Analytic bias correction for maximum likelihood estimators when the bias function is non-constant

Recently, many articles have obtained analytical expressions for the biases of various maximum likelihood estimators, despite their lack of closed-form solution. These bias expressions have provided an attractive alternative to the bootstrap. Unless the bias function is “flat,” however, the expressions are being evaluated at the wrong point(s). We propose an “improved” analytical bias-adjusted estimator, in which the bias expression is evaluated at a more appropriate point (at the bias adjusted estimator itself). Simulations illustrate that the improved analytical bias-adjusted estimator can eliminate significantly more bias than the simple estimator, which has been well established in the literature.     

On weighted composite scores for early Alzheimer's trials

Recent research on finding appropriate composite endpoints for preclinical Alzheimer's disease has focused considerable effort on finding “optimized” weights in the construction of a weighted composite score. In this paper, several proposed methods are reviewed. Our results indicate no evidence that these methods will increase the power of the test statistics, and some of these weights will introduce biases to the study. Our recommendation is to focus on identifying more sensitive items from clinical practice and appropriate statistical analyses of a large Alzheimer's data set. Once a set of items has been selected, there is no evidence that adding weights will generate more sensitive composite endpoints.     

Perception,Preference, and Patriotism: An Exploratory Analysis of the 1980 Winter Olympics

This article analyzes scores given by judges of figure skating at the 1980 Winter Olympics. Judges' scores are found to be highly correlated, with little evidence of scoring along political lines. However, an analysis of variance shows a small but consistent “patriotic” bias; judges tend to give higher scores to contestants from their own country. The influence of such effects on final placings is estimated.     

Interest-Rate Arbitrage in Currency Baskets: Forecasting Weights and Measuring Risk

This article addresses the problem of the bias of income and expenditure elasticities estimated on pseudopanel data caused by measurement error and unobserved heterogeneity. We gauge these biases empirically by comparing cross-sectional, pseudo-panel, and true panel data from both Polish and U.S. expenditure surveys. Our results suggest that unobserved heterogeneity imparts a downward bias to cross-section estimates of income elasticities of at-home food expenditures and an upward bias to estimates of income elasticities of away-from-home food expenditures. “Within” and first-difference estimators suffer less bias, but only if the effects of measurement error are accounted for with instrumental variables.     

Patterns of consent: evidence from a general household survey

Summary.  We analyse patterns of consent and consent bias in the context of a large general household survey, the 'Improving survey measurement of income and employment' survey, also addressing issues that arise when there are multiple consent questions. A multivariate probit regression model for four binary outcomes with two incidental truncations is used. We show that there are biases in consent to data linkage with benefit and tax credit administrative records that are held by the Department for Work and Pensions, and with wage and employment data held by employers. There are also biases in respondents' willingness and ability to supply their national insurance number. The biases differ according to the question that is considered. We also show that modelling questions on consent independently rather than jointly may lead to misleading inferences about consent bias. A positive correlation between unobservable individual factors affecting consent to Department for Work and Pensions record linkage and consent to employer record linkage is suggestive of a latent individual consent propensity.     

Errors-in-variables with systematic biases

A structural regression model is considered in which some of the variables are measured with error. Instead of additive measurement errors, systematic biases are allowed by relating true and observed values via simple linear regressions. Additional data is available, based on standards, which allows for “calibration” of the measuring methods involved. Using only moment assumptions, some simple estimators are proposed and their asymptotic properties are developed. The results parallel and extend those given by Fuller (1987) in which the errors are additive and the error covariance is estimated. Maximum likelihood estimation is also discussed and the problem is illustrated using data from an acid rain study in which the relationship between pH and alkalinity is of interest but neither variable is observed exactly.     

Least squares parameter estimation in multiplicative noise models

A simple multiplicative noise model with a constant signal has become a basic mathematical model in processing synthetic aperture radar images. The purpose of this paper is to examine a general multiplicative noise model with linear signals represented by a number of unknown parameters. The ordinary least squares (LS) and weighted LS methods are used to estimate the model parameters. The biases of the weighted LS estimates of the parameters are derived. The biases are then corrected to obtain a second-order unbiased estimator, which is shown to be exactly equivalent to the maximum log quasi-likelihood estimation, though the quasi-likelihood function is founded on a completely different theoretical consideration and is known, at the present time, to be a uniquely acceptable theory for multiplicative noise models. Synthetic simulations are carried out to confirm theoretical results and to illustrate problems in processing data contaminated by multiplicative noises. The sensitivity of the LS and weighted LS methods to extremely noisy data is analysed through the simulated examples.     

