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1. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 Asymptotic Normality of Estimator Variance Components for One-Way Repeated Measurements Model Hadeel Ismail Mustafa1, Abdul hussein Saber AL-Mouel2 1Computer Information System Department, Computer Scince and Information System collage, University of Basrah, Iraq, hadeelismu@gmail.com 2Mathematics Department, Education for Pure Science collage, University of Basrah, Iraq, abdulhusseinsaber@yahoo.com Abstract In this research, a modification was made to the mean bias reduction method to estimate the variance components in the repeated measurements model by replacing the bias function in the mean bias reduction method with another that is dependent on a sample of independent observations that simulates the study model with variance components. As a result, we obtained a modified function converging with the mean bias reduction function, and from it we find new estimators for the variance components of the repeated measurements model. The aim of this research is to study the behavior of the new estimator of the variance components in the repeated measurements model resulting from the modified method on the mean bias reduction method by studying the asymptotic normality of the modified method estimator. 1. Introduction Repeated measurements models are statistical models that are used to analyze data collected from the same subjects or objects at different points in time. These models are commonly used in various fields, including medical research, social sciences, and engineering, to study the changes in variables over time and to understand the relationship between these variables. One of the key advantages of repeated measurements models is their ability to account for the correlation between measurements taken from the same subject. This correlation arises due to the fact that the measurements are not independent of each other, and therefore, ignoring this correlation can lead to biased estimates of the model parameters. In practical life, repeated measurements models are used to analyze data from longitudinal studies, where subjects are followed up over a period of time to study the progression of a disease or the effect of an intervention. They are also used in quality control, where multiple measurements of the same product or process are taken to ensure consistency and reliability. The estimation of variance plays a critical role in repeated measurements models, as it is used to quantify the amount of variability in the data. In practice, the variance is often estimated using sample data, and it is important to find reliable and accurate estimators of the variance to ensure the validity of the statistical analysis. The Asymptotic normality of estimators is an important concept in repeated measurements models, as it refers to the fact that as the sample size increases, the estimator becomes more and more accurate, and its variability decreases. This property is important because it allows us to assess the precision of the estimator and to determine the sample size needed to achieve a desired level of precision. In 261

2. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 addition, the natural convergence of estimators helps us to understand the behavior of the estimator under different conditions, such as when the data is skewed or when there are outliers. The aim of this research is to study the behavior of the new estimator of the variance components in the repeated measurements model resulting from the modified method on the mean bias reduction method by studying the asymptotic normality of the modified method estimator. 2. Description the model The One-Way repeated measurements model is define as following ????= ? + ??+ ??+ (??)?? + ???(?)+ ???(?)+ ????…(?) Where ? = 1,2,…,?1 is a unite of experimentation index .? = 1,2,…,?2 is an indicator of the between units factor’s levels (Group). ? = 1,2,…,?3 is an indicator of the within unite factor’s levels (Time).???? = is the measurement of unit’s response over time in a group.????= is the random error.? = is the overall mean .In the table (1) classifies the effects of the math model (1) by type. Table (1): classifies the effects of One-Way repeated measurements model. The factor The Type Fixed The definition Is treatment between unite factor (group) The condition ?? ?? = ? ∑?? ?=? ?? Fixed Is treatment within unite factor (Time) ?? = ? ∑?? ?=? ?? Fixed Is the effect (between X within) unites = ? ∑???? (??)?? ?=? ?? = ? ∑???? ?=? Random ???(?) ?) Is the random effect to unit ? within Group (?) Is the random effect to unit? within time Is the random error ????∼ ?(?,??? Random ???(?) ?) ????∼ ?(?,??? Random ???? ?) ????∼ ?(?,?? To formulate our model as matrix formula, Where ? is the identity matrix, 1is the vector one’s, ⊗ be the Kronecker product . first we written the model (1) in as below : ???= ?? + ?????(?)+ ?????+ ??? …(2 ) where ???= [???1,,…,????3]′is a vector of response, ?? = [??1,,…,???3]′ is a vector of the fixed treatments,??(?)= ?1?(?) is the random effect of unit ? within a group (j), ??= 262

3. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 [??(1),,…,??(?3)]′ is the random effect to unit ? within time(?2?(?))and ??? = [???1,,…,????3]′is a vector of random error. Let бij represent the indicators of a group such that бij= {1 0 o.w if unit i is from group j ′ = [??1,,…,???2]′ , we can rewrite model (2) as τi = Xi ? +ZiW + εi j = 1,…, ?2 . By setting б? … (3) ′= [ti1 ,…, t??3], Xi=??3⊗бi, ?′= [?11,,…,??2?3], Zi = 1?3 ⊗ 1?2 , W= where t? ?1?3 ⋮ ??2?3 ?11 ⋮ ??21 ⋯ ⋯ ⋯ ′ = [εi1 ,…, ε??3] and ? = Vec [ ] .The final formulation of [?1?(?) ,?2?(?) ], ?? the study model is an alternative form of the model(4) as shown in the relationship (5), let ?′ = [ ?1 ′,…,??1 ′,…,X?1 ′,…,z?1 ′], ?