Improved Likelihood Inference on Testing the Difference of Non Centrality Parameters of Two Independent Non central t Distributions with Identical Degrees of Freedom

Kim and Hayter (2008 Kim , J. , Hayter , A. J. ( 2008 ). Testing the equality of the non centrality parameters of the two non central t-distributions with identical degrees of freedom . Communications in Statistics: Simulation and Computation 37 : 1709 – 1717 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) showed that testing the difference of the non centralities of two non central t distributions, which have identical degrees of freedom, is the equivalent to testing the difference of two signal-to-noise ratios for two independent samples from different normal distributions. Moreover, from a biomedical research point of view, the above test is also equivalent to testing the existence of the effect size when the sample sizes of the control group and treatment group are the same. In this article, two third-order likelihood-based methods are proposed to approximate the p-value for this significance test problem. A real life example is used to illustrate that results from the proposed methods and the existing methods can be quite different. Simulation results show that the proposed methods give remarkable accuracy even when sample size reaches the smallest possible value.     

Improved likelihood-based inference for the stationary AR(2) model

An improved likelihood-based method based on Fraser et al. (1999) is proposed in this paper to test the significance of the second lag of the stationary AR(2) model. Compared with the test proposed by Fan and Yao (2003) and the signed log-likelihood ratio test, the proposed method has remarkable accuracy. Simulation studies are performed to illustrate the accuracy of the proposed method. Application of the proposed method on historical data is presented to demonstrate the implementation of this method. Furthermore, the method can be extended to the general AR(p) model.     

A note on inference for P(X < Y) for right truncated exponentially distributed data

In this paper, a likelihood based analysis is developed and applied to obtain confidence intervals and p values for the stress-strength reliability R  =  P(X  <  Y) with right truncated exponentially distributed data. The proposed method is based on theory given in Fraser et al. (Biometrika 86:249–264, 1999) which involves implicit but appropriate conditioning and marginalization. Monte Carlo simulations are used to illustrate the accuracy of the proposed method.     

Likelihood based inference for the ratio of gamma means

Inference concerning the ratio of two means based two independent two-parameter gamma models with common shape parameter was examined in Booth et al. (1999 Booth, J. G., Hobert, J. P. and Ohman, P. A. (1999). On the probable error of the ratio of two gamma means. Biometrika, 86: 439–452.  [Google Scholar]) and a computationally intensive bootstrap calibration method was developed. In this paper, a likelihood based method is proposed for small sample inference about the ratio of two means of the two-parameter gamma models when the shape parameters may or may not be equal. The proposed method is very simple to use and, as illustrated in simulation studies, gives extremely accurate results.     

Third-order likelihood-based inference for the log-normal regression model

This paper examines the general third-order theory to the log-normal regression model. The interest parameter is its conditional mean. For inference, traditional first-order approximations need large sample sizes and normal-like distributions. Some specific third-order methods need the explicit forms of the nuisance parameter and ancillary statistic, which are quite complicated. Note that this general third-order theory can be applied to any continuous models with standard asymptotic properties. It only needs the log-likelihood function. With small sample settings, the simulation studies for confidence intervals of the conditional mean illustrate that the general third-order theory is much superior to the traditional first-order methods.     

Testing the Difference of Two Binomial Proportions: Comparison of Continuity Corrections for Saddlepoint Approximation

We carried out a simulation study based on the methodology of Newcombe (1998 Newcombe , R. G. ( 1998 ). Interval estimation for the difference between independent proportions: comparison of eleven methods . Statist. Med. 17 : 873 – 890 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to compare tests for the difference of two binomial proportions by applying different continuity corrections on saddlepoint approximation to tail probabilities. In this article, we proposed a new continuity correction based on the least common multiple of two sample sizes. We evaluated that the best test should have the actual Type I error rates that are, on the whole, closest to α, but not exceeding α, where α is nominal level of significance.     

Saddlepoint Approximation for the Distribution of the Modified Signed Root of Likelihood Ratio Statistics Near the Mean

The Tracy-Singh product, which can be viewed as generalized Kronecher product, is studied and used widely in matrix theory and statistics. Therefore, some well-known Kantorovich inequalities are generalized to Tracy-Singh product in this article. Furthermore, some applications to statistics are presented.     

A Note on the Noncentral Beta Distribution Function

The noncentral beta and the related noncentral F distributions have received much attention during the last decade, as is evident from the works of Norton, Lenth, Frick, Lee, Posten, Chattamvelli, and Chattamvelli and Shanmugam. This article reviews the existing algorithms for computing the cumulative distribution function (cdf) of a noncentral beta random variable, and proposes a simple algorithm, based on a sharp error bound, for computing the cdf. A variation of the noncentral beta random variable when the noncentrality is associated only with the denominator χ² and its computational details are also discussed.     

Converting observed likelihood to levels of significance for transformation models

Inference for a scalar parameter in the pressence of nuisance parameters requires high dimensional integrations of the joint density of the pivotal quantities. Recent development in asymptotic methods provides accurate approximations for significance levels and thus confidence intervals for a scalar component parameter. In this paper, a simple, efficient and accurate numerical procedure is first developed for the location model and is then extended to the location-scale model and the linear regression model. This numerical procedure only requires a fine tabulation of the parameter and the observed log likelihood function, which can be either the full, marginal or conditional observed log likelihood function, as input and output is the corresponding significance function. Numerical results showed that this approximation is not only simple but also very accurate. It outperformed the usual approximations such as the signed likelihood ratio statistic, the maximum likelihood estimate and the score statistic.     

