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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 商科代写|计量经济学代写Econometrics代考|B-Spline Calibration Estimator

The calibration approach (Deville and Särndal 1992) is a method widely used in national statistical agencies. It consists of finding new weights $\left(w_{k s}^{\mathrm{cal}}\right){k \in s}$ that are as close as possible to the sampling weights $\left(d{k}\right){k \in s}$ and such that $\sum{k \in s} w_{k s}^{\mathrm{cal}} \mathbf{x}{k}$ perfectly estimates the known population total of auxiliary information $\sum{k \in U} \mathbf{x}{k}$. The calibrated estimator $\sum{k \in s} w_{k s}^{\text {cal }} y_{k}$ is highly efficient for estimating $t_{y}$ if the relationship $f$ between $y$ and $x$ is close to a linear relationship but its efficiency may be worse than the HT estimator if $f$ is nonlinear. In order to overcome this issue, Goga and Ruiz-Gazen (2019) suggest the $B$-spline calibration: they suggest finding the calibration weights $\left(w_{k s}^{\mathrm{cal}}\right){k \in s}$ that minimize a distance measure $\Upsilon{s}$ to the sampling weights $\left(d_{k}\right){k \in s}$ and subject to the following calibration constraints: $$\sum{k \in s} w_{k s}^{\mathrm{cal}} \mathbf{b}\left(x_{k}\right)=\sum_{k \in U} \mathbf{b}\left(x_{k}\right)$$
Constraints are now on the $B$-splines function values $\left{B_{j}\left(x_{k}\right)\right}_{j=1}^{q}$ and not directly on $x_{k}$ as it is the case in the classical calibration as suggested by Deville and Särndal (1992), we need for that to know $x_{k}$ for all $k \in U$. However, polynomials $x^{\ell}$ belong to the space spanned by $\left{B_{j}(\cdot)\right}_{j=1}^{q}$ for all $\ell=0, \ldots, q-1$. As a result, weights satisfying (21) will also satisfy

$\sum_{k \in s} w_{k s}^{\mathrm{cal}}=N, \quad \sum_{k \in s} w_{k s}^{\mathrm{cal}} x_{k}=\sum_{k \in U} x_{k}$
$\sum_{k \in s} w_{k s}^{\mathrm{cal}} x_{k}^{\ell}=\sum_{k \in U} x_{k}^{\ell}, \quad \ell=2, \ldots, q-1$

## 商科代写|计量经济学代写Econometrics代考|B-Spline Model-Assisted Estimator for Complex Parameters

The estimation of nonlinear parameters $\Phi$ in finite populations has become a crucial problem in many recent surveys. For example, in the European Statistics on Income and Living Conditions (EU-SILC) survey, several indicators for studying social inequalities and poverty are considered; these include the Gini index, the atrisk-of-poverty rate, the quintile share ratio and the low-income proportion. Thus, $\mathrm{~ d e ̉ r i v i n g ~ e ̀ s t i m a a t o r s ~ a a n d ~ c o n n f i d e n c e}$
Consider now a parameter $\Phi$ which is more complicated than a total or a mean. Broadly speaking, linearization techniques consist in obtaining an expansion of an estimator $\widehat{\Phi}$ of $\Phi$ as follows:
$$\widehat{\Phi}-\Phi \simeq \sum_{k \in s} d_{k} u_{k}-\sum_{k \in U} u_{k}=\hat{t}{u d}-t{u},$$
where $u_{k}$ is a kind of artificial variable called the linearized variable of $\Phi$ by Deville (1999b). The way it is derived depends on the type of linearization method used which could include Taylor series (Särndal et al. 1992), estimating equations (Binder 1983) or influence function (Deville $1999 \mathrm{~b}$ ) approaches. The right hand-side of $(22)$ is the difference between the HT estimator and the corresponding population total of the variable $u_{k}$ over the population $U$. Consequently, the variance of the right hand-side is easily obtained and given by
$$\sum_{k \in U} \sum_{l \in U}\left(\pi_{k l}-\pi_{k} \pi_{l}\right) d_{k} d_{l} u_{k} u_{l}$$
We can see from above that we will achieve a small approximate variance and gond precision for $\widehat{\Phi}$ if we estimate $t_{u}=\sum_{k \in U} u_{k}$ in an efficient way, namely the variance given in (23) is small. However, linearized variables may have complicated mathematical expressions and it is not obvious how to improve efficiently the estimation of $t_{u}$. In particular, fitting a linear model onto a linearized variable may not be the most appropriate choice.

