A Distribution-free Approach in Statistical Modelling with Repeated Measurements and Missing Values

Abstract

Mixed effect models, which contain both fixed effects and random effects, are frequently used in dealing with correlated data arising from repeated measurements (made on the same statistical units). In mixed effect models, the distributions of the random effects need to be specified and they are often assumed to be normal. The analysis of correlated data from repeated measurements can also be done with GEE by assuming any type of correlation as initial input. Both mixed effect models and GEE are approaches requiring distribution specifications (likelihood, score function). In this article, we consider a distribution-free least square approach under a general setting with missing value allowed. This approach does not require the specifications of the distributions and initial correlation input. Consistency and asymptotic normality of the estimation are discussed.

Keywords:

• Asymptotic normality
• Consistent estimator
• Mean-square distance
• Mixed effect models
• Moment estimator.

Mathematics Subject Clssification:

• 62F12
• 62F86

Appendix

Proof of Lemma 2.1. Recall that the diagonal matrix in expression (8) is denoted as . We first show that (21) which is the key to proving the lemma. From assumption (II) and the properties of conditional expectation, we have for any j ∈ 1, …, p and hence This proves equality (19(19) ), from which we have This relationship and the definition of β⁽⁰⁾ in (9(9) ) give the lemma immediately.

Proof of Theorem 3.1. Under the assumptions (i), (ii), and (iii) of the theorem, Theorem 4.1 of Jennrich (1969) gives that Q_n(β) converges to Q(β) uniformly on Θ almost surely. The theorem now follows directly from this and the assumption (iv).

Proof of Theorem 4.1. It is clear that is asymptotically normal from the central limit theorem, and that n^{− 1}▽g⁽ⁿ⁾_j(β⁽⁰⁾) converges to a constant almost surely for each j = 1, …, r by the law of large numbers. Also note that each component of the Hessian matrix , n = 1, 2, …, is uniformly integrable under the assumptions of the theorem. Hence, and moreover, It follows, therefore, which is a deterministic matrix. Now the asymptotic multivariate normality of follows directly from the expression immediately above Theorem 4.1 and the assertions established here.