Testing for interactions in generalized additive models: Application to SO<Subscript>2</Subscript> pollution data

In this paper we considered a generalized additive model with second-order interaction terms. A local scoring algorithm (with backfitting) based on local linear kernel smoothers was used to estimate the model. Our main aim was to obtain procedures for testing second-order interaction terms. Backfitting theory is difficult in this context, and a bootstrap procedure is therefore provided for estimating the distribution of the test statistics. Given the high computational cost involved, binning techniques were used to speed up the computation in the estimation and testing process. A simulation study was carried out in order to assess the validity of the bootstrap-based tests. Lastly, our method was applied to real data drawn from an SO₂ binary time series.     

Local Likelihood Estimation in Generalized Additive Models

ABSTRACT.  Generalized additive models are a popular class of multivariate non-parametric regression models, due in large part to the ease of use of the local scoring estimation algorithm. However, the theoretical properties of the local scoring estimator are poorly understood. In this article, we propose a local likelihood estimator for generalized additive models that is closely related to the local scoring estimator fitted by local polynomial regression. We derive the statistical properties of the estimator and show that it achieves the same asymptotic convergence rate as a one-dimensional local polynomial regression estimator. We also propose a wild bootstrap estimator for calculating point-wise confidence intervals for the additive component functions. The practical behaviour of the proposed estimator is illustrated through a simulation experiment.     

Bootstrap adjustments of signed scoring rule root statistics

Scoring rules give rise to methods for statistical inference and are useful tools to achieve robustness or reduce computations. Scoring rule inference is generally performed through first-order approximations to the distribution of the scoring rule estimator or of the ratio-type statistic. In order to improve the accuracy of first-order methods even in simple models, we propose bootstrap adjustments of signed scoring rule root statistics for a scalar parameter of interest in presence of nuisance parameters. The method relies on the parametric bootstrap approach that avoids onerous calculations specific of analytical adjustments. Numerical examples illustrate the accuracy of the proposed method.     

Bootstrap bandwidth selection method for local linear estimator in exponential family models

Many biological experiments involve data whose distribution belongs to the exponential family. Such data are often analysed using generalised linear models but this method requires specification of the link function which can have strong influence on the resulting estimate. Instead a local method based on quasi-likelihood can be used, but the choice of the smoothing parameter is crucial for its performance. A bootstrap bandwidth selection method is proposed and shown to be consistent. Examples of application to data from biological and psychometric experiments are given.     

Summer temperature effects on deaths and hospital admissions among the elderly population in two Italian cities

In developed countries the effects of climate on health status are mainly due to temperature. Our analysis is aimed to deepen statistically the relationship between summer climate conditions and daily frequency of health episodes: deaths or hospital admissions. We expect to find a U-shaped relationship between temperature and frequencies of events occurring in summer regarding the elderly population resident in Milano and Brescia. We use as covariates hourly records of temperature recorded at observation sites located in Milano and Brescia. The analysis is performed using Generalized Additive Models (GAM), where the response variable is the daily number of events, which varies as a possibly non-linear function of meteorological variables measured on the same or previous day. We consider separate models for Milano and Brescia and then we compare temperature effects among the two towns and among different age classes. Moreover we consider separate models for all diagnosed events, for those due to respiratory disease and those due to circulatory pathologies. Model selection is a central problem, the basic methods used are the UBRE and GCV criteria but, instead of conditioning all final conclusions on the best model according to the chosen criterion, we investigated the effect of model selection by implementing a bootstrap procedure.     

Bootstrap procedures in a spatial-temporal model

Two bootstrap procedures are introduced into the hybrid of the backfitting algorithm and the Cochrane–Orcutt procedure in the estimation of a spatial-temporal model. The use of time blocks of consecutive observations in resampling steps proved to be optimal in terms of stability and efficiency of estimates. Between iterations, there were minimal changes in the empirical distributions of the parameter estimates associated with the covariate and temporal effects indicating convergence of the algorithm. Crop yield data are used to illustrate the proposed methods.

The simulation study indicated that prediction error from the fitted model (estimated from either Method 1 or Method 2) is very low. Also, the prediction error is relatively robust to the number of spatial units and the number of time points.     