A note on admissibility of some tests of independence against “Positive Dependence” in R×C contingeney table

Admissibility of some asymptotically optimal tests of independence against “positive dependence” in a R × C contingency table is deduced from earlier results of the author. “Positive dependence” is specified to be positive regression dependence and positive quadrant dependence.     

Using Bayesian graphical models to model biases in observational studies and to combine multiple sources of data: application to low birth weight and water disinfection by-products

Summary.  Data in the social, behavioural and health sciences frequently come from observational studies instead of controlled experiments. In addition to random errors, observational data typically contain additional sources of uncertainty such as missing values, unmeasured confounders and selection biases. Also, the research question is often different from that which a particular source of data was designed to answer, and so not all relevant variables are measured. As a result, multiple sources of data are often necessary to identify the biases and to inform about different aspects of the research question. Bayesian graphical models provide a coherent way to connect a series of local submodels, based on different data sets, into a global unified analysis. We present a unified modelling framework that will account for multiple biases simultaneously and give more accurate parameter estimates than standard approaches. We illustrate our approach by analysing data from a study of water disinfection by-products and adverse birth outcomes in the UK.     

Estimating Long and Short Run Effects in Static Panel Models

This paper assesses the biases of four different estimators with respect to the short run and the long run parameters if a static panel model is used, although the data generating process is a dynamic error components model. We analytically derive the associated biases and provide a discussion of the determinants thereof. Our analytical and numerical results as well as Monte Carlo simulations illustrate that the asymptotic bias of both the within and the between parameter with respect to the short run and long run impact can be substantial, depending on the memory of the data generating process, the length of the time series and the importance of the cross-sectional variation in the explanatory variables.     

Estimating Long and Short Run Effects in Static Panel Models

Abstract

This paper assesses the biases of four different estimators with respect to the short run and the long run parameters if a static panel model is used, although the data generating process is a dynamic error components model. We analytically derive the associated biases and provide a discussion of the determinants thereof. Our analytical and numerical results as well as Monte Carlo simulations illustrate that the asymptotic bias of both the within and the between parameter with respect to the short run and long run impact can be substantial, depending on the memory of the data generating process, the length of the time series and the importance of the cross-sectional variation in the explanatory variables.     

An Analysis of Nonlinearities,Heteroscedasticity, and Functional Form in the Market Model

Using a generalized specification of the single-index market model, this study examines the sources of statistical anomalies previously found in estimating the market model. Two generalized models are developed for juxtaposition with the traditional linear specification. The most general model is a Box–Cox model with different λ's and heteroscedastic errors. The empirical results indicate that previous findings of significant “nonlinearities” are primarily attributable to nonnormalities and unequal variance.     

Econometric Applications of the Forward Search in Regression: Robustness,Diagnostics, and Graphics

The article illustrates the use of the forward search to provide robust analyses of econometric data. The emphasis is on informative plots that reveal the inferential importance of each observation. The division of observations into “good” and “bad” leverage points is shown to be potentially misleading.     

Bayesian break-point forecasting in parallel time series,with application to university admissions

A regular supply of applicants to Queen's University in Kingston, Ontario is provided by 65 high schools. Each high school can be characterized by a series of grading standards which change from year to year. To aid admissions decisions, it is desirable to forecast the current year's grading standards for all 65 high schools using grading standards estimated from past year's data. We develop and apply a Bayesian break-point time-series model that generates forecasts which involve smoothing across time for each school and smoothing across schools. “Break point” refers to a point in time which divides the past into the “old past” and the “recent past” where the yearly observations in the recent past are exchangeable with the observations in the year to be forecast. We show that this model works fairly well when applied to 11 years of Queen's University data. The model can be applied to other data sets with the parallel time-series structure and short history, and can be extended in several ways to more complicated structures.     

