Estimating density ratio with application to discriminant analysis

The estimator ? for the ratio of densities based on grade transformation is introduced. Under certain conditions the strong uniform consistency of ? is proved and its asymptotic law is determined. The result is applied to study the behaviour of divergence curve used in discriminant analysis.     

Estimated Generalized Least Squares for Random Coefficient Regression Models

ABSTRACT. This paper considers a general class of random coefficient regression (RCR) models to represent pooled cross-sectional and time series data. A new method is given to estimate the covariance matrix of the error component in these RCR models. Also, the asymptotic and small sample properties of the estimated generalized least squares estimator of the regression coefficient vector are established. Procedures for testing a linear restriction on the mean vector of the random coefficients are derived. Finally, a test for non-randomness in the RCR model is devised, and the asymptotic distribution of the test statistic is obtained.     

Asymptotic distribution for the order statistics of a truncated one-parameter family of distributions

In this article we have presented some of the asymptotic theorems related to one-truncation parameter family of distributions ? Comparison of performance of different estimators and other inferential problems have been tackled - Also applications of the main results have been given and illustrated their uses with examples.     

Estimation of non-occurrence probability in periodic model

Dimitrov and Khalil (1992) introduced a class of new probability distributions for modeling environmental evolution with periodic behavior. One of the key parameters in these distributions is α, the probability that the event being studied does not occur. In that article the authors derive an estimator for this parameter assuming a series of conditions. In this article it is shown that the estimator is valid under more general conditions, i.e. same of the assumptions are not necessary. It is shown that under the assumption that the elapsed time measured from the starting point of a period until the first occurrence time of the event given that the event occurred in this cycle is related to α, an approximate maximum likelihood estimator of a is proposed. The large sample properties of the estimator are discussed. Monte Carlo study is done for supporting the theoretical results.     

On huntseerger type shrinkage estimator

This paper is concerned with Hintsberger type weighted shrinkage estimator of a parameter when a target value of the same is available. Expressions for the bias and the mean squared error of the estimator are derived. Some results concerning the bias, existence of uniformly minimum mean squared error estimator etc. are proved. For certain c to ices of the weight function, numerical results are presented for the pretest type weighted shrinkage estimator of the mean of normal as well as exponential distributions.     

Two-stage james-stein estimators of the mean based on prior knowledge

We suggest five types of two-stage James-Stein type estimators of the mean vector μ based on prior knowledge about μ and two-stage sampling scheme proposed by Waikar and Katti(1971) Their risks are evaluated and calculated to compare with two-stage estimator suggested by Waikar and Katti(1971) when the prior form of an initial estimate of μ is 0. We find that the five estimators suggested here all have high efficiencies in large dimensions and/or in large value of ratio of two sample sizes at each stage.     

Non-parametric inference for the effect of a treatment on survival times with application in the health and social sciences

In this paper we perform inference on the effect of a treatment on survival times in studies where the treatment assignment is not randomized and the assignment time is not known in advance. Two such studies are discussed: a heart transplant program and a study of Swedish unemployed eligible for employment subsidy. We estimate survival functions on a treated and a control group which are made comparable through matching on observed covariates. The inference is performed by conditioning on waiting time to treatment, that is, time between the entrance in the study and treatment. This can be done only when sufficient data are available. In other cases, averaging over waiting times is a possibility, although the classical interpretation of the estimated survival functions is lost unless hazards are not functions of waiting time. To show unbiasedness and to obtain an estimator of the variance, we build on the potential outcome framework, which was introduced by J. Neyman in the context of randomized experiments, and adapted to observational studies by D.B. Rubin. Our approach does not make parametric or distributional assumptions. In particular, we do not assume proportionality of the hazards compared. Small sample performance of the estimator and a derived test of no treatment effect are studied in a Monte Carlo study.     

Parameter estimation for ihe birnbaum-saunders distribution based on symmetrically censored samples

Ihe Bimbaum-Saunders distribution was derived to model fatigue life. Frequently, it becomes necessary to stop a life testing process before all the test items have failed. Therefore, estimation procedures need to be developed for use when censoring occurs. In this article, we have derived estimators for the parameters of this distribution which may be used for complete samples or Type II symmetrically censored samples A simulation study was also conducted to examine the performance of these estimators.     

Optimal Sequential Design in a Controlled Non-parametric Regression

Abstract. In a non‐parametric regression, the heteroscedasticity (dependence of the variance of the regression error on the predictor) can be a serious complication in estimation or visualization of an underlying regression function. If a controlled sampling is permitted, then the statistician can choose the design of predictors which attenuates the effect of heteroscedasticity. It is proposed to use a design which minimizes the mean integrated squared error of the regression function estimation. Then the corresponding optimal design density is proportional to the standard deviation of the regression error (the so‐called scale function). Because in general the statistician does not know an underlying scale function, the natural question is as follows: is it possible to suggest a sequential design which performs as well as an oracle that knows the underlying scale function? The answer is ‘yes’, and a corresponding sequential procedure is developed. It is proved, for the first time in the literature, that a data‐driven sequential design, together with an adaptive regression estimator, can mimic the oracle and be sharp minimax. Further, it is shown that the suggested method is feasible for small samples.     

A linear regression with unobserved dependent variables

Let (?,X) be a random vector such that E(X|?) = ? and Var(x|?) a + b? + c?² for some known constants a, b and c. Assume X₁,…,X_n are independent observations which have the same distribution as X. Let t(X) be the linear regression of ? on X. The linear empirical Bayes estimator is used to approximate the linear regression function. It is shown that under appropriate conditions, the linear empirical Bayes estimator approximates the linear regression well in the sense of mean squared error.