A pool-adjacent-violators-algorithm approach to detect infinite parameter estimates in one-regressor dose–response models with asymptotes

Binary response models are often applied in dose–response settings where the number of dose levels is limited. Commonly, one can find cases where the maximum likelihood estimation process for these models produces infinite values for at least one of the parameters, often corresponding to the ‘separated data’ issue. Algorithms for detecting such data have been proposed, but are usually incorporated directly into in the parameter estimation. Additionally, they do not consider the use of asymptotes in the model formulation. In order to study this phenomenon in greater detail, we define the class of specifiably degenerate functions where this can occur (including the popular logistic and Weibull models) that allows for asymptotes in the dose–response specification. We demonstrate for this class that the well-known pool-adjacent-violators algorithm can efficiently pre-screen for non-estimable data. A simulation study demonstrates the frequency with which this problem can occur for various response models and conditions.     

Modern likelihood inference for the maximum/minimum of a bivariate normal vector

ABSTRACT

We consider the use of modern likelihood asymptotics in the construction of confidence intervals for the parameter which determines the skewness of the distribution of the maximum/minimum of an exchangeable bivariate normal random vector. Simulation studies were conducted to investigate the accuracy of the proposed methods and to compare them to available alternatives. Accuracy is evaluated in terms of both coverage probability and expected length of the interval. We furthermore illustrate the suitability of our proposals by means of two data sets, consisting of, respectively, measurements taken on the brains of 10 mono-zygotic twins and measurements of mineral content of bones in the dominant and non-dominant arms for 25 elderly women.     

Implementing particle swarm optimization algorithm to estimate the mixture of two Weibull parameters with censored data

We present the maximum likelihood estimation (MLE) via particle swarm optimization (PSO) algorithm to estimate the mixture of two Weibull parameters with complete and multiple censored data. A simulation study is conducted to assess the performance of the MLE via PSO algorithm, quasi-Newton method and expectation-maximization (EM) algorithm for different parameter settings and sample sizes in both uncensored and censored cases. The simulation results showed that the PSO algorithm outperforms the quasi-Newton method and the EM algorithm in most cases regarding bias and root mean square errors. Two numerical examples are used to demonstrate the performance of our proposed method.     

LS estimation of periodic autoregressive models with non-Gaussian errors: a simulation study

In this article, the least squares (LS) estimates of the parameters of periodic autoregressive (PAR) models are investigated for various distributions of error terms via Monte-Carlo simulation. Beside the Gaussian distribution, this study covers the exponential, gamma, student-t, and Cauchy distributions. The estimates are compared for various distributions via bias and MSE criterion. The effect of other factors are also examined as the non-constancy of model orders, the non-constancy of the variances of seasonal white noise, the period length, and the length of the time series. The simulation results indicate that this method is in general robust for the estimation of AR parameters with respect to the distribution of error terms and other factors. However, the estimates of those parameters were, in some cases, noticeably poor for Cauchy distribution. It is also noticed that the variances of estimates of white noise variances are highly affected by the degree of skewness of the distribution of error terms.     

A Hellinger distance approach to MCMC diagnostics

Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions f and g based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the Anguilla australis, an eel native to New Zealand.     

Smooth tail-index estimation

The two parametric distribution functions appearing in the extreme-value theory – the generalized extreme-value distribution and the generalized Pareto distribution – have log-concave densities if the extreme-value index γ∈[?1, 0]. Replacing the order statistics in tail-index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density ? f _n leads to novel smooth quantile and tail-index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.     

Comparing diagnostic tests: test of hypothesis for likelihood ratios

Likelihood ratios (LRs) are used to characterize the efficiency of diagnostic tests. In this paper, we use the classical weighted least squares (CWLS) test procedure, which was originally used for testing the homogeneity of relative risks, for comparing the LRs of two or more binary diagnostic tests. We compare the performance of this method with the relative diagnostic likelihood ratio (rDLR) method and the diagnostic likelihood ratio regression (DLRReg) approach in terms of size and power, and we observe that the performances of CWLS and rDLR are the same when used to compare two diagnostic tests, while DLRReg method has higher type I error rates and powers. We also examine the performances of the CWLS and DLRReg methods for comparing three diagnostic tests in various sample size and prevalence combinations. On the basis of Monte Carlo simulations, we conclude that all of the tests are generally conservative and have low power, especially in settings of small sample size and low prevalence.     

Rank invariant tests with left truncated and interval censored data

Peto and Peto (1972) have studied rank invariant tests to compare two survival curves for right censored data. We apply their tests, including the logrank test and the generalized Wilcoxon test, to left truncated and interval censored data. The significance levels of the tests are approximated by Monte Carlo permutation tests. Simulation studies are conducted to show their size and power under different distributional differences. In particular, the logrank test works well under the Cox proportional hazards alternatives, as for the usual right censored data. The methods are illustrated by the analysis of the Massachusetts Health Care Panel Study dataset.     

Parameter identification for the discretely observed geometric fractional Brownian motion

This paper deals with the problem of estimating all the unknown parameters of geometric fractional Brownian processes from discrete observations. The estimation procedure is built upon the marriage of the quadratic variation and the maximum likelihood approach. The asymptotic properties of the estimators are provided. Moveover, we compare our derived method with the approach proposed by Misiran et al. [Fractional Black-Scholes models: complete MLE with application to fractional option pricing. In International conference on optimization and control; Guiyang, China; 2010. p. 573–586.], namely the complete maximum likelihood estimation. Simulation studies confirm theoretical findings and illustrate that our methodology is efficient and reliable. To show how to apply our approach in realistic contexts, an empirical study of Chinese financial market is also presented.     

A new two-parameter lifetime distribution with decreasing,increasing or upside-down bathtub-shaped failure rate

In this paper, we introduce a generalization of the Bilal distribution, where a new two-parameter distribution is presented. We show that its failure rate function can be upside-down bathtub shaped. The failure rate can also be decreasing or increasing. A comprehensive mathematical treatment of the new distribution is provided. The estimation by maximum likelihood is discussed, and a closed-form expression for Fisher’s information matrix is obtained. Asymptotic interval estimators for both of the two unknown parameters are also given. A simulation study is conducted and applications to real data sets are presented.