Semiparametric Estimation of a Two-component Mixture Model where One Component is known

Abstract.  We consider a two-component mixture model where one component distribution is known while the mixing proportion and the other component distribution are unknown. These kinds of models were first introduced in biology to study the differences in expression between genes. The various estimation methods proposed till now have all assumed that the unknown distribution belongs to a parametric family. In this paper, we show how this assumption can be relaxed. First, we note that generally the above model is not identifiable, but we show that under moment and symmetry conditions some 'almost everywhere' identifiability results can be obtained. Where such identifiability conditions are fulfilled we propose an estimation method for the unknown parameters which is shown to be strongly consistent under mild conditions. We discuss applications of our method to microarray data analysis and to the training data problem. We compare our method to the parametric approach using simulated data and, finally, we apply our method to real data from microarray experiments.     

A Semiparametric Model for Recapture Experiments

Abstract. A two-step procedure based on the conditional likelihood is proposed to estimate the population size of a closed population using a semiparametric model for recapture studies. An asymptotic variance estimate and numerical results are presented. The method is applied to a bird banding dataset in Hong Kong.     

Bootstrap hypothesis testing in generalized additive models for comparing curves of treatments in longitudinal studies

The study of the effect of a treatment may involve the evaluation of a variable at a number of moments. When assuming a smooth curve for the mean response along time, estimation can be afforded by spline regression, in the context of generalized additive models. The novelty of our work lies in the construction of hypothesis tests to compare two curves of treatments in any interval of time for several types of response variables. The within-subject correlation is not modeled but is considered to obtain valid inferences by the use of bootstrap. We propose both semiparametric and nonparametric bootstrap approaches, based on resampling vectors of residuals or responses, respectively. Simulation studies revealed a good performance of the tests, considering, for the outcome, different distribution functions in the exponential family and varying the correlation between observations along time. We show that the sizes of bootstrap tests are close to the nominal value, with tests based on a standardized statistic having slightly better size properties. The power increases as the distance between curves increases and decreases when correlation gets higher. The usefulness of these statistical tools was confirmed using real data, thus allowing to detect changes in fish behavior when exposed to the toxin microcystin-RR.     

A test for the linearity of the nonparametric part of a semiparametric logistic regression model

A semiparametric logistic regression model is proposed in which its nonparametric component is approximated with fixed-knot cubic B-splines. To assess the linearity of the nonparametric component, we construct a penalized likelihood ratio test statistic. When the number of knots is fixed, the null distribution of the test statistic is shown to be asymptotically the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. We set the asymptotic null expectation of this test statistic equal to a value to determine the smoothing parameter value. Monte Carlo experiments are conducted to investigate the performance of the proposed test. Its practical use is illustrated with a real-life example.     

Single-index modelling of conditional probabilities in two-way contingency tables

When analysing a contingency table, it is often worth relating the probabilities that a given individual falls into different cells from a set of predictors. These conditional probabilities are usually estimated using appropriate regression techniques. In particular, in this paper, a semiparametric model is developed. Essentially, it is only assumed that the effect of the vector of covariates on the probabilities can entirely be captured by a single index, which is a linear combination of the initial covariates. The estimation is then twofold: the coefficients of the linear combination and the functions linking this index to the related conditional probabilities have to be estimated. Inspired by the estimation procedures already proposed in the literature for single-index regression models, four estimators of the index coefficients are proposed and compared, from a theoretical point-of-view, but also practically, with the aid of simulations. Estimation of the link functions is also addressed.     

Semiparametric model average prediction in panel data analysis

Forecasting in economic data analysis is dominated by linear prediction methods where the predicted values are calculated from a fitted linear regression model. With multiple predictor variables, multivariate nonparametric models were proposed in the literature. However, empirical studies indicate the prediction performance of multi-dimensional nonparametric models may be unsatisfactory. We propose a new semiparametric model average prediction (SMAP) approach to analyse panel data and investigate its prediction performance with numerical examples. Estimation of individual covariate effect only requires univariate smoothing and thus may be more stable than previous multivariate smoothing approaches. The estimation of optimal weight parameters incorporates the longitudinal correlation and the asymptotic properties of the estimated results are carefully studied in this paper.