Semiparametric Mixtures of Symmetric Distributions

type="main" xml:id="sjos12015-abs-0001"> We consider in this paper the semiparametric mixture of two unknown distributions equal up to a location parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. To insure the identifiability of the model, it is assumed that the mixed distribution is zero symmetric, the model being then defined by the mixing proportion, two location parameters and the probability density function of the mixed distribution. We propose a new class of M-estimators of these parameters based on a Fourier approach and prove that they are n -consistent under mild regularity conditions. Their finite sample properties are illustrated by a Monte Carlo study, and a benchmark real dataset is also studied with our method.

[1]  A. Tsybakov,et al.  Wavelets, approximation, and statistical applications , 1998 .

[2]  L. Bordes,et al.  SEMIPARAMETRIC ESTIMATION OF A TWO-COMPONENT MIXTURE MODEL , 2006, math/0607812.

[3]  Cristina Butucea,et al.  Goodness-of-fit testing and quadratic functional estimation from indirect observations , 2006, math/0612361.

[4]  Xiao-Hua Zhou,et al.  NONPARAMETRIC ESTIMATION OF COMPONENT DISTRIBUTIONS IN A MULTIVARIATE MIXTURE , 2003 .

[5]  A. Bowman,et al.  A look at some data on the old faithful geyser , 1990 .

[6]  P. Hall.  On the Non‐Parametric Estimation of Mixture Proportions , 1981 .

[7]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[8]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[9]  D. Hunter,et al.  Inference for mixtures of symmetric distributions , 2007, 0708.0499.

[10]  E. Gassiat,et al.  Testing the order of a model using locally conic parametrization : population mixtures and stationary ARMA processes , 1999 .

[11]  David W. Scott,et al.  Multivariate Density Estimation: Theory, Practice, and Visualization , 1992, Wiley Series in Probability and Statistics.

[12]  P. Cerrito.  Using stratification to estimate multimodal density functions with applications to regression , 1992 .

[13]  J. Birch,et al.  Interactive Data Analysis , 1978 .

[14]  G. Imbens,et al.  Case-control studies with contaminated controls☆ , 1996 .

[15]  Jing Qin,et al.  Empirical likelihood ratio based confidence intervals for mixture proportions , 1999 .

[16]  Jiahua Chen.  Optimal Rate of Convergence for Finite Mixture Models , 1995 .

[17]  R. Beran.  An Efficient and Robust Adaptive Estimator of Location , 1978 .

[18]  D. Titterington,et al.  Estimation Problems with Data from a Mixture , 1978 .

[19]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[20]  Mohamed Lemdani,et al.  Likelihood ratio tests in contamination models , 1999 .

[21]  B. Everitt,et al.  Finite Mixture Distributions , 1981 .

[22]  B. Leroux.  Consistent estimation of a mixing distribution , 1992 .