## 统计代写|生存模型代写survival model代考|MATH97183

statistics-lab™ 为您的留学生涯保驾护航 在代写生存模型survival model方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写生存模型survival model代写方面经验极为丰富，各种代写生存模型survival model相关的作业也就用不着说。

• 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

## 统计代写|生存模型代写survival model代考|Cumulative Distribution and Survivor Functions

Definition: The cumulative distribution function (CDF) of $T$ is
$$F(t)=P(T \leq t)$$
the probability of death by age $t$.
Definition: The survivor (or reliability) function of $T$ is
$$S(t)=\mathrm{P}(T>t)=1-F(t)$$
the probability of surviving beyond age $t$.

$S(t)$ is a valid survivor function iff

1. $S(0)=1, \lim _{t \rightarrow \infty} S(t)=0$
2. $0 \leq S(t) \leq 1, \forall t \in \mathbb{R}^{+}$;
3. Monotonicity: $\forall t_1, t_2 \in \mathbb{R}^{+}, t_1<t_2 \Longrightarrow S\left(t_1\right) \geq S\left(t_2\right)$;
4. $S$ is càdlàg: $\forall t \in \mathbb{R}^{+}$,
(a) $S\left(t^{+}\right) \equiv \lim {u \downarrow t} S(u)=S(t)$ ( $S$ is right-continuous); (b) $S\left(t^{-}\right) \equiv \lim {u \uparrow t} S(u)$ exists. (left limits exist for $S$ ).

Definition: Let $T_x$ be the future lifetime of an individual who has survived to age $x$, for $0 \leq x \leq \omega$, so $T_x \in[0, \omega-x]$. So clearly

• $T_0 \equiv T_1$
• The distribution of $T_x$ is the same as $T-x \mid T>x$, in other words: for all measurable sets $A$,
$$\mathrm{P}\left(T_x \in A\right)=\mathrm{P}(T-x \in A \mid T>x)$$
Definition: The cumulative distribution and survivor functions of $T_x$ are
\begin{aligned} & F_x(t)=P\left(T_x \leq t\right) \ & S_x(t)=P\left(T_x>t\right)=1-F_x(t) \end{aligned}

## 统计代写|生存模型代写survival model代考|Density Function

Definition: The probability density function (PDF) of the random variable $T_x$ is
$$f_x(t)=\frac{d}{d t} F_x(t)=\lim _{h \downarrow 0} \frac{F_x(t+h)-F_x(t)}{h} .$$
Since we are assuming $T_x$ is a continuous random variable, by definition this density exists.

For individuals who have currently survived to age $x, f_x(t)$ is the rate of death $t$ further units of time into the future. That is, for such an individual, the probability of death within the interval $[t, t+h]$ for a small interval width $h$ is approximately $f_x(t) h$.

The hazard function plays a central role in survival analysis. We denote the hazard function (or force of mortality) at age $x, 0 \leq x \leq \omega$, by $h(x)$.
Definition: The hazard function of $T$ is defined as
$$h(x)=\lim _{h \downarrow 0} \frac{\mathrm{P}(T \leq x+h \mid T>x)}{h} .$$
We will always assume this limit exists.

Note
$$h(x)=\lim _{h \downarrow 0} \frac{F_x(h)}{h}=f_x(0) .$$
The interpretation of $h(x)$ is important. It represents the instantaneous death rate for an individual who has survived to time $x$.
Or, approximately, for small $h$
$$\mathrm{P}(T \leq x+\Delta \mid T>x) \approx \Delta h(x)$$
Given an individual has reached age $x$, the probability of death in the next short period of time of length $\Delta$ is roughly proportional to $\Delta$, the constant of proportionality being $h(x)$.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Cumulative Distribution and Survivor Functions

$$F(t)=P(T \leq t)$$

$$S(t)=\mathrm{P}(T>t)=1-F(t)$$

$S(t)$ 是一个有效的幸存者函数当且仅当

1. $S(0)=1, \lim _{t \rightarrow \infty} S(t)=0$
2. $0 \leq S(t) \leq 1, \forall t \in \mathbb{R}^{+}$;
3. 单调性: $\forall t_1, t_2 \in \mathbb{R}^{+}, t_1<t_2 \Longrightarrow S\left(t_1\right) \geq S\left(t_2\right)$;
4. $S$ 和目录: $\forall t \in \mathbb{R}^{+}$，
(一) $S\left(t^{+}\right) \equiv \lim u \downarrow t S(u)=S(t)$ ( $S$ 是右连续的)；（乙) $S\left(t^{-}\right) \equiv \lim u \uparrow t S(u)$ 存 在。 (左极限存在于 $S$ ).
定义: 让 $T_x$ 是一个活到老的人的末来一生 $x$ ，为了 $0 \leq x \leq \omega$ ，所以 $T_x \in[0, \omega-x]$. 如此清晰
$T_0 \equiv T_1$
• 的分布 $T_x$ 是相同的 $T-x \mid T>x$ ，换句话说: 对于所有可测量的集合 $A$,
$$\mathrm{P}\left(T_x \in A\right)=\mathrm{P}(T-x \in A \mid T>x)$$
定义：的侽积分布和幸存者函数 $T_x$ 是
$$F_x(t)=P\left(T_x \leq t\right) \quad S_x(t)=P\left(T_x>t\right)=1-F_x(t)$$