The Consistency of Estimators in Finite Mixture Models

The parameters of a finite mixture model cannot be consistently estimated when the data come from an embedded distribution with fewer components than that being fitted, because the distribution is represented by a subset in the parameter space, and not by a single point. Feng & McCulloch (1996) give conditions, not easily verified, under which the maximum likelihood (ML) estimator will converge to an arbitrary point in this subset. We show that the conditions can be considerably weakened. Even though embedded distributions may not be uniquely represented in the parameter space, estimators of quantities of interest, like the mean or variance of the distribution, may nevertheless actually be consistent in the conventional sense. We give an example of some practical interest where the ML estimators are root of n -consistent.
Similarly consistent statistics can usually be found to test for a simpler model vs a full model. We suggest a test statistic suitable for a general class of model and propose a parameter-based bootstrap test, based on this statistic, for when the simpler model is correct.     

A case study of green tree frog population size estimation by repeated capture–mark–recapture method with individual tagging: a parametric bootstrap method vs Jolly–Seber method

This paper deals with estimation of a green tree frog population in an urban setting using repeated capture–mark–recapture (CMR) method over several weeks with an individual tagging system which gives rise to a complicated generalization of the hypergeometric distribution. Based on the maximum likelihood estimation, a parametric bootstrap approach is adopted to obtain interval estimates of the weekly population size which is the main objective of our work. The method is computation-based; and programming intensive to implement the algorithm for re-sampling. This method can be applied to estimate the population size of any species based on repeated CMR method at multiple time points. Further, it has been pointed out that the well-known Jolly–Seber method, which is based on some strong assumptions, produces either unrealistic estimates, or may have situations where its assumptions are not valid for our observed data set.     

Confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. A result of Efron in 2014 is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron considered a confidence interval centred on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post‐model‐selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of the confidence interval centred on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probability, based on the full model.     

Corrected local polynomial estimation in varying‐coefficient models with measurement errors

The authors study a varying‐coefficient regression model in which some of the covariates are measured with additive errors. They find that the usual local linear estimator (LLE) of the coefficient functions is biased and that the usual correction for attenuation fails to work. They propose a corrected LLE and show that it is consistent and asymptotically normal, and they also construct a consistent estimator for the model error variance. They then extend the generalized likelihood technique to develop a goodness of fit test for the model. They evaluate these various procedures through simulation studies and use them to analyze data from the Framingham Heart Study.     

超越对数生产函数的半参数变系数估计模型

提出超越对数生产函数的半参数变系数模型,利用Profile方法给出产出弹性函数系数的局部加权最小二乘估计,并利用非参数条件自助法对有限样本的近似分布进行模拟,给出相对精确的广义似然比检验。规模报酬约束下中国1953--2008年的实证结果拒绝超越对数生产函数模型假设,产出弹性不可简单线性化而是对数劳均资本的非线性函数,时变资本弹性表现为倒U型变化趋势,时变劳动力弹性表现为U型变化趋势。     

A simple bootstrap bandwidth selector for local polynomial fitting

A new, fully data-driven bandwidth selector with a double smoothing (DS) bias term and a data-driven variance estimator is developed following the bootstrap idea. The data-driven variance estimation does not involve any additional bandwidth selection. The proposed bandwidth selector convergences faster than a plug-in one due to the DS bias estimate, whereas the data-driven variance improves its finite sample performance clearly and makes it stable. Asymptotic results of the proposals are obtained. A comparative simulation study was done to show the overall gains and the gains obtained by improving either the bias term or the variance estimate, respectively. It is shown that the use of a good variance estimator is more important when the sample size is relatively small.     

Efficient estimation of non parametric simultaneous equations models

This paper defines a new procedure to efficiently estimate non parametric simultaneous equations models. The proposed estimation procedure exploits the additive structure and achieves oracle efficiency without the knowledge of unobserved error terms. Furthermore, simulation results show that our new estimator outperforms the existing estimator in terms of mean squared error.     