′ = [ X1 ′] , ? = [?1?(?) ,?2?(?),?? ] and Z =[z1 ′] then ? = X ? + Z ?… (4) ? ? ⋮ ∑?1+ ∑?2+ ∑? ? ⋮ ? ⋯ ? ⋯ ⋯ ? ∑?1+ ∑?2+ ∑? ⋮ ? ] The variance matrix of ? is Σ = [ ∑?1+ ∑?2+ ∑? Where 2[??1⊗ ??2 2[??1 ⊗ ??2 ⨂??3] , where ? denoted the matrix of ∑?1=??1 ones. ??3], ∑?2=??2 ⊗ ∑?= ??2 [??1 ⊗ ??2 ⊗ ??3] 3. Reduction Estimation of variance components To estimate the variance components of the repeated measurements model shown in Equation (4) using the maximum likelihood method, by deriving the maximum likelihood function for the Θ = (??1 and by solving equation (5) we obtain estimators for the variance components that are biased. 2 ,??2 2 ,??2) = Θ?i.e ( Θ? is ??ℎ component of vector of variance Θ ) defined by formula (5), 1 2[?′ Σ−1( Θ) Σ0 Σ−1( Θ) ? − ??(Σ−1( Θ)Σ0)] 1 2[?′ Σ−1( Θ) Σ1 Σ−1( Θ) ? − ??(Σ−1( Θ)Σ1)] 1 2[?′ Σ−1( Θ) Σ2 Σ−1( Θ) ? − ??(Σ−1( Θ)Σ2)]] …(5) ΓΘ= [ To reduce the bias resulting from the above method, we use the method of Mean bias reduction where the derivative is defined with respect to the variance components in the formula (6), through the solution of equation (6) we get Θ ̃ is the estimators of the variance components that are less biased than maximum likelihood method. 263

4. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ?2? ?2? ?2? ??(?,?) ?? −1 ?Θ0?Θ0 ?2? ?Θ0?Θ1 ?2? ?Θ0?Θ2 ?2? ? Θ0 ??(?,?) ?Θ0 ?? 0 E01 0 E21 E02 E12 0 + 1 1 2 tr Σ−1 Σ−1 ΓΘ ̃ = 2 tr E10 E20 [ ] ? Θ1 ??(?,?) ? Θ2] ?Θ1 ?? ?Θ2] ?Θ1?Θ0 ?2? ?Θ1?Θ1 ?2? ?Θ1?Θ2 ?2? ?Θ2?2 ] [ [ [ [ ] ?Θ2?Θ0 ?Θ2?Θ1 [ ] [ ] …(6) 0 E01 0 E21 E02 E12 0 ] E10 E20 where E(Θ) = [ E01 = E(?2?(?,?) - ? −1Σ0? −1Σ2). 2 tr (? −1Σ1? −1Σ0 - ? −1Σ0? −1Σ1), E02= E(?2?(?,?) ?Θ0?Θ1) = - 1 ?Θ0?Θ2) = - 1 2 tr (? −1Σ2? −1Σ0 E10 = E (?2?(?,?) - ? −1Σ1? −1Σ2). 2 tr (? −1Σ0? −1Σ1 - ? −1Σ1? −1Σ0),E12 = E( ?2?(?,?) ?Θ1?Θ0 )= - 1 ?Θ1?Θ2 ) = - 1 2 tr (? −1Σ2? −1Σ1 E20 = E(?2?(?,?) - ? −1Σ2? −1Σ1). 2 tr (? −1Σ0? −1Σ2 - ? −1Σ2? −1Σ0),E21 = E ?2?(?,?) ?Θ2?Θ0) = - 1 ?Θ2?Θ1) =- 1 2 tr (? −1Σ1? −1Σ2 ?2? ?2? ?2? ?? ?Θ0?Θ0 ?2? ?Θ0?Θ1 ?2? ?Θ0?Θ2 ?2? ?Θ0 ?? …(7)is bias of Θ ̃. Set M = tr Σ−1 Σ−1 ?Θ1 ?? ?Θ2] ?Θ1?Θ0 ?2? ?Θ1?Θ1 ?2? ?Θ1?Θ2 ?2? ?Θ2?2 ] [ [ [ ] ?Θ2?Θ0 ?Θ2?Θ1 4. Method based on Mean bias reduction In some cases, it is difficult to find the analytical solution for the amount of bias M defined by formula (7) so we defined M* (Θ) = ∑ ? and ?? are responses simulated by model with Θ . substituted M* (Θ ̃) in formula (6) and as a result, we got a new equation ΓΘ ̃* based on the mean bias reduction method ? ?=1 Θ ̃?(??) − Θ , where Θ ̃(??) is a solution of ΓΘ(??) =0 ??(?,?) Θ ̃0(??) Θ ̃1(??) ?=1 Θ ̃2(??) ?=1 ? ?=1 ? ?−1∑ ?−1∑ ?−1∑ −1 ? Θ0 ??(?,?) − Θ − Θ − Θ 0 E01 0 E21 E02 E12 0 + 1 ΓΘ ̃* = 2 tr [ [ ] ]…(8) E10 E20 ] [ ? Θ1 ??(?,?) ? Θ2] ? [ Theorem 1(asymptotically consistent). If Ä⊂ ℜ? is a compact set for parameters, ΓΘ and E(Θ) are continuous functions of Θ then ΓΘ ̃ and ΓΘ ̃* are converges in probability . Proof : 264

5. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ??(?,?) Θ ̃0(??) Θ ̃1(??) Θ ̃2(??) ? Θ0 ??(?,?) 0 E01 0 E21 E02 E12 0 Θ0 Θ1 Θ2 let ΓΘ ̃* = ?−1∑ ? ?=1 . E10 E20 − [ ] [ − ] ? Θ1 ??(?,?) ? Θ2] { [ } ??(?,?) Θ ̃0(??) Θ ̃1(??) Θ ̃2(??) ? Θ0 ??(?,?) 0 E01 0 E21 E02 E12 0 Θ0 Θ1 Θ2 set Θ ∊ Ä then ] is continuous for every Θ. E10 E20 − [ ] [ − ? Θ1 ??(?,?) ? Θ2] [ ??(?,?) Θ ̃0(??) Θ ̃1(??) Θ ̃2(??) ? Θ0 ??(?,?) 0 E01 0 E21 E02 E12 0 Θ0 Θ1 Θ2 ‖ ‖ + ‖[ By the triangle inequality with ?= ‖ ] ‖ such that E10 E20 ‖ ] [ − ? Θ1 ??(?,?) ? Θ2 : ??(?,?) ??(?,?) Θ ̃0(??) Θ ̃1(??) Θ ̃2(??) ? Θ0 ??(?,?) ? Θ0 ??(?,?) 0 E01 0 E21 E02 E12 0 Θ0 Θ1 Θ2 ‖ ‖ ‖ ≤ ?. Then E10 E20 − [ ] [ − ]‖ − ? Θ1 ??(?,?) ? Θ2] E01 E10 E20 | Θ − Θ0| < ? when Ȧ is partition of Ä then ‖ Ȧ‖ < ? Θ1 ??(?,?) ? Θ2] [ [ Θ ̃0(??) Θ ̃1(??) Θ ̃2(??) 0 E02 E12 0 Θ0 Θ1 Θ2 ] is bounded on Ä . ∀? > 0,∃? > 0 and Θ0 ,Θ⊂ Äsuch that 0 [ ] [ − E21 ? √p since ? is continuous on Ä there are Θ? ? , Θ0? in Ä? such that ?(Θ?) = sup?(Θ)Θ ∊Ä? and ?(Θ0) = inf?(Θ)Θ ∊Ä? since ‖ Ȧ‖ < | Θ?− Θ0?| < ? with Θ = Θ? and Θ0= Θ0? then |?(Θ) − ?(Θ0)| < ? i.e ? is continuous in ℜ?. By Heine-Borel theorem Ä is closed and bounded . Ä is closed subset in product of bounded integrable. Since ΓΘ ̃=ΓΘ when Θ ̃ = Θ such that M=0 then ΓΘ ̃ ∞ then ΓΘ ̃* →ΓΘ are such that √p and ? →ΓΘ Similarly ΓΘ ̃*=ΓΘ as R→ ? ‖ ΓΘ ̃− ΓΘ ̃∗ ‖≤‖ ΓΘ ̃− ΓΘ+ ΓΘ− ΓΘ ̃∗‖≤‖ ΓΘ ̃− ΓΘ‖ + ‖ ΓΘ− ΓΘ ̃∗‖ Sup ‖ ΓΘ ̃− ΓΘ ̃∗‖≤ sup ‖ ΓΘ ̃− ΓΘ‖ + sup‖ ΓΘ ̃∗ − ΓΘ‖→ 0 as R→ ∞ . Theorem 2.If Ä⊂ ℜ? is a compact set for parameters, ΓΘ and E(Θ) are continuous functions of Θ. ΓΘ ̃ has a unique solution at Θ ̃∊Ä and ΓΘ ̃* uniform → ΓΘ ̃ as R→ ∞ then any Θ ̃∗∊Ä ΓΘ ̃*= 0. ? →Θ ̃ when 265

6. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 Proof: Proof is easy from theorem (1) , ΓΘ ̃* uniform ℬ, → ΓΘ ̃ as D→ ∞ then ∀? > 0,∃ ℬ > 0 such that D > ? >supΘ ∊Ä‖ ΓΘ ̃− ΓΘ ̃∗‖≥‖ ΓΘ ̃(Θ ̃∗) − ΓΘ ̃∗(Θ ̃∗)‖ = ‖ ΓΘ ̃∗(Θ ̃∗)‖ So Θ ̃∗→ unique Θ ̃ . 5. Asymptotic normality of ? ̃∗ estimator Asymptotic normality is a fundamental concept in statistics that refers to the behavior of statistical estimators as the sample size approaches infinity. Theorem 3.If Ä⊂ ℜ? is a compact set for parameters, ΓΘ and E(Θ) are continuous functions of Θ, ? →lim →? when n= n1 n2 n3 and j= 1,2,…,p then ?−1 ? ?=1 ?ΓΘ ̃∗ ?Θ?| ? ?=1 ? ?=1 ?−1∑ ?→∞?−1∑ the matrix ?= -?−1∑ is positive definite with ΓΘΘ E(Θ)? E(Θ)? ? 2 (Θ ̃∗− Θ) ~ N(0, (1+D-1) E(Θ)-1). ?−1|− Proof: From asymptotically consistent theorem Θ ̃∗ is consistent estimator of Θ .By Taylor`s theorem to ΓΘ ̃∗ a bout Θ ̃∗ such that : ΓΘ ̃∗(Θ) + ∇ΓΘ ̃∗(Θ ̌) (Θ ̃∗ - Θ) = 0 where Θ ̌ = Θ + ?(Θ ̃∗ - Θ) , ?∊ (0,1) −1 (Θ ̃∗ - Θ) = - ΓΘ ̃∗(Θ) {∇ ΓΘ ̃∗(Θ ̌)} −1 ⁄ΓΘ ̃∗(Θ) {−∇ ΓΘ ̃∗(Θ ̌)} ⁄ (Θ ̃∗- Θ) = ?1 2 ?1 2 −1 ⁄ (Θ ̃∗ - Θ) = {−∇ ΓΘ ̃∗(Θ ̌) ⁄ΓΘ ̃∗(Θ)). ?1 2 (?−1 2 } ? d → N( 0p , E(Θ)) as n → ∞ .Then ?1 2 d → N( 0p ⁄ (Θ ̃∗ - Θ) ⁄ΓΘ From the central limit theorem ?−1 2 , E(Θ)−1) ⁄ (Θ ̃∗?- Θ) are independent for ?= 1,2,…,D we have the following joint as n → ∞ . where ?1 2 limit 266

7. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 ⁄ (Θ ̃∗1− Θ) ⁄ (Θ ̃∗2− Θ) ⋮ ?1 2 ?1 2 ?1 2 d → N( 0pD ,Ξ) such that Ξ is diagonal blocks of E(Θ)−1. From continuous ⁄ (Θ ̃∗?− Θ)] [ mapping theorem d → N( 0p , ?−1E(Θ)−1) ⁄ (Θ ̃∗?− Θ) ⁄M(Θ)? ? ?=1 ?1 2 ∗ = ?−1∑ ?1 2 ⁄ (Θ ̃∗?− Θ) ? ?=1 ⁄ ∑ ? ?=1 Because ?1 2 and ?−1∑ ?1 2 are independent ΓΘ ? ?=1 ⁄ ∑ ?1 2 d → N (02? , E(Θ) 0?×? then [?1 2 ΓΘ ∗] ?−1E(Θ)−1) ⁄ M(Θ)? 0?×? d → N( 0p , (1 + ?−1)E(Θ)). From the ∗ = ?−1 2 ⁄ΓΘ ̃ ⁄ΓΘ - ?−1 E(Θ)?1 2 ⁄M(Θ)? 2 (Θ ̃∗− Θ) ~ N(0, (1+D-1) E(Θ)-1) and we have ?−1 2 consistency of Θ ̌ and slutusky`s lemma we have ?−1 ∗ 6. Conclusions The mean bias reduction method was modified to estimate the variance components in the repeated measurement model by replacing the bias function in a mean bias reduction method with another that is dependent on a sample of independent observations that simulates the study model with variance components. Then proved the modified method converged with the mean bias reduction method, it was found that the estimator of the variance components in the modified method is consistent and finally, it was proved that the asymptotic normality of the estimator is achieved. References [1] Al-Mouel , A. H. S. and Wang, J. L., (2004) .One –Way Multivariate Repeated Measurements Analysis of Variance Model. Applied Mathematics a Journal of Chinese Universities 19(4),pp.435-448. [2] AL-Mouel, A. H. S and Mustafa, H. I., (2014). The Sphericity Test for OneWay Multivariate Repeated Measurements Analysis of Variance Mode. Journal of Kufa for Mathematics and Computer Vol.2, no.2, Dec,2014,pp 107-115. [3] Al-Mouel, A. H. S., (2004). Multivariate Repeated Measures Models and Comparison of Estimators, Ph.D. Thesis, East China Normal University, China. [4] Jassim N O and Al-Mouel, A H S., (2020) .Lasso Estimation for High-Dimensional Repeated Measurement Model, AIP Conf. Proc., 2292, 020002, pp. 1-10. [5] Jiang, J.,(2007). Linear and Generalized Linear Mixed Models and Their Applications. Springer Series in Statistics. 267

8. UtilitasMathematica ISSN 0315-3681 Volume 120, 2023 [6] Kori H. A. and AL-Mouel, A. H. S, (2021). Expected mean square rate estimation of repeated measurements model, Int. J. Nonlinear Anal. Appl. 12 (2021) No. 2, 75-83. [7] Kosmidis I, Guolo A and Varin C.,( 2017). Improving the accuracy of likelihood-based inference in meta-analysis and metaregression. Biometrika; 104: 489–496. [8] Kosmidis, I., (2014a). Bias in parametric estimation: reduction and useful sideeffects. Wiley Interdisciplinary Reviews: Computational Statistics 6, 185–196. [9] Vonesh, E.F. and Chinchilli, V.M., (1997). Linear and Nonlinear Models for the Analysis of Repeated Measurements, Marcel Dakker, Inc., New York. [10] Wand, M., (2007). Fisher information for generalised linear mixed models. Journal of Multivariate Analysis 98, 1412–1416. 268