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. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..

Analysis of a simulated microarray dataset: Comparison of methods for data normalisation and detection of differential expression (Open Access publication)

“…Correction for the false discovery rate (FDR) [2] was the most commonly used technique for adjusting P-values. However, a direct comparison of multiple testing procedures occurred in the INRA_T analyses, with the two novel methods presented out-performing the FDR procedure proposed by Benjamini and Hochberg [2] in terms of error rate; the mixture model described by Bordes et al [3] performed particularly well. The performance of the INRA_T methods is of note given that similar gene-by-gene methods have been shown to lack power in comparison to shrinkage methods such as limma [17] and the structural model [11].…”

“…INRA_J used a structural mixed model, more completely described in Jaffrézic et al [11]. CDB employed a cut-off value for the mean log ratio to define the proportion of differentially expressed genes [10,18] [3]. Apart from INRA_T, those methods that corrected P-values for multiple testing did so using the FDR as described by Benjamini and Hochberg [2].…”

Semiparametric Two-Component Mixture Models When One Component Is Defined Through Linear Constraints

“…This model appears in the study of gene expression data coming from microarray analysis. An application to two bovine gestation mode comparison is performed in [1]. The authors suppose that θ is known, f 0 is symmetric around an unknown µ and that r = 1.…”

“…Another application of model (1) in genetics can be found in [3]. The semiparametric model was employed in [4] (supposing that θ is known) for multiple testing procedures in order to estimate the posterior population probabilities and the local false rate discovery.…”

“…A method was proposed in [1] and was later improved in [8] where the authors assume that the unknown density is symmetric around an unknown value µ and that the parametric component is fully known, i.e. θ is known.…”