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统计代写|广义线性模型代写Generalized linear model代考|STAT7430 Asymptotic Behavior of ML and REML Estimators in Non-Gaussian Mixed ANOVA Models

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统计代写|广义线性模型代写Generalized linear model代考|Asymptotic Behavior of ML and REML Estimators in Non-Gaussian Mixed ANOVA Models

Asymptotic properties of ML estimators under the normality assumption have been studied by Hartley and Rao (1967), Anderson (1969, 1971a), and Miller (1977), among others. Asymptotic behavior of REML estimators has been studied by Das (1979) and Cressie and Lahiri (1993) under the normality assumption and by Richardson and Welsh (1994) and Jiang (1996, 1997a) without the normality assumption. All but Jiang (1996, 1997a) have assumed that the rank $p$ of the matrix $X$ is fixed or bounded, which turns out to be a critical assumption. This is because, under such an assumption, the ML and REML estimators are asymptotically equivalent in the sense that they are both consistent and asymptotically normal with the same asymptotic covariance matrix. On the other hand, earlier examples, including the famous Neyman-Scott problem (Neyman and Scott 1948; see Example 1.7), showed apparent asymptotic superiority of REML over ML in cases where the number of fixed effects increases with the sample size. In other words, to uncover the true superiority of REML one has to look into the case where the number of fixed effects grows with the sample size.

For simplicity of presentation, here we state some results for the balanced mixed ANOVA models with the Hartley-Rao form of variance components $\lambda$ and $\gamma_r, 1 \leq$ $r \leq s$ (see Sect. 1.2.1.1) and refer further and more general results to Jiang (1996, 1997a).

First introduce some notation for a general linear mixed model (not necessarily with balanced data). The model is called un-confounded if (i) the fixed effects are not confounded with the random effects and errors, that is, $\operatorname{rank}\left(X, Z_r\right)>$ $p, \forall r$ and $X \neq I$, and (ii) the random effects and errors are not confounded, that is, the matrices $I$ and $Z_r Z_r^{\prime}, 1 \leq r \leq s$ are linearly independent (e.g., Miller 1977). The model is called non-degenerate if $\operatorname{var}\left(\alpha_{r 1}^2\right), 0 \leq r \leq s$ are bounded away from zero, where $\alpha_{r 1}$ is the first component of $\alpha_r$. Note that if $\operatorname{var}\left(\alpha_{r 1}^2\right)=0, \alpha_{r 1}=-c$ or $c$ with probability one for some constant $c$. A sequence of estimators $\hat{\lambda}n, \hat{\gamma}{1, n}, \ldots, \hat{\gamma}{s, n}$ is asymptotically normal if there are sequences of positive constants $p{r, n} \rightarrow \infty, 0 \leq r \leq s$ and a sequence of matrices $\mathcal{M}n$ such that $\lim \sup \left(\left|\mathcal{M}_n^{-1}\right| \vee\left|\mathcal{M}_n\right|\right)<\infty$ and $$\mathcal{M}_n\left(p{0, n}\left(\hat{\lambda}n-\lambda\right), p{1, n}\left(\hat{\gamma}{1, n}-\gamma_1\right), \ldots, p{s, n}\left(\hat{\gamma}{s, n}-\gamma_s\right)\right)^{\prime} \stackrel{\mathcal{D}}{\longrightarrow} N\left(0, I{s+1}\right)$$

统计代写|广义线性模型代写Generalized linear model代考|Truncated Estimator

For non-Gaussian linear mixed models, the REML estimator is defined as the solution to the (Gaussian) REML equations, if the solution lies within the parameter space. If the solution is out of the parameter space, it is customary to truncate the solution at the boundary of the parameter space. For example, for ANOVA models, let $\dot{\theta}=\left(\dot{\tau}^2, \dot{\sigma}_1^2, \ldots, \dot{\sigma}_s^2\right)^{\prime}$ be the solution to the REML equation. Suppose that $\dot{\tau}^2>0, \dot{\sigma}_1^2<0$, and $\dot{\sigma}_i^2 \geq 0,2 \leq i \leq s$. Then, the truncated REML estimator is $\left(\dot{\tau}^2, 0, \dot{\sigma}_2^2, \ldots, \dot{\sigma}_s^2\right)^{\prime}$

Again, we focus on REML estimation. Similar results for ML can be found in Jiang (2005a). This case is relatively simpler (compared to ML) because only estimation of the variance components is involved. Furthermore, as shown below, the QUIM in this case does not involve the third moments of the random effects and errors.

