Maximum likelihood estimation in mixed normal models with two variance Components

Gnot, S.; Stemann, Dietmar; Trenkler, Götz; ́ska-Motyka, A. Urban

doi:10.1080/02331880213197

Cited by 4 publications

(8 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hoeschele also showed that the corresponding restricted likelihood must have one maximum. Gnot et al () also showed that the likelihood equation for the two‐variance problem with two groups can have up to two solutions, with one being a saddle point. Their restricted likelihood and likelihood have nearly identical forms, and for the former, they show that there is a single solution for their two‐group problem.…”

Section: Previous Workmentioning

confidence: 99%

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Henn

Hodges

2014

Int Statistical Rev

View full text Add to dashboard Cite

Scattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying in addition an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak-finding algorithms to estimate parameters.

show abstract

Section: Previous Workmentioning

confidence: 99%

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Henn

Hodges

2014

Int Statistical Rev

View full text Add to dashboard Cite

show abstract

“…It can be checked that l 0 (β, s, Y ) < l 0 ( β, s, Y ) for β = β. It can be thus seen that the problem of computing the ML estimator of (β, s) reduces to finding the maximizer of l(s, Y ) := − log |Σ(s)| − Y ′ R(s)Y over s ∈ S, which we will refer to as the ML estimator of s, compare [11, p. 230] and [4,284]. It can be also observed that for a given value y of the vector Y the ML estimate of s exists if and only if the ML estimate of (β, s) exists.…”

Section: Model With Two Variance Componentsmentioning

confidence: 99%

On maximum likelihood estimation in mixed normal models with two variance components

Grządziel¹

2014

Discuss. Math. Probability and Statistics

View full text Add to dashboard Cite

We extend the results concerning the upper bounds for the maximum likelihood degree and the REML degree of the one-way random effects model presented in Gross et al. [Electron. J. Stat. 6 (2012), pp. 993-1016] to the case of the normal linear mixed model with two variance components. Then we prove that both parts of Conjecture 1 in the paper of Gross et al., which concerns a certain extension of the one-way random effects model, are true under fairly mild conditions.

show abstract

“…It is well known that ML estimators, when they exist, are generally heavily biased. For the oneand two-way classification models this problem has been considered in detail by Swallow and Monahan [17] and by Gnot et al [5]. An adjusted version of the profile log likelihood l can be constructed by adding to l the term − 1 2 log|X Σ −1 X|.…”

Section: Introductionmentioning

confidence: 99%

“…0 be the eigenvalues of V with the multiplicities s 1 , s 2 , ..., s d 0 , respectively (s d 0 = n − rank(V )). It has been shown by Gnot et al [5] that the log likelihood functions l and l 0 can be presented in a spectral form given in the following propositions.…”

Section: Introductionmentioning

confidence: 99%

“…A next important remark is that (3) and ( 4) is linear with respect to the variance components iff d = d 0 = 2 and m 1 = α 1 , while the system ( 5) and ( 6) becomes linear iff d = 2 (cf. Szatrowski [18], Szatrowski and Miller [19], Gnot et al [4], [5]). If the above conditions hold, then ML and REML estimators can be expressed in explicit forms given by the following theorems.…”

mentioning

confidence: 98%

See 1 more Smart Citation

On some properties of ML and REML estimators in mixed normal models with two variance components

Gnot

Michalski

Urbańska-Motyka

2023

Discuss. Math. Probability and Statistics

View full text Add to dashboard Cite

In the paper, the problem of estimation of variance components σ 2 1 and σ 2 2 by using the ML-method and REML-method in a normal mixed linear model N Y, E(Y ) = Xβ, Cov(Y ) = σ 2 1 V + σ 2 2 I n is considered. This paper deal with properties of estimators of variance components, particularly when an explicit form of these estimators is unknown. The conditions when the ML and REML estimators can be expressed in explicit forms are given, too. The simulation study for one-way classification unbalanced random model together with a new proposition of approximation of expectation and variances of ML and REML estimators are shown. Numerical calculations with reference to the generalized Fisher's information are also given.

show abstract

Maximum likelihood estimation in mixed normal models with two variance Components

Cited by 4 publications

References 7 publications

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

On maximum likelihood estimation in mixed normal models with two variance components

On some properties of ML and REML estimators in mixed normal models with two variance components

Contact Info

Product

Resources

About