## 统计代写|生存模型代写survival model代考|Density Function

$$f_x(t)=\frac{d}{d t} F_x(t)=\lim {h \downarrow 0} \frac{F_x(t+h)-F_x(t)}{h} .$$ 因为我们假设 $T_x$ 是一个连续的随机变量，根据定义这个密度是存在的。 对于目前活到老年的人 $x, f_x(t)$ 是死亡率 $t$ 末来的更多时间单位。即对于这样的个体，在区间内死亡的概 率 $[t, t+h]$ 对于小间隔宽度 $h$ 大约是 $f_x(t) h$. 风险函数在生存分析中起着核心作用。我们表示年龄的风险函数（或死亡率力) $x, 0 \leq x \leq \omega$ ，经过 $h(x)$. 定义: 的风险函数 $T$ 定义为 $$h(x)=\lim {h \downarrow 0} \frac{\mathrm{P}(T \leq x+h \mid T>x)}{h} .$$

$$h(x)=\lim _{h \downarrow 0} \frac{F_x(h)}{h}=f_x(0)$$

$$\mathrm{P}(T \leq x+\Delta \mid T>x) \approx \Delta h(x)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|生存模型代写survival model代考|MATH96048

statistics-lab™ 为您的留学生涯保驾护航 在代写生存模型survival model方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写生存模型survival model代写方面经验极为丰富，各种代写生存模型survival model相关的作业也就用不着说。

• 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

## 统计代写|生存模型代写survival model代考|Principles of Modelling Time to Event Data

A stochastic or statistical model of a system is a mathematical model which represents inherent uncertainties in the system as random variables. Any model which does not make such allowances for uncertainty, on the other hand, is said to be deterministic. In actuarial science, statistical models are especially useful for dealing with uncertainty in the time until a specific event will occur, such as predicting the number of premium payments that will be received before paying out on a life assurance contract. These statistical models for time to event data also have much wider applications in fields such as medicine, biostatistics and engineering.

Mathematical models are an imperfect representation of reality. Their utility comes from being able to approximately learn the consequences of hypothetically changing certain experimental inputs or actions. Statistical models extend this utility by capturing the uncertainty surrounding unknown future outcomes, providing the possibility of searching for an optimal decision under this uncertainty.

The complexity of a good statistical model is often constrained for several reasons. The analyst might be restricted by the inputs for which data are readily available, or by computational or statistical limitations to the models which can be reliably fitted. Additionally, it is important that the chosen model satisfies the purposes for which it will be used; usually this means that the mathematical structure of the model need be easily interpretable for purposes of understanding and communication, and that the resulting fitted model will not overfit and understate uncertainty and risk. For these reasons, the statistical analyst will often seek to fit the most parsimonious model that the data will sensibly allow.

## 统计代写|生存模型代写survival model代考|Sensitivity

Model choice and fitting should be viewed as an iterative procedure. Once a particular model has been fitted to the available data, the quality of the fit should be inspected; hypothesis tests such as goodness of fit tests can determine the suitability of the selected model. If the structure of the data is not well captured by the model, this may suggest that the model chosen is inadequate and the analyst should return to the model selection stage to consider other alternatives.
Note however that performance of the chosen model will also depend heavily on the quality of the data, with poor quality inputs likely to lead to unreliable output inference (garbage in, garbage out). If there are questions about the accuracy of the data being used, then it becomes particularly important to carry out a sensitivity analysis. This investigates the magnitude of change in the model outputs when small perturbations are applied to the model inputs.

Finally, it should be noted that the suitability of a well fitted model may not be comfortably relied upon outside the range of the data used; for example, an exponential relationship can appear fairly linear in the short term, before ballooning away from such a fit in the longer term.

Consider a homogeneous population of individuals, who each have an associated event time which is initially unknown and treated as a random variable. We will often refer to the period of time until the event occurs as the lifetime of the individual.

Throughout the course, let $T$ denote the future lifetime of a new-born individual (aged 0 ). Unless otherwise stated, we shall assume $T$ is a continuous random variable which takes values on the positive part of the real line $\mathbb{R}^{+}=[0, \infty)$, with associated probability measure $P$.

More specifically, in this chapter we might assume the existence of a limiting age $\omega$ which $T$ cannot exceed, so $T \in[0, \omega]$. For human life calculations, typically we take $\omega \approx 120$ years.

We first define the cumulative distribution function, $F$, the survivor function $S$, the density $f$, the hazard $h$ and the cumulative hazard $H$ for a lifetime, and derive relationships between them.

This will provide us with a range of tools for specifying the distribution of a lifetime, and rules for moving between them.

# 生存模型代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。