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# 统计代写|广义线性模型代写Generalized linear model代考|STAT7430 Jackknife Method

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## 统计代写|广义线性模型代写Generalized linear model代考|Jackknife Method

Although, at convergence, the I-WLS leads to estimators of $\beta$ and $V$, its main purpose is to produce an efficient estimator for $\beta$. One problem with I-WLS is that it only produces point estimators. A naive estimator of $\Sigma=\operatorname{Var}(\hat{\beta})$, where $\hat{\beta}$ is the I-WLS estimator of $\beta$, may be obtained by replacing $V$ in the covariance matrix of the BLUE (1.36), which is $\left(X^{\prime} V^{-1} X\right)^{-1}$, by $\hat{V}$, an estimator of $V$, say, the limiting I-WLS estimator. However, such an estimator of $\Sigma$ is likely to underestimate the true variation, because it does not take into account the additional variation due to estimation of $V$.

Furthermore, in some cases the covariance structure of the data is specified up to a set of parameters, that is, $V=V(\theta)$, where $\theta$ is a vector of unknown variance components. In such cases the problems of interest may include estimation of both $\beta$ and $\theta$. Note that the I-WLS is developed under a non-parametric covariance structure, and therefore does not apply directly to this case. On the other hand, a similar quasi-likelihood method to that discussed in Sect. 1.4.1 may apply to this case. In particular, the quasi-likelihood is obtained by first assuming that the longitudinal data have a (multivariate) Gaussian distribution. Note that, under the longitudinal model, the observations can be divided into independent blocks (i.e., $y_1, \ldots, y_m$ ). Therefore, asymptotic results for quasi-likelihood estimators with independent observations may apply (see Heyde 1997). The asymptotic covariance matrix of the estimator may be estimated by the POQUIM method of Sect. 1.4.2.
Alternatively, the asymptotic covariance matrix may be estimated by the jackknife method. The jackknife was first introduced by Quenouille (1949) and later developed by Tukey (1958). It has been used in estimating the bias and variation of estimators, mostly in the i.i.d. case. See Shao and Tu (1995). In the case of correlated observations with general M-estimators of parameters, the method was developed in the context of small area estimation (see Jiang et al. 2002). One advantage of the method is that it applies in the same way to different estimating procedures, including I-WLS and quasi-likelihood, and to generalized linear mixed models as well (see Sect. 3.6.2). We describe such a method below, but keep in mind that the method is not restricted to linear models. On the other hand, it is necessary that the data can be divided into independent groups or clusters.

## 统计代写|广义线性模型代写Generalized linear model代考|High-Dimensional Misspeciﬁﬁed Mixed Model Analysis

Recall the GWAS example of Sect. 1.1.2. Statistically, the heritability estimation based on the GWAS data can be casted as a problem of variance component estimation in high-dimensional regression, where the response vector is the phenotypic values and the design matrix is the standardized genotype matrix (to be detailed below). One needs to estimate the residual variance and the variance that can be attributed to all of the variables in the design matrix. In a typical GWAS dataset, although there may be many weak-effect SNPs (e.g., $\sim 10^3$ ) that are associated with the phenotype, they are still only a small portion of the total number SNPs (e.g., $10^5 \sim 10^6$ ). In other words, using a statistical term, the true underlying model is sparse. However, the LMM-based approach used by Yang et al. (2010) assumes that the effects of all the SNPs are nonzero. It follows that the assumed LMM is misspecified.
Consider a mixed ANOVA model that can be expressed as
$$y=X \beta+\tilde{Z} \alpha+\epsilon,$$
where $y$ is an $n \times 1$ vector of observations; $X$ is a $n \times q$ matrix of known covariates; $\beta$ is a $q \times 1$ vector of unknown regression coefficients (the fixed effects); and $\tilde{Z}=p^{-1 / 2} Z$, where $Z$ is an $n \times p$ matrix whose entries are random variables. Furthermore, $\alpha$ is a $p \times 1$ vector of random effects that is distributed as $N\left(0, \sigma^2 I_p\right)$, and $\epsilon$ is an $n \times 1$ vector of errors that is distributed as $N\left(0, \tau^2 I_n\right)$, and $\alpha, \epsilon$, and $Z$ are independent. There are a couple of notable differences here from the previous sections. The first is in terms of notation: Here, $p$ represents the total number of random effects, rather than that of the fixed effects; however, the number of random effects that are nonzero, $m$, is usually much smaller than $p$ (the notation $p$ is chosen in view of the notion of “large $\mathrm{p}$ small $\mathrm{n}$ ” problems in high-dimensional data analysis). The second difference is that the design matrix $Z$ is not only random but also high-dimensional: In GWAS, $p$ is typically much larger than $n$.

The estimation of $\tau^2$ is among the main interests. Without loss of generality, assume that $X$ is full rank. Another variance component of interest is the heritability, as mentioned earlier (see below for detail).

## 统计代写|广义线性模型代写Generalized linear model代考|High-Dimensional Misspecifified Mixed Model Analysis

$$y=X \beta+\bar{Z} \alpha+\epsilon,$$

## MATLAB代写

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