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统计代写|生存模型代考Survival Models代写|Partial Data

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统计代写|生存模型代考Survival Models代写|Partial Data

As defined in Section 6.2.2, this situation simply states that, of $n_x$ lives exactly age $x, d_x$ of them die in $(x, x+1]$, and $n_x-d_x$ survive to age $x+1$. We recognize this as a binomial model, so the likelihood is simply the binomial probability of obtaining the sample result actually obtained. That is,
$$L\left(q_x \mid n_x, d_x\right)=\frac{n_{x} !}{d_{x} !\left(n_x-d_x\right) !}\left(q_x\right)^{d_x}\left(1-q_x\right)^{n_x-d_x} .$$
One of the basic properties of MLE is that any multiplicative constants can be ignored, and the same estimate of $q_x$ will still result. When this is done, the likelihood is no longer the probability of the sample per se, but rather is proportional to it. Thus many writers prefer to write
$$L\left(q_x \mid n_x, d_x\right) \propto\left(q_x\right)^{d_x} \cdot\left(1-q_x\right)^{n_x-d_x},$$
where $\alpha$ is read “is proportional to.” We wish to take the point of view that it is just as reasonable to call the right side of (7.2) the likelihood itself, as to call it something to which the likelihood is proportional. Thus we would write simply
$$L\left(q_x \mid n_x, d_x\right)=\left(q_x\right)^{d_x} \cdot\left(1-q_x\right)^{n_x-d_x} .$$
The notation $L\left(q_x \mid n_x, d_x\right)$ reminds us that the likelihood is a function of the unknown $q_x$, and that $n_x$ and $d_x$ are given values, namely those observed in the sample upon which our estimate of $q_x$ is to be based. When there is no doubt as to the unknown and the given values, we will simply use $L$ instead of $L\left(q_x \mid n_x, d_x\right)$. Finally, for convenience we will frequently suppress the subscript $x$. Thus we will write the likelihood for the Special Case A partial data situation as
$$L=q^d \cdot(1-q)^{n-d}$$

统计代写|生存模型代考Survival Models代写|Full Data

Now ive assume that we have the precise age at death for each of the $d_x$ deaths in the interval. Since this age is different for each death, we consider them individually, and take the product of each death’s contribution to the likelihood function.

The likelihood for the $i^{\text {th }}$ death is given by the probability density function (PDF) for death at that particular age, given alive at age $x$. That is, for death at age $x_i$,
$$L_i=f\left(x_i \mid X>x\right)=\frac{f\left(x_i\right)}{S(x)}=\frac{S\left(x_i\right) \cdot \lambda\left(x_i\right)}{S(x)}$$
is the contribution to $L$ of the $i^{t h}$ death. If we let $s_i=x_i-x$ be the time of the $i^{t h}$ death within $(x, x+1]$, where $0<s_i \leq 1$, then we have
$$L_i=\frac{S\left(x+s_i\right) \cdot \lambda\left(x+s_i\right)}{S(x)}=s_{s_i} p_x \mu_{x+s_i}$$
in standard actuarial notation. The contribution to $L$ for all deaths combined is
$$\prod_{i=1}^d s_i p_x \mu_{x+s_i},$$
which is commonly written as $\prod_D s_s p_x \mu_{x+s_1}$, and read as “multiplied over all deaths.”

Of course the $n_x-d_x$ survivors contribute $\left(p_x\right)^{n_x-d_x}=\left(1-q_x\right)^{n_x-d_x}$ to $L$, so we have the total likelihood
$$L=\left(1-q_x\right)^{n_x-d_s} \cdot \prod_{\mathcal{D}} p_i p_x \mu_{x+s_i}$$
for our Special Case A full data situation.
To solve (7.11) for $\hat{q}x$ it is necessary to make a distribution assumption which will express $s_i p_x \mu{x+s_t}$ in terms of $q_x$. We will consider two such assumptions.

生存模型代考

统计代写|生存模型代考Survival Models代写|Partial Data

$$L\left(q_x \mid n_x, d_x\right)=\frac{n_{x} !}{d_{x} !\left(n_x-d_x\right) !}\left(q_x\right)^{d_x}\left(1-q_x\right)^{n_x-d_x} .$$
MLE的一个基本特性是可以忽略任何乘法常数，并且仍然会得到相同的$q_x$估计。当这样做时，可能性不再是样本本身的概率，而是与之成正比。因此，许多作家更喜欢写作
$$L\left(q_x \mid n_x, d_x\right) \propto\left(q_x\right)^{d_x} \cdot\left(1-q_x\right)^{n_x-d_x},$$
$\alpha$的意思是“与…成正比”。我们认为，把式7.2的右边称为似然本身，正如把它称为与似然成正比的某种东西一样，都是合理的。因此，我们可以简单地写
$$L\left(q_x \mid n_x, d_x\right)=\left(q_x\right)^{d_x} \cdot\left(1-q_x\right)^{n_x-d_x} .$$

$$L=q^d \cdot(1-q)^{n-d}$$

统计代写|生存模型代考Survival Models代写|Full Data

$i^{\text {th }}$死亡的可能性由该特定年龄的死亡概率密度函数(PDF)给出，给定其存活年龄为$x$。也就是说，在$x_i$岁时死亡
$$L_i=f\left(x_i \mid X>x\right)=\frac{f\left(x_i\right)}{S(x)}=\frac{S\left(x_i\right) \cdot \lambda\left(x_i\right)}{S(x)}$$

$$L_i=\frac{S\left(x+s_i\right) \cdot \lambda\left(x+s_i\right)}{S(x)}=s_{s_i} p_x \mu_{x+s_i}$$

$$\prod_{i=1}^d s_i p_x \mu_{x+s_i},$$

$$L=\left(1-q_x\right)^{n_x-d_s} \cdot \prod_{\mathcal{D}} p_i p_x \mu_{x+s_i}$$

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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。