Under the ANOVA model with normality, we have (1.18), which can be further expressed as $\partial l_{\mathrm{R}} / \partial \theta_r=u^{\prime} B_r u-b_r, 0 \leq r \leq s$, where $\theta_0=\lambda, \theta_r=\gamma_r, 1 \leq r \leq s$; $u=y-X \beta ; B_0=(2 \lambda)^{-1} P, B_r=(\lambda / 2) P Z_r Z_r^{\prime} P, b_0=(n-p) / 2 \lambda$, and $b_r=(\lambda / 2) \operatorname{tr}\left(P Z_r Z_r^{\prime}\right), 1 \leq r \leq s$. Note that $b_r=\mathrm{E}\left(u^{\prime} B_r u\right), 0 \leq r \leq s$.

Let $u_i=y_i-x_i^{\prime} \beta$ be the $i$ th component of $u$, where $x_i^{\prime}$ is the $i$ th row of $X$. The kurtoses of the random effects and errors are defined as $\kappa_t=\mathrm{E}\left(\alpha_{t 1}^4\right)-3 \sigma_t^4=$ $\mathrm{E}\left(\alpha_{t 1}^4\right)-3\left(\lambda \gamma_t\right)^2, 0 \leq t \leq s$, where $\alpha_0=\epsilon$ and $\gamma_0=1$. Also, with a slight abuse of the notation, let $z_{i t}^{\prime}$ and $z_{t l}$ be the $i$ th row and $l$ th column of $Z_t$, respectively, $0 \leq t \leq s$, where $Z_0=I$. Define $\Gamma\left(i_1, i_2\right)=\sum_{t=0}^s \gamma_t\left(z_{i_1 t} \cdot z_{i_2 t}\right)$. Here, the dot product of vectors $a_1, \ldots, a_k$ of the same dimension is defined as $a_1 \cdot a_2 \cdots a_k=$ $\sum_l a_{1 l} a_{2 l} \cdots a_{k l}$, where $a_{r j}$ is the $j$ th component of $a_r, 1 \leq r \leq k$. Also, let $m_t$ be the dimension of $\alpha_t, 0 \leq t \leq s$ (so that $m_0=n$ ). We begin with an expression for $\operatorname{cov}\left(u_{i_1} u_{i_2}, u_{i_3} u_{i_4}\right)$ as well as one for $\operatorname{cov}\left(\partial l_{\mathrm{R}} / \partial \theta_j, \partial l_{\mathrm{R}} / \partial \theta_k\right)$, the $(j, k)$ element of $\mathcal{I}_1$

统计代写|广义线性模型代写Generalized linear model代 芸|Asymptotic Behavior of ML and REML Estimators in NonGaussian Mixed ANOVA Models

Hartley 和 Rao (1967)、Anderson (1969, 1971a) 和 Miller (1977) 等研究了正态假设下 ML 估计 量的渐近特性。Das (1979) 和 Cressie 和 Lahiri (1993) 在正态性假设下以及 Richardson 和 Welsh (1994) 和 Jiang $(1996,1997 a)$ 在没有正态性假设的情况下研究了 REML 估计量的渐近行 为。除了 Jiang $(1996,1997 a)$ 之外的所有人都认为 $p$ 矩阵的 $X$ 是固定的或有界的，这被证明是一个 关键的假设。这是因为，在这样的假设下， $M L$ 和 REML 估计量是渐近等价的，因为它们与相同的渐 近协方差矩阵一致且渐近正态。另一方面，较早的示例，包括著名的 Neyman-Scott 问题 (Neyman 和 Scott 1948；参见示例 1.7) ，在固定效应的数量随样本大小增吅的情况下显示 REML 明显优于 ML。换句话说，要揭示 REML 的真正优势，必须研究固定效应的数量随样本大小而 增长的情况。

$$\mathcal{M}_n\left(p 0, n(\hat{\lambda} n-\lambda), p 1, n\left(\hat{\gamma} 1, n-\gamma_1\right), \ldots, p s, n\left(\hat{\gamma} s, n-\gamma_s\right)\right)^{\prime} \stackrel{\mathcal{D}}{\longrightarrow} N(0, I s+1)$$

MATLAB代写

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