Nonnegative estimates of variance components in a two-way random model

  • 투고 : 2018.01.22
  • 심사 : 2018.06.05
  • 발행 : 2019.07.31


This paper discusses a method for obtaining nonnegative estimates for variance components in a random effects model. A variance component should be positive by definition. Nevertheless, estimates of variance components are sometimes given as negative values, which is not desirable. The proposed method is based on two basic ideas. One is the identification of the orthogonal vector subspaces according to factors and the other is to ascertain the projection in each orthogonal vector subspace. Hence, an observation vector can be denoted by the sum of projections. The method suggested here always produces nonnegative estimates using projections. Hartley's synthesis is used for the calculation of expected values of quadratic forms. It also discusses how to set up a residual model for each projection.


연구 과제 주관 기관 : National Research Foundation of Korea (NRF)


  1. EI-Leithy AH, Abdel-Wahed AZ, and Abdallah SM (2016). On non-negative estimation of variance components in mixed linear models, Journal of Advanced Research, 7, 59-68.
  2. Graybill FA (1983). Matrices with Applications in Statistics, Wadsworth, California.
  3. Hartley HO (1967). Expectations, variances and covariances of ANOVA mean squares by "synthesis", Biometrics, 23, 105-114.
  4. Henderson CR (1953). Estimation of variance and covariance components, Biometrics, 9, 226-252.
  5. Johnson RA and Wichern DW (1988). Applied Multivariate Statistical Analysis, Prentice-Hall, New Jersey.
  6. Milliken GA and Johnson DE (1984). Analysis of Messy Data, Van Nostrand Reinhold, New York.
  7. Searle SR (1971). Linear Models, John Wiley & Sons, New York.
  8. Searle SR, Casella G, and McCulloch CE (2009). Variance Components, John Wiley & Sons, New York.