), under the assumption that the number of periods is less than the number of treatments and that each treatment precedes each other treatment once. These designs are then shown to be connected for both residual and direct treatment effects.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

On the ABLUE of the normal mean from a censored sample

The asymptotically best linear unbiased estimate (ABLUE) of the normal mean is discussed. The estimate is based on k selected order statistics chosen from a singly or doubly censored large sample of size n(>k). The coefficients, the asymptotic relative efficiency of the estimate, and the optimum spacing of k real numbers between 0 and 1 which determines the optimum ranks of order statistics, are provided. A comparison between the ABLUE and the iterated maximum likelihood estimate is made.     

Estimation of a residual distribution with small numbers of repeated measurements

The authors consider the estimation of a residual distribution for different measurement problems with a common measurement error process. The problem is motivated by issues arising in the analysis of gene expression data but should have application in other similar settings. It is implicitly assumed throughout that there are large numbers of measurements but small numbers of repeated measurements. As a consequence, the distribution of the estimated residuals is a biased estimate of the residual distribution. The authors present two methods for the estimation of the residual distribution with some restriction on the form of the distribution. They give an upper bound for the rate of convergence for an estimator based on the characteristic function and compare its performance with that of another estimator with simulations.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

Comparison of two methods in estimating standard error of the method of simulated moments estimators for generalized linear mixed models

ABSTRACT

The likelihood of a generalized linear mixed model (GLMM) often involves high-dimensional integrals, which in general cannot be computed explicitly. When direct computation is not available, method of simulated moments (MSM) is a fairly simple way to estimate the parameters of interest. In this research, we compared parametric bootstrap (PB) and nonparametric bootstrap methods (NPB) in estimating the standard errors of MSM estimators for GLMM. Simulation results show that when the group size is large, the PB and NPB perform similarly; when group size is medium, NPB performs better than PB in estimating standard errors of the mean.     

An L-statistic approach to a test of exponentiality against IFR alternatives

In this note we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a test statistic constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distribution of the test statistic is obtained and the test is shown to be consistent.     

Sequence-compound estimation in scale-exponential families and speed of convergence

This paper deals with a sequence-compound estimation. The component problem is the squared error loss estimation of θ?[a,b] based on an observation X whose p.d.f. is of the form $\text{u(x)c(θ)}\text{exp}\text{(?}\text{x}\text{θ}\text{)}$. For each $\text{0<t<}\text{1}\text{2}$ a class of sequence-compound estimators $\text{ψ}\text{?}\text{=}\text{ψ}\text{?}{}_1\text{,}\text{ψ}\text{?}{}_2\text{,…)}$ is exhibited whose compound risk (average of risks) up to stage n differs from the Bayes envelope (in the component problem) w.r.t. the empiric distribution G_n of the parameters involved up to stage n by a quantity of order O(n^?δt) for a δ>0. It is also shown that at any stage i the difference of the risk of ψ?_i and the risk of the Bayes response w.r.t. G_i?1 is O(i^?δt). Examples of the above type of families are given where δ is min{1,$\text{2a}\text{b}$} and t is arbitrarily close to $\text{1}\text{2}$. Here it may be worthwhile to mention that a rate $\text{O}\text{(n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$ or better has not yet been obtained even in a very special family of densities.     

Non-parametric recursive estimates of a probability density function and its derivatives

Recursive estimates of a probability density function (pdf) are known. This paper presents recursive estimates of a derivative of any desired order of a pdf. Let f be a pdf on the real line and p?0 be any desired integer. Based on a random sample of size n from f, estimators f^(p)_n of f^(p), the pth order derivatives of f, are exhibited. These estimators are of the form $\text{n}{}^{\mathrm{?1}}\text{∑n}\text{j=1}\text{δ}{}_{\mathrm{jp}}$, where δ_jp depends only on p and the jth observation in the sample, and hence can be computed recursively as the sample size increases. These estimators are shown to be asymptotically unbiased, mean square consistent and strongly consistent, both at a point and uniformly on the real line. For pointwise properties, the conditions on f^(p) have been weakened with a little stronger assumption on the kernel function.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

A class of count models and a new consistent test for the Poisson distribution

We define a class of count distributions which includes the Poisson as well as many alternative count models. Then the empirical probability generating function is utilized to construct a test for the Poisson distribution, which is consistent against this class of alternatives. The limit distribution of the test statistic is derived in case of a general underlying distribution, and efficiency considerations are addressed. A simulation study indicates that the new test is comparable in performance to more complicated omnibus tests.     

A new approach for joint modelling of longitudinal measurements and survival times with a cure fraction

The joint analysis of longitudinal measurements and survival data is useful in clinical trials and other medical studies. In this paper, we consider a joint model which assumes a linear mixed $tt$ model for longitudinal measurements and a promotion time cure model for survival data and links these two models through a latent variable. A semiparametric inference procedure with an EM algorithm implementation is developed for the parameters in the joint model. The proposed procedure is evaluated in a simulation study and applied to analyze the quality of life and time to recurrence data from a clinical trial on women with early breast cancer. The Canadian Journal of Statistics 40: 207–224; 2012 © 2012 Statistical Society of Canada     

Nonnegative Unbiased Estimation of Scale Parameters and Associated Quantiles Based on a Ranked Set Sample

In this article, we are interested in estimating the scale parameter in location and scale families. It is well known that the best linear unbiased estimator (BLUE) of scale parameter based on a simple random sample (SRS) is nonnegative. However, the BLUE of scale parameter based on a ranked set sample (RSS) can assume negative values. We suggest various modifications of BLUE of scale parameter based on RSS so that the resulting estimators are unbiased as well as nonnegative. Their performances in terms of relative efficiencies are compared and some recommendations are made for normal, logistic, double exponential, two-parameter exponential and Weibull distributions. We also briefly discuss an application of the proposed nonnegative BLUE of scale parameter for quantile estimation for the above populations.     

On-line control procedures under the random-walk model with measurement error and attribute observations

Assuming the quality characteristic process follows the random-walk model, the economic online control procedures are investigated by using the boundary correction technique when (1) the noise or measurement error follows either the random-walk model or the pure random model, and (2) the observations are taken by attributes only. Simple approximations for the long-run cost rate and optimal-control parameters are obtained.