The exact density of nonparametric regression estimators: fixed design case

This paper studies the exact density of a general nonparametric regression estimator when the errors are non-normal. The fixed design case is considered. The density function is derived by an application of the technique of Davis (1976)     

An extension of the ranked set sampling theory

This paper is concerned with ranked set sampling theory which is useful to estimate the population mean when the order of a sample of small size can be found without measurements or with rough methods. Consider n sets of elements each set having size m. All elements of each set are ranked but only one is selected and quantified. The average of the quantified elements is adopted as the estimator. In this paper we introduce the notion of selective probability which is a generalization of a notion from Yanagawa and Shirahata (1976). Uniformly optimal unbiased procedures are found for some (n,m). Furthermore, procedures which are unbiased for all distributions and are good for symmetric distributions are studied for (n,m) which do not allow uniformly optimal unbiased procedures.     

Forecasting Economic Time Series With Structural and Box-Jenkins Models: A Case Study

The basic structural model is a univariate time series model consisting of a slowly changing trend component, a slowly changing seasonal component, and a random irregular component. It is part of a class of models that have a number of advantages over the seasonal ARIMA models adopted by Box and Jenkins (1976). This article reports the results of an exercise in which the basic structural model was estimated for six U.K. macroeconomic time series and the forecasting performance compared with that of ARIMA models previously fitted by Prothero and Wallis (1976).     

A NOTE ON THE HOMOGENETIC ESTIMATE FOR THE VARIANCE OF THE KAPLAN–MEIER ESTIMATE

ABSTRACT

The Greenwood estimate (GE) is commonly employed for estimating the variance of the Kaplan–Meier estimate (KME) even though it underestimates the variance. To reduce the bias of the GE, Zhao (1996) proposed an alternative, called the homogenetic estimate (HE). In this note, we point out that the HE actually esimates the variance of the reduced sample estimate (RE) and can seriously overestimate that of the KME. We also derive the explict relationship between the HE and the GE and discuss the use of the HE.     

Empirical bayes estimation of functionals of unknown probability measures

The empirical Dayes approach to one and two sal-npie problcrns has beeir considered by Korwar and Hollander (1976), Holiander and Korwar (1976) and Phadia and Susarla (1979). In this article we essen- tially generalize their empirical Bayes results by replacing the inlicaro-functions of. the sets (?∞,x) and {X≦Y} by arbitrary mea5, irable functions h(x) and h(x,y). More speclfically, the ernpiricaion yes estimation of esrimabie paramerers of degree one ani KG,I;ti kliown probability measure Pon (R,R) is considered. The asymptotic optimality of the these estimators, obtaining the exact risk expressions, is established. Also the results of Dalal and Phad (1983) we extended to the estimation of an estimable parametric function of an unknow probability measure P on (R² , B²)     

Some methods for constructing C-designs

Block designs having a property described by Caliński (1971) and Saha (1976), referred to as C-property, are further considered. A necessary and sufficient condition for a block design to have the C-property is given by Saha (1976) and another by Ceranka and Koz?owska (1983). In this paper some methods for constructing such block designs are presented.     

On integrals of dirichlet distributions and their applications

The purpose of this paper is to extend results obtained by Sahai and Anderson (1973) and Hurlburt and Spiegel (1976) computationally as well as theoretically. We apply results given in Yassaee (1974), (1976) and apply our computer program to show that one can evaluate the probability of type one error for conditional tests in linear model, in general. We show that one can compute the probability integral of Dirichlet distribution by the use of our program which computes the probability integral of inverted Dirichlet distribution.     

The asymptotic properties of the maximum-relevance weighted likelihood estimators

We define the maximum-relevance weighted likelihood estimator (MREWLE) using the relevance-weighted likelihood function introduced by Hu and Zidek (1995). Furthermore, we establish the consistency of the MREWLE under a wide range of conditions. Our results generalize those of Wald (1948) to both nonidentically distributed random variables and unequally weighted likelihoods (when dealing with independent data sets of varying relevance to the inferential problem of interest). Asymptotic normality is also proven. Applying these results to generalized smoothing model is discussed.     

Excursion and meander in random walk

Bernoulli bridge, excursion and meander are defined on the symmetric random walk similarly to Brownian bridge, excursion and meander (cf. Chung 1976). Distributions of certain characteristics defined on these Bernoulli processes, which are of a combinatorial nature, and their limits are obtained. Using weak convergence, these derivations give a verification of some of the earlier results on Brownian excursion and Brownian meander, as well as some new results.     

