A simple method for estimating parameters of the location–scale distribution family

The purpose of this paper is to estimate the parameters of the location–scale distribution family. As a special case, the method is used for estimating the parameters of the normal distribution and Cauchy distribution. For the Cauchy distribution, neither the moment estimation method nor the maximum likelihood estimation method works properly for estimating the parameters. The quantiles for obtaining confidence intervals and point estimates for the parameters of the two-parameter Cauchy distribution are given in the paper. It is shown that the estimators obtained in this paper are unbiased with respect to the median and possess some optimal properties.     

A modification of the Box–Meyer method for finding the active factors in screening experiments

Screening experiments are conducted to identify a few active factors among a large number of factors. For the objective of identifying active factors, Box and Meyer provided an innovative approach, the Box–Meyer method (BMM). With the use of means models, we propose a modification of the BMM in this paper. Compared with the original BMM, the modified BMM (MBMM) can circumvent the problem that the original BMM runs into, namely that it may fail to identify some active factors due to the ignorance of higher order interactions. Furthermore, the number of explanatory variables in the MBMM is smaller. Therefore, the computational complexity is reduced. Finally, three examples with different types of designs are used to demonstrate the wide applicability of the MBMM.     

A general reduction method for n–variate normal orthant probability

Let X₁,X₂,…,X_n be n normal variates with zero means, unit variances and correlation matrix {p_ij). The orthant probability is the probability that all of the X₁'s are simultaneously positive. This paper presents a general reduction method by extending the method of Childs (1967), and shows that the probability can be represented by a linear combination of some multivariate integrals of order([n/2]?1). As illustrations, we apply the proposed method to the quadrivariate and six–variate cases. Some numerical results are also given.     

A generalised Box–Cox transformation for the parametric estimation of clinical reference intervals

Parametric methods for the calculation of reference intervals in clinical studies often rely on the identification of a suitable transformation so that the transformed data can be assumed to be drawn from a Gaussian distribution. In this paper, the two-stage transformation recommended by the International Federation for Clinical Chemistry is compared with a novel generalised Box–Cox family of transformations. Investigation is also made of sample sizes needed to achieve certain criteria of reliability in the calculated reference interval. Simulations are used to show that the generalised Box–Cox family achieves a lower bias than the two-stage transformation. It was found that there is a possibility that the two-stage transformation will result in percentile estimates that cannot be back-transformed to obtain the required reference intervals, a difficulty not observed when using the generalised Box–Cox family introduced in this paper.     

A modification of the Whittaker–Henderson method of graduation

This paper presents a modified Whittaker–Henderson (WH) Method of Graduation. After giving a closed-form solution, we show that it is of practical use because it provides not only a smoothed series identical to that of the WH graduation, but also an extrapolation beyond the sample limit of current data. In addition, we introduce two other penalized least squares problems and show that they provide the same results as those of the modified WH graduation.     

A version of the Kolmogrov–Feller weak law of large numbers for maximal weighted sums of random variables

Abstract

In this paper we establish Kolmogrov–Feller weak law of large numbers for maximal weighted sums of i.i.d. random variables.     

An improved method of estimation for the parameters of the Birnbaum–Saunders distribution

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution, frequently used for analysing lifetime data. In this paper, we propose a simple method of estimation for the parameters of the two-parameter BS distribution by making use of some key properties of the distribution. Compared with the maximum likelihood estimators and the modified moment estimators, the proposed method has smaller bias, but having the same mean square errors as these two estimators. We also discuss some methods of construction of confidence intervals. The performance of the estimators is then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.     

A pathwise inference method for the parameters of diffusion terms

We consider inference of the parameters of the diffusion term for continuous time stochastic processes with a power-type dependence of the diffusion coefficient from the underlying process such as Cox–Ingersoll–Ross, CKLS, and similar processes. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex-valued process generated by the underlying real-valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

A note on expressing wilks’ a as the product of independent beta variables

The purpose of this note is to show that the null distribution of Wilks’ A can be expressed as the product of p independent beta variables regardless of the relationship of p and q, where p is the number of variables and q is the degrees of freedom associated with the hypothesis.     

