On estimation of measurement error models with replication under heavy-tailed distributions

Measurement error (errors-in-variables) models are frequently used in various scientific fields, such as engineering, medicine, chemistry, etc. In this work, we consider a new replicated structural measurement error model in which the replicated observations jointly follow scale mixtures of normal (SMN) distributions. Maximum likelihood estimates are computed via an EM type algorithm method. A closed expression is presented for the asymptotic covariance matrix of those estimators. The SMN measurement error model provides an appealing robust alternative to the usual model based on normal distributions. The results of simulation studies and a real data set analysis confirm the robustness of SMN measurement error model.

[1]  Felipe Osorio,et al.  On estimation and influence diagnostics for the Grubbs' model under heavy-tailed distributions , 2009, Comput. Stat. Data Anal..

[2]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[3]  F. Vaida,et al.  Conditional Akaike information for mixed-effects models , 2005 .

[4]  Tak K. Mak,et al.  Maximum Likelihood Estimation of a Linear Structural Relationship with Replication , 1979 .

[5]  E. Cornish,et al.  On the Secular Variation of Rainfall at Adelaide , 1954 .

[6]  Ying Nian Wu,et al.  Efficient Algorithms for Robust Estimation in Linear Mixed-Effects Models Using the Multivariate t Distribution , 2001 .

[7]  Alexander Kukush,et al.  Measurement Error Models , 2011, International Encyclopedia of Statistical Science.

[8]  Ea Coornish,et al.  The Multivariate t-Distribution Associated with a Set of Normal Sample Deviates , 1954 .

[9]  H. Akaike.  A new look at the statistical model identification , 1974 .

[10]  K. Lange,et al.  Normal/Independent Distributions and Their Applications in Robust Regression , 1993 .

[11]  Filidor V. Labra,et al.  Multivariate measurement error models based on scale mixtures of the skew–normal distribution , 2010 .

[12]  D. F. Andrews,et al.  Scale Mixtures of Normal Distributions , 1974 .

[13]  R. Hoover,et al.  Dietary fat and breast cancer in the National Health and Nutrition Examination Survey I Epidemiologic Follow-up Study. , 1987, Journal of the National Cancer Institute.

[14]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[15]  E A Frongillo,et al.  Sources of fiber and fat in diets of US women aged 19 to 50: implications for nutrition education and policy. , 1992, American journal of public health.

[16]  V. H. Lachos,et al.  On estimation and local influence analysis for measurement errors models under heavy-tailed distributions , 2011 .

[17]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[18]  C. Dunnett,et al.  A BIVARIATE GENERALIZATION OF STUDENT'S t-DISTRIBUTION, WITH TABLES FOR CERTAIN SPECIAL CASES , 1954 .

[19]  W. Rogers,et al.  Understanding some long-tailed symmetrical distributions , 1972 .

[20]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[21]  Yoshiko Isogawa.  Estimating a Multivariate Linear Structural Relationship with Replication , 1985 .

[22]  D. Hall.  Measurement Error in Nonlinear Models: A Modern Perspective , 2008 .

[23]  Sudhir Gupta,et al.  Statistical Regression With Measurement Error , 1999, Technometrics.

[24]  Nan Lin,et al.  Adjustment of Measuring Devices With Linear Models , 2004, Technometrics.

[25]  D. Harville.  Matrix Algebra From a Statistician's Perspective , 1998 .

[26]  Francisco José de A. Cysneiros,et al.  Assessment of diagnostic procedures in symmetrical nonlinear regression models , 2010, Comput. Stat. Data Anal..

[27]  O. Reiersøl.  Identifiability of a Linear Relation between Variables Which Are Subject to Error , 1950 .