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Consistency, bias and efficiency of the normal-distribution-based MLE: The role of auxiliary variables

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  • Yuan, Ke-Hai
  • Savalei, Victoria

Abstract

Normal-distribution-based maximum likelihood (NML) is most widely used for missing data analysis although real data seldom follow a normal distribution. When missing values are missing at random (MAR), recent results indicate that NML estimates (NMLEs) are still consistent for nonnormally distributed populations as long as the variables are linearly related. However, NMLEs are generally not consistent when the variables are nonlinearly related in the population. Similarly, NMLEs are generally not consistent when data are missing not at random (MNAR). It is well-known that including proper auxiliary variables mitigates the bias in MLEs caused by MNAR mechanism. With nonlinear relationships underlying the manifest variables and under MAR mechanism, the article contains a theoretical result showing that NMLEs are still consistent when proper nonlinear functions of the observed variables are included as auxiliary variables. Empirical results indicate that including auxiliary variables reduces bias in the estimates, but may also increase their standard errors substantially when sample size is small and the proportion of missing data is not trivial. Empirical results also imply that bias in NMLEs due to a nonnormally distributed population and MAR mechanism can be considerably greater when compared to bias caused by MNAR mechanism with a normally distributed population. How to select auxiliary variables in practice is also discussed.

Suggested Citation

  • Yuan, Ke-Hai & Savalei, Victoria, 2014. "Consistency, bias and efficiency of the normal-distribution-based MLE: The role of auxiliary variables," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 353-370.
    • Handle: RePEc:eee:jmvana:v:124:y:2014:i:c:p:353-370
    DOI: 10.1016/j.jmva.2013.11.006
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    References listed on IDEAS

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    1. Yuan, Ke-Hai, 2009. "Normal distribution based pseudo ML for missing data: With applications to mean and covariance structure analysis," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1900-1918, October.
    2. Kano, Yutaka & Takai, Keiji, 2011. "Analysis of NMAR missing data without specifying missing-data mechanisms in a linear latent variate model," Journal of Multivariate Analysis, Elsevier, vol. 102(9), pages 1241-1255, October.
    3. Roderick J. A. Little, 1988. "Robust Estimation of the Mean and Covariance Matrix from Data with Missing Values," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 37(1), pages 23-38, March.
    4. Ke-Hai Yuan & Zhiyong Zhang, 2012. "Robust Structural Equation Modeling with Missing Data and Auxiliary Variables," Psychometrika, Springer;The Psychometric Society, vol. 77(4), pages 803-826, October.
    5. Liu, Chuanhai, 1997. "ML Estimation of the MultivariatetDistribution and the EM Algorithm," Journal of Multivariate Analysis, Elsevier, vol. 63(2), pages 296-312, November.
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