An approximation of logarithmic functions in the regression setting

We consider a method of moments approach for dealing with censoring at zero for data expressed in levels when researchers would like to take logarithms. A Box–Cox transformation is employed. We explore this approach in the context of linear regression where both dependent and independent variables are censored. We contrast this method to two others, (1) dropping records of data containing censored values and (2) assuming normality for censored observations and the residuals in the model. Across the methods considered, where researchers are interested primarily in the slope parameter, estimation bias is consistently reduced using the method of moments approach.     

GMM tests for the Katz family of distributions

Generalized method of moments (GMM) is used to develop tests for discriminating discrete distributions among the two-parameter family of Katz distributions. Relationships involving moments are exploited to obtain identifying and over-identifying restrictions. The asymptotic relative efficiencies of tests based on GMM are analyzed using the local power approach and the approximate Bahadur efficiency. The paper also gives results of Monte Carlo experiments designed to check the validity of the theoretical findings and to shed light on the small sample properties of the proposed tests. Extensions of the results to compound Poisson alternative hypotheses are discussed.     

Markov chain Monte Carlo methods for high dimensional inversion in remote sensing

Summary.  We discuss the inversion of the gas profiles (ozone, NO₃, NO₂, aerosols and neutral density) in the upper atmosphere from the spectral occultation measurements. The data are produced by the 'Global ozone monitoring of occultation of stars' instrument on board the Envisat satellite that was launched in March 2002. The instrument measures the attenuation of light spectra at various horizontal paths from about 100 km down to 10–20 km. The new feature is that these data allow the inversion of the gas concentration height profiles. A short introduction is given to the present operational data management procedure with examples of the first real data inversion. Several solution options for a more comprehensive statistical inversion are presented. A direct inversion leads to a non-linear model with hundreds of parameters to be estimated. The problem is solved with an adaptive single-step Markov chain Monte Carlo algorithm. Another approach is to divide the problem into several non-linear smaller dimensional problems, to run parallel adaptive Markov chain Monte Carlo chains for them and to solve the gas profiles in repetitive linear steps. The effect of grid size is discussed, and we present how the prior regularization takes the grid size into account in a way that effectively leads to a grid-independent inversion.     

Goodness of fit for the inverse Gaussian distribution

For testing the fit of the inverse Gaussian distribution with unknown parameters, the empirical distribution-function statistic A² is studied. Two procedures are followed in constructing the test statistic; they yield the same asymptotic distribution. In the first procedure the parameters in the distribution function are directly estimated, and in the second the distribution function is estimated by its Rao-Blackwell distribution estimator. A table is given for the asymptotic critical points of A². These are shown to depend only on the ratio of the unknown parameters. An analysis is provided of the effect of estimating the ratio to enter the table for A². This analysis enables the proposal of the complete operating procedure, which is sustained by a Monte Carlo study.     

Assessing accuracy of a continuous screening test in the presence of verification bias

Summary.  In studies to assess the accuracy of a screening test, often definitive disease assessment is too invasive or expensive to be ascertained on all the study subjects. Although it may be more ethical or cost effective to ascertain the true disease status with a higher rate in study subjects where the screening test or additional information is suggestive of disease, estimates of accuracy can be biased in a study with such a design. This bias is known as verification bias. Verification bias correction methods that accommodate screening tests with binary or ordinal responses have been developed; however, no verification bias correction methods exist for tests with continuous results. We propose and compare imputation and reweighting bias-corrected estimators of true and false positive rates, receiver operating characteristic curves and area under the receiver operating characteristic curve for continuous tests. Distribution theory and simulation studies are used to compare the proposed estimators with respect to bias, relative efficiency and robustness to model misspecification. The bias correction estimators proposed are applied to data from a study of screening tests for neonatal hearing loss.     

Almost Consistent Estimation of Panel Probit Models with “Small” Fixed Effects

We propose four different GMM estimators that allow almost consistent estimation of the structural parameters of panel probit models with fixed effects for the case of small Tand large N. The moments used are derived for each period from a first order approximation of the mean of the dependent variable conditional on explanatory variables and on the fixed effect. The estimators differ w.r.t. the choice of instruments and whether they use trimming to reduce the bias or not. In a Monte Carlo study, we compare these estimators with pooled probit and conditional logit estimators for different data generating processes. The results show that the proposed estimators outperform these competitors in several situations.     

Expected sample moments of concomitants of selected order statistics

In this paper, the task of determining expected values of sample moments, where the sample members have been selected based on noisy information, is considered. This task is a recurring problem in the theory of evolution strategies. Exact expressions for expected values of sums of products of concomitants of selected order statistics are derived. Then, using Edgeworth and Cornish-Fisher approximations, explicit results that depend on coefficients that can be determined numerically are obtained. While the results are exact only for normal populations, it is shown experimentally that including skewness and kurtosis in the calculations can yield greatly improved results for other distributions.     

A relation for the covariances of order statistics from n independent and non-identically distributed random variables

We derive a simple relation satisfied by the covariances of order statistics in the i.i.d. case and then generalize it to the case when the variables are independent and non-identically distributed. This relation could be employed successfully either to check the calculations or to reduce the amount of direct computations involved in evaluating the covariances of order statistics from an outlier model.     

Record-breaking data: a parametric comparison of the inverse-sampling and the random-sampling schemes

In many industrial quality control experiments and destructive stress testing, the only available data are successive minima (or maxima)i.e., record-breaking data. There are two sampling schemes used to collect record-breaking data: random sampling and inverse sampling. For random sampling, the total sample size is predetermined and the number of records is a random variable while in inverse-sampling the number of records to be observed is predetermined; thus the sample size is a random variable. The purpose of this papper is to determinevia simulations, which of the two schemes, if any, is more efficient. Since the two schemes are equivalent asymptotically, the simulations were carried out for small to moderate sized record-breaking samples. Simulated biases and mean square errors of the maximum likelihood estimators of the parameters using the two sampling schemes were compared. In general, it was found that if the estimators were well behaved, then there was no significant difference between the mean square errors of the estimates for the two schemes. However, for certain distributions described by both a shape and a scale parameter, random sampling led to estimators that were inconsistent. On the other hand, the estimated obtained from inverse sampling were always consistent. Moreover, for moderated sized record-breaking samples, the total sample size that needs to be observed is smaller for inverse sampling than for random sampling.     

Tests for generalized exponential laws based on the empirical Mellin transform

Recently, many standard families of distributions have been generalized by exponentiating their cumulative distribution function (CDF). In this paper, test statistics are constructed based on CDF–transformed observations and the corresponding moments of arbitrary positive order. Simulation results for generalized exponential distributions show that the proposed test compares well with standard methods based on the empirical distribution function.