Estimating population size with truncated sampling

Let X₁, X₂,…,X_n be independent, indentically distributed random variables with density f(x,θ) with respect to a σ-finite measure μ. Let R be a measurable set in the sample space X. The value of X is observable if X ? (X?R) and not otherwise. The number J of observable X’s is binomial, N, Q, Q = 1?P(X ? R). On the basis of J observations, it is desired to estimate N and θ. Estimators considered are conditional and unconditional maximum likelihood and modified maximum likelihood using a prior weight function to modify the likelihood before maximizing. Asymptotic expansions are developed for the [Ncirc]’s of the form [Ncirc] = N + α√N + β + o_p(1), where α and β are random variables. All estimators have the same α, which has mean 0, variance σ² (a function of θ) and is asymptotically normal. Hence all are asymptotically equivalent by the usual limit distributional theory. The β’s differ and Eβ can be considered an “asymptotic bias”. Formulas are developed to compare the asymptotic biases of the various estimators. For a scale parameter family of absolutely continuous distributions with X = (0,∞) and R = (T,∞), special formuli are developed and a best estimator is found.     

Local functional data model for characterizing shape variation of similar curves

“Similar curves” in the present article refers to a family of curves whose major shape are similar, but who have variation coming from curve-specific sources. The goal here is to develop a general methodology to describe small changes among similar curves. Previous methods mainly focus on dimension reduction through FPCA, which are not appropriate for quantifying local variation. Here, we consider a local functional data model which divides data into segments adaptively and models each segment with a shape invariant model. Such model has great flexibility in characterizing local variation of curves, as illustrated by simulation and real data examples.     

Rejoinder

Kaplan–Meier graphs of survival and other timed events are an extremely effective way to summarize outcome data from some types of clinical studies, but information on dates and the interaction among events is lost. A new mode for presenting such data is proposed, an Eventchart, that aids in looking at the study as a whole by simultaneously displaying the timing of 2–3 different types of events, including censoring and time-dependent covariate events. The chart helps to monitor accrual and can be used to “guesstimate” the amount of information resulting from additional enrollment and/or follow-up. Eventcharts are designed to supplement, not replace, quantitative methods of analysis or conventional survival graphs. Data from bone marrow transplantation and AIDS studies are used to illustrate the method. Terminology is suggested for describing the concepts and distinguishing among the complex of events studied in modern clinical trials, including timed events, loss and date censoring, and time-braided events.     

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

Empirical Bayes is a versatile approach to “learn from a lot” in two ways: first, from a large number of variables and, second, from a potentially large amount of prior information, for example, stored in public repositories. We review applications of a variety of empirical Bayes methods to several well‐known model‐based prediction methods, including penalized regression, linear discriminant analysis, and Bayesian models with sparse or dense priors. We discuss “formal” empirical Bayes methods that maximize the marginal likelihood but also more informal approaches based on other data summaries. We contrast empirical Bayes to cross‐validation and full Bayes and discuss hybrid approaches. To study the relation between the quality of an empirical Bayes estimator and p, the number of variables, we consider a simple empirical Bayes estimator in a linear model setting. We argue that empirical Bayes is particularly useful when the prior contains multiple parameters, which model a priori information on variables termed “co‐data”. In particular, we present two novel examples that allow for co‐data: first, a Bayesian spike‐and‐slab setting that facilitates inclusion of multiple co‐data sources and types and, second, a hybrid empirical Bayes–full Bayes ridge regression approach for estimation of the posterior predictive interval.     

Alleviating linear ecological bias and optimal design with subsample data

Summary.  We illustrate that combining ecological data with subsample data in situations in which a linear model is appropriate provides two main benefits. First, by including the individual level subsample data, the biases that are associated with linear ecological inference can be eliminated. Second, available ecological data can be used to design optimal subsampling schemes that maximize information about parameters. We present an application of this methodology to the classic problem of estimating the effect of a college degree on wages, showing that small, optimally chosen subsamples can be combined with ecological data to generate precise estimates relative to a simple random subsample.     

Missing plot technique in linear models

This paper extends the missing plot substitution technique to the case where the missing observations-cause some previously estimable functions to become non-estimable. It is shown that with appropriate modifications, the usual methods of analysis remain valid. We also obtain necessary and sufficient conditions under which the sum of squares due to a hypothesis can be calculated without “re-estimating” the missing observations