• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.543-564
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• 1999

Consider a symmetric bilinear form defined on $\prod\times\prod$ by $_{\lambda\mu}$ = $<\sigma,fg>\;+\;\lambdaL[f](a)L[g](a)\;+\;\muM[f](b)m[g](b)$ ,where $\sigma$ is a quasi-definite moment functional, L and M are linear operators on $\prod$, the space of all real polynomials and a,b,$\lambda$ , and $\mu$ are real constants. We find a necessary and sufficient condition for the above bilinear form to be quasi-definite and study various properties of corresponding orthogonal polynomials. This unifies many previous works which treated cases when both L and M are differential or difference operators. finally, infinite order operator equations having such orthogonal polynomials as eigenfunctions are given when $\mu$=0.

• Communications for Statistical Applications and Methods
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• v.21 no.2
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• pp.169-181
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• 2014

Marginalized random effects models (MREM) are commonly used to analyze longitudinal categorical data when the population-averaged effects is of interest. In these models, random effects are used to explain both subject and time variations. The estimation of the random effects covariance matrix is not simple in MREM because of the high dimension and the positive definiteness. A relatively simple structure for the correlation is assumed such as a homogeneous AR(1) structure; however, it is too strong of an assumption. In consequence, the estimates of the fixed effects can be biased. To avoid this problem, we introduce one approach to explain a heterogenous random effects covariance matrix using a modified Cholesky decomposition. The approach results in parameters that can be easily modeled without concern that the resulting estimator will not be positive definite. The interpretation of the parameters is sensible. We analyze metabolic syndrome data from a Korean Genomic Epidemiology Study using this method.