Simultaneous Inferences on the Cumulative Distribution Function of a Normal Distribution

This paper addresses the problem of constructing simultaneous confidence intervals for the cumulative distribution function of a normal distribution at several specified points. The procedure is based upon the observation of a random sample of independent observations from a normal distribution with an unknown mean and variance. A new methodology is proposed for obtaining confidence intervals with a specified overall simultaneous confidence level through the inversion of acceptance sets. Both one-sided and two-sided confidence intervals are considered. Some illustrations of the new method are provided, and comparisons are made with other approaches to the problem.     

A Higher Order Approximation to a Percentage Point of the Distribution of a Noncentral t-Statistic Without the Normality Assumption

Noncentral distributions appear in two sample problems and are often used in several fields, for example, in biostatistics. A higher order approximation for a percentage point of the noncentral t-distribution under normality is given by Akahira (1995 Akahira, M. 1995. A higher order approximation to a percentage point of the non-central t-distribution. Communications in Statistics–Simulation, 24(3): 595–605. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and is also shown to be numerically better than others. In this article, without the normality assumption, we obtain a higher order approximation to a percentage point of the distribution of a noncentral t-statistic, in a similar way to Akahira (1995 Akahira, M. 1995. A higher order approximation to a percentage point of the non-central t-distribution. Communications in Statistics–Simulation, 24(3): 595–605. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) where the statistic based on a linear combination of a normal random variable and a chi-statistic takes an important role. Its application to the confidence limit and the confidence interval for a noncentrality parameter are also given. Further, a numerical comparison of the higher order approximation with the limiting normal distribution is done and the former one is shown to be more accurate. As a result of the numerical calculation, the higher order approximation seems to be useful in practical situations, when the size of sample is not so small.     

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.     

Location and Scale Parameter Family of Distributions Attaining the Bhattacharyya Bound

It is well known that, under appropriate regularity conditions, the variance of an unbiased estimator of a real-valued function of an unknown parameter can coincide with the Cramér–Rao lower bound only if the family of distributions is a one-parameter exponential family. But it seems that the necessary conditions about the probability distribution for which there exists an unbiased estimator whose variance coincides with the Bhattacharyya lower bound are not completely known. The purpose of this paper is to specify the location, scale, and location-scale parameter family of distributions attaining the general order Bhattacharyya bound in certain class.     

A Computer Program to Improve LR Tests for Generalized Linear Models

Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion distributions. Improved likelihood ratio tests for these models were developed by Cordeiro (1983 Cordeiro , G. M. (1983). Improved likelihood ratio statistics for generalized linear models. Journal of the Royal Statistical Society, Series B: Methodological 45:404–413. [Google Scholar])Cordeiro (1987 Cordeiro , G. M. ( 1987 ). On the corrections to the likelihood ratio statistics . Biometrika 74 : 265 – 274 .[Crossref], [Web of Science ®] , [Google Scholar]). We present a simple R program source for calculating Bartlett corrections to improve likelihood ratio tests in these models. The program was tested on some special models, confirming all of the previously reported numerical results for the Bartlett corrections.     

Approximation of the Distribution of the Location Parameter in the Growth Curve Model

Abstract.  In this paper an Edgeworth-type approximation of order O(n ⁻² ) to the density of the estimator of the location parameter in the growth curve model has been derived. The approximation is a mixture of a normal and a Kotz-type distribution, thus being an elliptical distribution. A condition for unimodality of the mixture was found and marginal distribution of a subvector of the mixture distribution was derived. Finally, a small example was given to demonstrate an application of the approximation.     

Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏_γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏_γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏_γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏_γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏_γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏_γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.     

A Saddlepoint Approximation to the Distribution of the Half-Life Estimator in a Stationary Autoregressive Model

We derive saddlepoint approximations for the distribution and density functions of the half-life estimated by OLS from autoregressive time-series models. Our results are used to prove that none of the integer-order moments of these half-life estimators exist. This provides an explanation for the very large estimates of persistency, and the extremely wide confidence intervals, that have been reported by various authors, i.e., in the empirical economics literature relating to purchasing power parity.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

Doubly Robust Inference for the Distribution Function in the Presence of Missing Survey Data

Item non‐response in surveys occurs when some, but not all, variables are missing. Unadjusted estimators tend to exhibit some bias, called the non‐response bias, if the respondents differ from the non‐respondents with respect to the study variables. In this paper, we focus on item non‐response, which is usually treated by some form of single imputation. We examine the properties of doubly robust imputation procedures, which are those that lead to an estimator that remains consistent if either the outcome variable or the non‐response mechanism is adequately modelled. We establish the double robustness property of the imputed estimator of the finite population distribution function under random hot‐deck imputation within classes. We also discuss the links between our approach and that of Chambers and Dunstan. The results of a simulation study support our findings.