Opportunities of the minimum Anderson–Darling estimator as a variant of the maximum likelihood method

We reveal that the minimum Anderson–Darling (MAD) estimator is a variant of the maximum likelihood method. Furthermore, it is shown that the MAD estimator offers excellent opportunities for parameter estimation if there is no explicit formulation for the distribution model. The computation time for the MAD estimator with approximated cumulative distribution function is much shorter than that of the classical maximum likelihood method with approximated probability density function. Additionally, we research the performance of the MAD estimator for the generalized Pareto distribution and demonstrate a further advantage of the MAD estimator with an issue of seismic hazard analysis.     

Ties in one block comparison experiments: a generalization of the Mallows–Bradley–Terry ranking model

This study is concerned with the extension of the Mallows–Bradley–Terry ranking model for one block comparison consisting of all the items of interest to situations which allow an expression of no preference. We consider a modification of the Mallows–Bradley–Terry ranking model by introducing an additional parameter, called an index of discrimination, in the model. This permits ties in the model. The maximum likelihood estimates of the parameters are found using a Maximization–Minimization algorithm: the evaluation of the mathematical expectations involved in the log-likelihood equation is obtained by generating samples of Monte Carlo Markov chain from the stationary distribution. In addition, a simulation study for asymptotic properties assessment has been made. The proposed method is applied to analyze data election.     

Correction for bias in maximum likelihood estimators of in a right–censored normal distribution

It is assumed that the logs of the time to failure in a life test follow a normal distribution. If the test is terminated after r of a sample of n items fail, the test is said to be censored. If the sample size is small and censoring severe, the usual maximum likelihood estimator of a is downwardly biased. Monte Carlo techniques and regression analysis were used to develop an empirical correction factor. Applying the correction factor to the maximum likelihood estimator yields an unbiased estimate of σ.     

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.     

A Bayesian benchmarking of the Scott–Smith model for small areas

When the finite population ‘totals’ are estimated for individual areas, they do not necessarily add up to the known ‘total’ for all areas. Benchmarking (BM) is a technique used to ensure that the totals for all areas match the grand total, which can be obtained from an independent source. BM is desirable to practitioners of survey sampling. BM shifts the small-area estimators to accommodate the constraint. In doing so, it can provide increased precision to the small-area estimators of the finite population means or totals. The Scott–Smith model is used to benchmark the finite population means of small areas. This is a one-way random effects model for a superpopulation, and it is computationally convenient to use a Bayesian approach. We illustrate our method by estimating body mass index using data in the third National Health and Nutrition Examination Survey. Several properties of the benchmarked small-area estimators are obtained using a simulation study.     

Three-stage and ‘accelerated’ sequential procedures for the mean of a normal population with known coefficient of variation

Three-stage and ‘accelerated’ sequential procedures are developed for estimating the mean of a normal population when the population coefficient of variation (CV) is known. In spite of the usual estimator, i.e. the sample mean, Searls' (1964 Searls, DT. (1964). The utilization of a known coefficient of variation in the estimation procedure. J. Amer. Statist. Assoc, 50: 1225–1226.  ) estimator is utilized for the estimation purpose. It is established that Searls' estimator dominates the sample mean under the two sampling schemes.     

A simple test procedure in standardizing the power of Hosmer–Lemeshow test in large data sets

The Hosmer–Lemeshow (H–L) test is a widely used method when assessing the goodness-of-fit of a logistic regression model. However, the H–L test is sensitive to the sample sizes and the number of groups in H–L test. Cautions need to be taken for interpreting an H–L test with a large sample size. In this paper, we propose a simple test procedure to evaluate the model fit of logistic regression model with a large sample size, in which a bootstrap method is used and the test result is determined by the power of H–L test at the target sample size. Simulation studies show that the proposed method can effectively standardize the power of the H–L test under the pre-specified level of type I error. Application to the two datasets illustrates the usefulness of the proposed model.