## 商科代写|计量经济学代写Econometrics代考|B-Spline Imputation for Handling Item Nonresponse

The theory presented in the above sections supposed that all the sampled individuals respond, so we have full sample data $\left{y_{k}\right}_{k \in s}$. In practice however, due to various reasons, some individuals do not respond to the survey questionnaire (unit nonresponse) or respond only partially (item nonresponse). Unit nonresponse is treated by weighting methods while item nonresponse is treated by imputation. We focus here on item nonresponse and the estimation of finite population total $t_{y}$.

Let $s_{r}$ be the subset of the original sample $s$ containing the individuals that responded to item $y$ and $s_{m}=s-s_{r}$, the subset of $s$ containing the nonrespondents. To estimate $t_{y}$, we use an imputed estimator $\hat{t}{I}$ which is obtained from the HT estimator given in (1) by replacing or imputing the missing values $y{k}, k \in s_{m}$ by values $\hat{y}{k}$ $$\hat{t}{I}=\sum_{k \in s_{r}} d_{k} y_{k}+\sum_{k \in s_{m}} d_{k} \hat{y}{k}$$ The imputed valnes are ohtained by fitting an imputation model. It is usually assumed that the response mechanism is MAR (missing at random), namely the distribution of $\mathcal{Y}$ is the same within respondents and nonrespondents given fully observed covariates. Under the MAR assumption and provided that the auxiliary information $x{k}$ is available for all $k \in s$, the respondent data $\left{\left(y_{k}, x_{k}\right)\right}_{k \in r}$ may be used to build imputation models and to predict $y_{k}$ for the nonrespondents. Goga et al. (2020) suggested a $B$-spline imputation procedure. We consider the model (2) as the imputation model and we estimate $f$ by $B$-splines from the respondent data, the imputed value $\hat{y}_{k}$ is given by $$\hat{y}{k}=\mathbf{b}^{T} \hat{\boldsymbol{\theta}}{r}, \quad k \in s_{m}$$
where $\hat{\boldsymbol{\theta}}{r}=\left(\sum{k \in s_{r}} d_{k} \mathbf{b}\left(x_{k}\right) \mathbf{b}^{T}\left(x_{k}\right)\right)^{-1} \sum_{k \in s_{r}} d_{k} \mathbf{b}\left(x_{k}\right) y_{k}$. Using the same techniques as in Sect. 2, we can show that $\sum_{k \in s_{r}} d_{k}\left(y_{k}-\hat{y}{k}\right)=0$ so the imputed estimator can be also written in a projection form, namely $\hat{t}{I}=\sum_{k \in s} d_{k} \hat{y}{k}$. Under the assumptions described in the Appendix and assuming that the response probabilities are all bounded away from 0 , the imputed estimator is consistent for $t{y}$ but with a consistency rate which is slower than in the full response case. The imputed estimator can be written as a weighted sum of $y_{k}$ ‘s values with weights not depending on $y$, so the approach supposed by Beaumont and Bissonnette (2011) can be used to compute and estimate the variance of $\hat{t}{I}$. Goga et al. (2020) also suggest random $B$-spline imputation which consists in replacing the missing $y{k}$ by
$$\hat{y}{k}=\hat{f}\left(x{k}\right)+\epsilon_{k}^{*}, \quad k \in s_{m}$$

## 商科代写|计量经济学代写Econometrics代考|B-Spline Calibration Estimator

$$\sum k \in s w_{k s}^{\mathrm{cal}} \mathbf{b}\left(x_{k}\right)=\sum_{k \in U} \mathbf{b}\left(x_{k}\right)$$

\begin{aligned} &\sum_{k \in s} w_{k s}^{\mathrm{cal}}=N, \quad \sum_{k \in s} w_{k s}^{\mathrm{cal}} x_{k}=\sum_{k \in U} x_{k} \ &\sum_{k \in s} w_{k s}^{\mathrm{cal}} x_{k}^{\ell}=\sum_{k \in U} x_{k}^{\ell}, \quad \ell=2, \ldots, q-1 \end{aligned}

## 商科代写|计量经济学代写Econometrics代考|B-Spline Model-Assisted Estimator for Complex Parameters

$$\widehat{\Phi}-\Phi \simeq \sum_{k \in s} d_{k} u_{k}-\sum_{k \in U} u_{k}=\hat{t} u d-t u$$

$$\sum_{k \in U} \sum_{l \in U}\left(\pi_{k l}-\pi_{k} \pi_{l}\right) d_{k} d_{l} u_{k} u_{l}$$

## 商科代写|计量经济学代写Econometrics代考|B-Spline Imputation for Handling Item Nonresponse

$$\hat{t} I=\sum_{k \in s_{r}} d_{k} y_{k}+\sum_{k \in s_{m}} d_{k} \hat{y} k$$

$$\hat{y} k=\hat{f}(x k)+\epsilon_{k}^{*}, \quad k \in s_{m}$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师