Assessing inter- and intra-agreement for dependent binary data: a Bayesian hierarchical correlation approach

Agreement measures are designed to assess consistency between different instruments rating measurements of interest. When the individual responses are correlated with multilevel structure of nestings and clusters, traditional approaches are not readily available to estimate the inter- and intra-agreement for such complex multilevel settings. Our research stems from conformity evaluation between optometric devices with measurements on both eyes, equality tests of agreement in high myopic status between monozygous twins and dizygous twins, and assessment of reliability for different pathologists in dysplasia. In this paper, we focus on applying a Bayesian hierarchical correlation model incorporating adjustment for explanatory variables and nesting correlation structures to assess the inter- and intra-agreement through correlations of random effects for various sources. This Bayesian generalized linear mixed-effects model (GLMM) is further compared with the approximate intra-class correlation coefficients and kappa measures by the traditional Cohen’s kappa statistic and the generalized estimating equations (GEE) approach. The results of comparison studies reveal that the Bayesian GLMM provides a reliable and stable procedure in estimating inter- and intra-agreement simultaneously after adjusting for covariates and correlation structures, in marked contrast to Cohen’s kappa and the GEE approach.     

Kernel Density-Based Linear Regression Estimate

For linear regression models with non normally distributed errors, the least squares estimate (LSE) will lose some efficiency compared to the maximum likelihood estimate (MLE). In this article, we propose a kernel density-based regression estimate (KDRE) that is adaptive to the unknown error distribution. The key idea is to approximate the likelihood function by using a nonparametric kernel density estimate of the error density based on some initial parameter estimate. The proposed estimate is shown to be asymptotically as efficient as the oracle MLE which assumes the error density were known. In addition, we propose an EM type algorithm to maximize the estimated likelihood function and show that the KDRE can be considered as an iterated weighted least squares estimate, which provides us some insights on the adaptiveness of KDRE to the unknown error distribution. Our Monte Carlo simulation studies show that, while comparable to the traditional LSE for normal errors, the proposed estimation procedure can have substantial efficiency gain for non normal errors. Moreover, the efficiency gain can be achieved even for a small sample size.     

Statistical inference for α-series process with the inverse Gaussian distribution

Statistical inferences for the geometric process (GP) are derived when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). An α-series process, as a possible alternative to the GP, is introduced since the GP is sometimes inappropriate to apply some reliability and scheduling problems. In this study, statistical inference problem for the α-series process is considered where the distribution of first occurrence time is IG. The estimators of the parameters α, μ, and σ² are obtained by using the maximum likelihood (ML) method. Asymptotic distributions and consistency properties of the ML estimators are derived. In order to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators, Monte Carlo simulations are performed. The results showed that the ML estimators are more efficient than the MM estimators. Moreover, two real life datasets are given for application purposes.     

Some new group divisible designs

The paper lists fourteen new group divisible PBIB/2 designs, which were obtained using the computer program described in John (1976).     

Progressive censoring with fixed censoring times

In this paper, we revisit the progressive Type-I censoring scheme as it has originally been introduced by Cohen [Progressively censored samples in life testing. Technometrics. 1963;5(3):327–339]. In fact, original progressive Type-I censoring proceeds as progressive Type-II censoring but with fixed censoring times instead of failure time based censoring times. Apparently, a time truncation has been added to this censoring scheme by interpreting the final censoring time as a termination time. Therefore, not much work has been done on Cohens's original progressive censoring scheme with fixed censoring times. Thus, we discuss distributional results for this scheme and establish exact distributional results in likelihood inference for exponentially distributed lifetimes. In particular, we obtain the exact distribution of the maximum likelihood estimator (MLE). Further, the stochastic monotonicity of the MLE is verified in order to construct exact confidence intervals for both the scale parameter and the reliability.     

Dose-response modeling of quantitative response data for risk assessment

The current regulation of non-carcinogenic effects has generally been based on dividing a safety factor into an experimental no-observed-effect-level (NOEL), giving a regulatory reference dose (RfD). This approach does not attempt to estimate the risk as a function of dose; it assumes no significant risk for the dose below the RfD. This paper proposes a mathematical model for finding the upper confidence limit on risk and lower confidence limit on dose for quantitative risk assessment when the responses follow a normal distribution. The proposed procedure appears to be conservative; this is supported by results of a simulation study. The procedure is illustrated by application to real data.     

A test for homogeneity when the sample is a positive lagrangian poisson distribution

In this note, we obtain, based on the sample sum, a statistic to test the homogeneity of a random sample from a positive (zero truncated) Lagrangian Poisson distribution given in Consul and Jain (1973). This test statistic conforms, in a special case, to Singh (1978). A goodness-of-fit test statistic for the Borel-Tanner distribution is obtained as a particular case cf our results.     

Empitical bayes estimations for some distribution families

Some estimates of prior density based on orthogonal expansions are proposed for some family of conditional densities. Their related properties are studied. The associated empirical Bayes estimators are also proposed. Three examples are illustrated and some of its Monte Carlo results are also given.     

Some new group divisible designs

The paper lists four new group divisible designs which are obtained by trial and error. These designs are believed to be new, since they are not listed in Clatworthy (1973), Freeman (1976) or John and Turner (1977).