Component Selection in the Additive Regression Model

Abstract. Similar to variable selection in the linear model, selecting significant components in the additive model is of great interest. However, such components are unknown, unobservable functions of independent variables. Some approximation is needed. We suggest a combination of penalized regression spline approximation and group variable selection, called the group‐bridge‐type spline method (GBSM), to handle this component selection problem with a diverging number of correlated variables in each group. The proposed method can select significant components and estimate non‐parametric additive function components simultaneously. To make the GBSM stable in computation and adaptive to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.     

Maximum entropy estimation of income share function from generalized Gini index

Following Sir Anthony and Atkinson who started thinking about the insensitivity of the Gini index to income shares of the lower and the upper income groups, a generalization of the classical Gini index was introduced by Kakwani, Donaldson, Weymark and Yitzhaki which is sensitive to both high and low incomes. In this paper, the maximum entropy method is used to estimate the underlying true income share function based on the limited information of the generalized Gini index about the income shares of a population's percentiles. The income share function is estimated through maximizing both the Shannon entropy and the second-order entropy. In the end, through parametric bootstrap and analyzing a real dataset, the results are compared with the estimator of the share function, which is obtained based on the total information. In contrast to the classic Gini index, the derived share function based on the generalized Gini index provides more accurate approximations for income shares of the lower and the upper percentiles.     

A dynamic system for Gompertz model

Two-parameter Gompertz distribution has been introduced as a lifetime model for reliability inference recently. In this paper, the Gompertz distribution is proposed for the baseline lifetimes of components in a composite system. In this composite system, failure of a component induces increased load on the surviving components and thus increases component hazard rate via a power-trend process. Point estimates of the composite system parameters are obtained by the method of maximum likelihood. Interval estimates of the baseline survival function are obtained by using the maximum-likelihood estimator via a bootstrap percentile method. Two parametric bootstrap procedures are proposed to test whether the hazard rate function changes with the number of failed components. Intensive simulations are carried out to evaluate the performance of the proposed estimation procedure.     

Estimation of reliability with cumulative stress and strength degradation

In many real-life scenarios, system reliability depends on dynamic stress–strength interference where strength degrades and stress accumulates concurrently over time. In this paper, we consider the problem of estimating reliability of a system under deterministic strength degradation and cumulative damage due to shocks arriving according to a point process. Maximum likelihood estimation under two different sampling plans has been considered. Large sample properties in general are discussed. The method is illustrated through simulation and real-life data analysis.     

Smoothing fertility trends in agricultural field experiments

Fertility trend within blocks and local variations are the major obstacles to estimate cultivar contrasts in agricultural field trials. This paper examines methods of smoothing fertility trends in field trials using the P-spline. We begin by smoothing trend within block and for each block, and proceeds to demonstrate how it can be extended to smooth trends in trials with two-dimensional setting. We propose simultaneous modelling of trends and local variation. We use Papadakis [J.S. Papadakis, Comparison de differentes methds d'expermentation phytotechnique, Rev. Argen. Agronom. 7 (1940), pp. 297–362.] and kriged covariate to model local variation. We emphasize on the benefit of using P-spline to compromise between parametric and non-parametric approaches. Data sets from wheat and barley trials, designed as randomized complete block design and row-column, are analyzed. We set out a simple strategy of choosing between additive model and two-dimensional setting. We explore different estimation methods and offer some generalizations. The results show importance of the P-spline in modelling trend and the need to choose between additive and two-dimensional settings.     

Polynomial Spline Estimation for a Generalized Additive Coefficient Model

Abstract.  We study a semiparametric generalized additive coefficient model (GACM), in which linear predictors in the conventional generalized linear models are generalized to unknown functions depending on certain covariates, and approximate the non-parametric functions by using polynomial spline. The asymptotic expansion with optimal rates of convergence for the estimators of the non-parametric part is established. Semiparametric generalized likelihood ratio test is also proposed to check if a non-parametric coefficient can be simplified as a parametric one. A conditional bootstrap version is suggested to approximate the distribution of the test under the null hypothesis. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed methods. We further apply the proposed model and methods to a data set from a human visceral Leishmaniasis study conducted in Brazil from 1994 to 1997. Numerical results outperform the traditional generalized linear model and the proposed GACM is preferable.