We study rare events data, binary dependent variables with dozens to thousands of times fewer ones (events, such as wars, vetoes, cases of political activism, or epidemiological infections) than zeros (“nonevents”). In many literatures, these variables have proven difficult to explain and predict, a problem that seems to have at least two sources. First, popular statistical procedures, such as logistic regression, can sharply underestimate the probability of rare events. We recommend corrections that outperform existing methods and change the estimates of absolute and relative risks by as much as some estimated effects reported in the literature. Second, commonly used data collection strategies are grossly inefficient for rare events data. The fear of collecting data with too few events has led to data collections with huge numbers of observations but relatively few, and poorly measured, explanatory variables, such as in international conflict data with more than a quarter-million dyads, only a few of which are at war. As it turns out, more efficient sampling designs exist for making valid inferences, such as sampling all available events (e.g., wars) and a tiny fraction of nonevents (peace). This enables scholars to save as much as 99 % of their (nonfixed) data collection costs or to collect much more meaningful explanatory

Renshaw and Verrall (1994) specified the generalized linear model (GLM) underlying the chain-ladder technique and suggested some other GLMs which might be useful in claims reserving. The purpose of this paper is to construct bounds for the discounted loss reserve within the framework of GLMs. Exact calculation of the distribution of the total reserve is not feasible, and hence the determination of lower and upper bounds with a simpler structure is a possible way out. The paper ends with numerical examples illustrating the usefulness of the presented approximations.

Some of the most important phenomena in international conflict are coded as "rare events data," binary dependent variables with dozens to thousands of times fewer events, such as wars, coups, etc., than "nonevents". Unfortunately, rare events data are difficult to explain and predict, a problem that seems to have at least two sources. First, and most importantly, the data collection strategies used in international conflict are grossly inefficient. The fear of collecting data with too few events has led to data collections with huge numbers of observations but relatively few, and poorly measured, explanatory variables. As it turns out, more efficient sampling designs exist for making valid inferences, such as sampling all available events (e.g., wars) and a tiny fraction of non-events (peace). This enables scholars to save as much as 99% of their (non-fixed) data collection costs, or to collect much more meaningful explanatory variables. Second, logistic regression, and other commonly ...

Varying coefficient models result from generalized linear models by allowing the parameter of the linear predictor to vary across some additional explanatory quantity called effect modifier. While Hastie & Tibshirani (1993) have used spline smoothing techniques in varying-coefficient models with univariate response here the local likelihood approach is considered within the framework of multivariate generalized models. The local likelihood approach has several advantages. It allows the derivation of asymptotic properties under weak assumptions, consistency and asymptotic normality of the estimates are shown under rather general conditions. The estimation procedure may be performed with standard software. This holds even for the additive bias reduction method which is proposed and investigated. The results are given for discrete as well as for continuous effect modifiers and asymptotically optimal rates of smoothing are derived. An alternative normalization of weights is proposed which corresponds to the augmentation of the information supplied by the observation. The normalization results from theoretical considerations and is supported by simulations which show improved variance properties for finite sample size. A real data example demonstrates the applicability of the results.

of bias in exponential family models

Discipline is the highest of all virtues. Only so may strength and desire be counterbalanced and the endeavors of man bear fruit. N. KAZANTZAKIS,

iterative adjustment of responses for the reduction of bias in

A general iterative algorithm is developed for the computation of reduced-bias parameter estimates in regular statistical models using the adjusted score approach of Firth (1993, Biometrika 80, 27–38). The algorithm unifies and provides appealing new interpretation for iterative methods that have been published previously for some specific model classes. The new algorithm can usefully be viewed as a series of iterative bias corrections, thus facilitating the adjusted score approach to bias reduction in any model for which the first-order bias of the maximum likelihood estimator has already been derived. The method is tested by application to a logit-linear multiple regression model with beta-distributed responses; the results confirm the effectiveness of the new algorithm, and also reveal some important errors in the existing literature on beta regression.

This paper reviews the literature on Bartlett and Bartlett-type corrections. It focuses on the corrections to the likelihood ratio, score and Wald test statistics. Three di erent Bartlett-type corrections which are equivalent to order n 1, n being the sample size, are compared through simulation. One of the forms displayed superior behavior both in terms of size and power. We also use Monte Carlo simulation to examine the e ect of independent variables and the impact of the number of nuisance parameters on the nite-sample behavior of some asymptotic econometric criteria in regression models. 1

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