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We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.

This paper studies estimation of the conditional mean squared error of prediction, conditional on what is known at the time of prediction. The particular problem considered is the assessment of actuarial reserving methods given data in the form of run-off triangles (trapezoids), where the use of prediction assessment based on out-of-sample performance is not an option. The prediction assessment principle advocated here can be viewed as a generalisation of Akaike’s final prediction error. A direct application of this simple principle in the setting of a data-generating process given in terms of a sequence of general linear models yields an estimator of the conditional mean squared error of prediction that can be computed explicitly for a wide range of models within this model class. Mack’s distribution-free chain ladder model and the corresponding estimator of the prediction error for the ultimate claim amount are shown to be a special case. It is demonstrated that the prediction assessment principle easily applies to quite different data-generating processes and results in estimators that have been studied in the literature.

Although loss reserving has been deeply studied in the literature, there are still practical issues that have not been addressed a lot. One of them is the estimation of reserves during the year, which is necessary for forecasts or closings during the year. We will study the following question: What can be done for forecasts and closings during the year that goes along with the reserving at year end? In order to make it not too complicated, we will focus on the Chain-Ladder method introduced by Mack (1993). We will describe several methods that are used in practice. We will discuss advantages and disadvantages of these methods based on a simple deterministic example. Roughly spoken, we will see that you may shift development or accident periods, or may split development periods, but should not split accident periods.

Spatial statistics for very large spatial data sets is challenging. The size of the data set, n, causes problems in computing optimal spatial predictors such as kriging, since its computational cost is of order [graphic removed] . In addition, a large data set is often defined on a large spatial domain, so the spatial process of interest typically exhibits non-stationary behaviour over that domain. A flexible family of non-stationary covariance functions is defined by using a set of basis functions that is fixed in number, which leads to a spatial prediction method that we call fixed rank kriging. Specifically, fixed rank kriging is kriging within this class of non-stationary covariance functions. It relies on computational simplifications when n is very large, for obtaining the spatial best linear unbiased predictor and its mean-squared prediction error for a hidden spatial process. A method based on minimizing a weighted Frobenius norm yields best estimators of the covariance function parameters, which are then substituted into the fixed rank kriging equations. The new methodology is applied to a very large data set of total column ozone data, observed over the entire globe, where n is of the order of hundreds of thousands.

We introduce a stochastic model for the development of attritional and large claims in long-tail lines of business and present a corresponding “chain ladder-like” IBNR method which allows the use of claims payment data for attritional and claims incurred data for large losses. We derive formulas for the mean squared error of prediction and apply the method to a German motor third party liability portfolio.

We propose a simple, unified, Monte-Carlo-assisted approach (called ‘Sumca’) to second-order unbiased estimation of the mean-squared prediction error (MSPE) of a small area predictor.The MSPE estimator proposed is easy to derive, has a simple expression and applies to a broad range of predictors that include the traditional empirical best linear unbiased predictor, empirical best predictor and post-model-selection empirical best linear unbiased predictor and empirical best predictor as special cases. Furthermore, the leading term of the MSPE estimator proposed is guaranteed positive; the lower order term corresponds to a bias correction, which can be evaluated via a Monte Carlo method. The computational burden for the Monte Carlo evaluation is much less, compared with other Monte-Carlo-based methods that have been used for producing second-order unbiased MSPE estimators, such as the double bootstrap and Monte Carlo jackknife. The Sumca estimator also has a nice stability feature. Theoretical and empirical results demonstrate properties and advantages of the Sumca estimator.

Results of multi-environment trials to evaluate new plant cultivars may be displayed in a two-way table of genotypes by environments. Different estimators are available to fill the cells of such tables. It has been shown previously that the predictive accuracy of the simple genotype by environment mean is often lower than that of other estimators, e.g. least-squares estimators based on multiplicative models, such as the additive main effects multiplicative interaction (AMMI) model, or empirical best-linear unbiased predictors (BLUPs) based on a two-way analysis-of-variance (ANOVA) model. This paper proposes a method to obtain BLUPs based on models with multiplicative terms. It is shown by cross-validation using five real data sets (oilseed rape, Brassica napus L.) that the predictive accuracy of BLUPs based on models with multiplicative terms may be better than that of least-squares estimators based on the same models and also better than BLUPs based on ANOVA models.

