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# 统计代写|生存模型代考Survival Models代写|THE DISTRIBUTION OFT

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## 统计代写|生存模型代考Survival Models代写|THE DISTRIBUTION OFT

In Chapter 1 we chose to define and describe a survival model in terms of the function $S(t)$, which represents $\operatorname{Pr}(T>t)$, where $T$ is the failure time random variable. This function of the random variable $T$ is called the Survival Distribution Function (SDF). We recall that it gives the probability that failure (death) will occur after time $t$, which is the same as the probability that the entity, known to exist at time $t=0$, will survive to at least time $t$. We also recall that $S(0)=1$ and $S(\infty)=0$.
The Cumulative Distribution Function
The Cumulative Distribution Function (CDF) of $T$ is $F(t)$. The CDF gives the probability that the random variable will assume a value less than or equal to $t$. That is,
$$F(t)=\operatorname{Pr}(T \leq t)$$
In the special case of our failure time random variable, $F(t)$ gives the probability that failure (death) will occur not later than time $t$. It should be clear that
$$F(t)=1-S(t),$$
and that $F(0)=0$ and $F(\infty)=1$.
In most probability textbooks, the CDF, $F(t)$, is given greater emphasis than is the SDF, $S(t)$. But for our special kind of random variable, $S(t)$ will receive greater attention.

## 统计代写|生存模型代考Survival Models代写|The Probability Density Function

For the special case of a continuous random variable, the Probability Density Function (PDF), $f(t)$, is defined as the derivative of $F(t)$. Thus
$$f(t)=\frac{d}{d t} F(t)=-\frac{d}{d t} S(t), t \geq 0 .$$
Consequently, it is easy to see that
$$F(t)=\int_0^t f(y) d y$$
and
$$S(t)=\int_t^{\infty} f(y) d y .$$
Of course it must be true that
$$\int_0^{\infty} f(y) d y=1$$
Although we have given mathematical definitions of $f(t)$, it will be useful to describe $f(t)$ more fully in the context of the failure time random variable. Whereas $F(t)$ and $S(t)$ are probabilities which relate to certain time intervals, $f(t)$ relates to a point of time, and is not a probability, per se. We prefer to refer to $f(t)$ by its conventional description as “probability density.” It is the density of failure at time $t$, and is an instantaneous measure, as opposed to an interval measure.

It is important to recognize that $f(t)$ is the unconditional density of failure at time $t$. By this we mean that it is the density of failure at time $t$ given only that the entity existed at $t=0$. The significance of this point will become clearer in the next subsection.

# 生存模型代考

## 统计代写|生存模型代考Survival Models代写|Cox’s Proportional Hazards Model

$$h(t ; z)=h_0(t) \exp (\beta \cdot z)$$

$$S(t ; z)=S_0(t)^{\exp (\beta \cdot z)}, \quad f(t ; z)=\exp (\beta \cdot z) S_0(t)^{\exp (\beta \cdot z)-1} f_0(t),$$

## 统计代写|生存模型代考Survival Models代写|Model checking

$$2\left(\ell_{p+q}-\ell_p\right)$$

$$H_0: \beta_{p+1}=\ldots=\beta_{p+q}=0 \text { vs. } H_1:\left(\beta_{p+1}, \ldots, \beta_{p+q}\right) \in \mathbb{R}^q$$

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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。