A conventional linear model for functional data involves expressing a response variable Y in terms of the explanatory function X(t), via the model Y = a + ∫𝓘b(t)X(t)dt + error, where a is a scalar, b is an unknown function and 𝓘 = [0, α] is a compact interval. However, in some problems the support of b or X, 𝓘1 say, is a proper and unknown subset of 𝓘, and is a quantity of particular practical interest. Motivated by a real data example involving particulate emissions, we develop methods for estimating 𝓘1. We give particular emphasis to the case 𝓘1 = [0, θ], where θ ∈ (0, α], and suggest two methods for estimating a, b and θ jointly; we introduce techniques for selecting tuning parameters; and we explore properties of our methodology by using both simulation and the real data example mentioned above. Additionally, we derive theoretical properties of the methodology and discuss implications of the theory. Our theoretical arguments give particular emphasis to the problem of identifiability.

The nested-error regression model is one of the best-known models in small area estimation. A small area mean is often expressed as a linear combination of fixed effects and realized values of random effects. In such analyses, prediction is made by borrowing strength from other related areas or sources and mean-squared prediction error (MSPE) is often used as a measure of uncertainty. In this article, we propose a bias-corrected analytical estimation of MSPE as well as a moment-match jackknife method to estimate the MSPE without specific assumptions about the distributions of the data. Theoretical and empirical studies are carried out to investigate performance of the proposed methods with comparison to existing procedures. Le modèle de régression à erreur imbriquée est l’un des mieux connus pour l’estimation sur des petits domaines. La moyenne d’un petit domaine est souvent exprimée comme une combinaison linéaire d’effets fixes et de valeurs réalisées d’effets aléatoires. Pour de telles analyses, les prévisions sont effectuées en empruntant de l’information d’autres domaines associés ou d’autres sources, et l’erreur quadratique moyenne de prévision (EQMP) sert souvent à mesurer l’incertitude. Les auteurs proposent une estimation analytique de l’EQMP corrigée pour le biais ainsi qu’une méthode jackknife d’appariement des moments afin d’estimer l’EQMP sans formuler d’hypothèses spécifiques sur la distribution des données. Ils présentent des études théoriques et empiriques comparant la performance des méthodes proposées aux procédures existantes.

Objective: To assess which of the equations used to estimate mechanical power output for a wide aerobic range of exercise intensities gives the closest value to that measured with the SRM training system. Methods: Thirty four triathletes and endurance cyclists of both sexes (mean (SD) age 24 (5) years, height 176.3 (6.6) cm, weight 69.4 (7.6) kg and Vo2max 61.5 (5.9) ml/kg/min) performed three incremental tests, one in the laboratory and two in the velodrome. The mean mechanical power output measured with the SRM training system in the velodrome tests corresponding to each stage of the tests was compared with the values theoretically estimated using the nine most referenced equations in literature (Whitt (Ergonomics 1971;14:419–24); Di Prampero et al (J Appl Physiol 1979;47:201–6); Whitt and Wilson (Bicycling science. Cambridge: MIT Press, 1982); Kyle (Racing with the sun. Philadelphia: Society of Automotive Engineers, 1991:43–50); Menard (First International Congress on Science and Cycling Skills, Malaga, 1992); Olds et al (J Appl Physiol 1995;78:1596–611; J Appl Physiol 1993;75:730–7); Broker (USOC Sport Science and Technology Report 1–24, 1994); Candau et al (Med Sci Sports Exerc 1999;31:1441–7)). This comparison was made using the mean squared error of prediction, the systematic error and the random error. Results: The equations of Candau et al, Di Prampero et al, Olds et al (J Appl Physiol 1993;75:730–7) and Whitt gave a moderate mean squared error of prediction (12.7%, 21.6%, 13.2% and 16.5%, respectively) and a low random error (0.5%, 0.6%, 0.7% and 0.8%, respectively). Conclusions: The equations of Candau et al and Di Prampero et al give the best estimate of mechanical power output when compared with measurements obtained with the SRM training system.