如果你也在 怎样代写生存模型Survival Models这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。生存模型Survival Models在许多可用于分析事件时间数据的模型中，有4个是最突出的:Kaplan Meier模型、指数模型、Weibull模型和Cox比例风险模型。

生存模型Survival Models精算师和其他应用数学家使用预测人类或其他实体(有生命或无生命)生存模式的模型，并经常使用这些模型作为相当重要的财务计算的基础。具体来说，精算师使用这些模型来计算与个人人寿保险单、养老金计划和收入损失保险相关的财务价值。人口统计学家和其他社会科学家使用生存模型对该模型适用的人口的未来构成做出预测。

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统计代写|生存模型代写survival model代考|PROGRESSION OF A DISEASE

In Panjer’s model, a total of six states are defined. A healthy person not infected by the AIDS virus is said to be in State la, and a person infected by the virus (i.e., testing HIV+) but having no symptoms of the disease is in State $1 \mathrm{~b}$. The next three states, denoted $2 \mathrm{a}, 2 \mathrm{~b}$, and 3 , reflect progressive stages of the disease, and the final state is that of death. Without concerning ourselves with medical meanings, we identify the three stages of the disease as follows:
State 2a: Lymphadenopathy Syndrome (LAS)
State 2b: AIDS-Related Compex (ARC)
State 3: Acquired Immune Deficiency Syndrome (AIDS)
This classification system is consistent with the Walter Reed Staging Method (see Redfield, et al. [62]).

It is assumed that persons progress through the successive stages of the disease without returning to an earlier stage. For simplicity it is also assumed that death prior to State 3, although certainly possible, is a risk small enough to be ignored. This means that a person in any state can leave that state only by progressing to the next state in the model (i.e., a single-decrement environment), as illustrated in Figure 10.3.

Since the only decrement in a given state is progressing to the next state, then the model is completely defined by the hazard rates, which we shall call forces of progression, operating in each state. A simplifying assumption is that these forces of progression, denoted $\mu_j$ for $j=1 a, 1 b, 2 a$, $2 b, 3$, are constant within each state, not depending on the age or sex of the person nor the length of time the person has been in that particular state.

统计代写|生存模型代写survival model代考|Properties of the Model

Since the force of progression is constant, then the random variable for length of time spent in State $j$, denoted $T_j$, is exponentially distributed, with
$$
\begin{gathered}
S_{T_j}(t)=\operatorname{Pr}\left(T_j>t\right)=e^{-t \cdot \mu_j}, \
F_{T_j}(t)=\operatorname{Pr}\left(T_j \leq t\right)=1-e^{-t \cdot \mu_j}, \
f_{T_j}(t)=\frac{d}{d t} F_{T_j}(t)=\mu_j \cdot e^{-t \cdot \mu_j},
\end{gathered}
$$

$$
E\left[T_j\right]=\frac{1}{\mu_j},
$$
and
$$
\operatorname{Var}\left(T_j\right)=\frac{1}{\mu_j^2} .
$$
Recall the “memoryless” property of the exponential distribution, which says that $T_j$ is the random variable for future time spent in State $j$, whether measured from the time of entry into State $j$ or any later time while in State $j$. Furthermore, the random variable $T_j$ is independent of the random variables associated with the other states, so that progression to the next state does not depend on how long the person was in a previous state.

The model assumes that persons leave State 3 (AIDS) only through death, so $\mu_3$ is actually the force of mortality for persons with AIDS, and $T_3$ is the future lifetime random variable for such persons. Then the random variable
$$
R=T_{2 b}+T_3
$$
denotes the future lifetime of a person in State $2 b$ (ARC), with expected value
$$
E[R]=E\left[T_{2 b}\right]+E\left[T_3\right]=\frac{1}{\mu_{2 b}}+\frac{1}{\mu_3}
$$
and
$$
\operatorname{Var}(R)=\operatorname{Var}\left(T_{2 b}\right)+\operatorname{Var}\left(T_3\right)=\frac{1}{\mu_{2 b}^2}+\frac{1}{\mu_3^2},
$$

生存模型代考

统计代写|生存模型代写survival model代考|PROGRESSION OF A DISEASE

在Panjer的模型中，总共定义了六种状态。未感染艾滋病病毒的健康人被称为州la，感染病毒(即检测艾滋病毒阳性)但没有疾病症状的人被称为州$1 \mathrm{~b}$。接下来的三个状态，表示$2 \mathrm{a}, 2 \mathrm{~b}$和3，反映了疾病的进展阶段，最后的状态是死亡。在不考虑医学意义的情况下，我们将这种疾病分为以下三个阶段:
状态2a:淋巴结病综合征(LAS)
州2b:艾滋病相关复合物(ARC)
状态3:获得性免疫缺陷综合症(艾滋病)
该分类系统与Walter Reed分期法一致(见Redfield等人[62])。

假定人在疾病的连续阶段取得进展，而不返回到较早的阶段。为简单起见，我们还假设在状态3之前死亡，尽管这是可能的，但风险小到可以忽略不计。这意味着处于任何状态的人只能通过进入模型中的下一个状态(即，单减量环境)才能离开该状态，如图10.3所示。

由于给定状态下的唯一减量是向下一个状态前进，因此模型完全由在每个状态下运行的危险率定义，我们将其称为前进力。一个简化的假设是，这些进步的力量，表示为$\mu_j$ ($j=1 a, 1 b, 2 a$, $2 b, 3$)，在每个状态中都是恒定的，不取决于人的年龄或性别，也不取决于人在该特定状态中的时间长短。

统计代写|生存模型代写survival model代考|Properties of the Model

由于前进的力是恒定的，那么在状态$j$中花费的时间长度的随机变量，记为$T_j$，呈指数分布，为
$$
\begin{gathered}
S_{T_j}(t)=\operatorname{Pr}\left(T_j>t\right)=e^{-t \cdot \mu_j}, \
F_{T_j}(t)=\operatorname{Pr}\left(T_j \leq t\right)=1-e^{-t \cdot \mu_j}, \
f_{T_j}(t)=\frac{d}{d t} F_{T_j}(t)=\mu_j \cdot e^{-t \cdot \mu_j},
\end{gathered}
$$

$$
E\left[T_j\right]=\frac{1}{\mu_j},
$$
和
$$
\operatorname{Var}\left(T_j\right)=\frac{1}{\mu_j^2} .
$$
回想一下指数分布的“无记忆”属性，即$T_j$是未来在状态$j$中花费的时间的随机变量，无论是从进入状态$j$的时间还是在状态$j$的任何以后的时间测量。此外，随机变量$T_j$独立于与其他状态相关的随机变量，因此，到下一个状态的进展并不取决于该人在前一个状态中停留了多长时间。

该模型假设人们只有通过死亡才能离开状态3(艾滋病)，因此$\mu_3$实际上是艾滋病患者的死亡率，$T_3$是这些人未来寿命的随机变量。然后随机变量
$$
R=T_{2 b}+T_3
$$
表示在$2 b$ (ARC)州的人的未来寿命，具有期望值
$$
E[R]=E\left[T_{2 b}\right]+E\left[T_3\right]=\frac{1}{\mu_{2 b}}+\frac{1}{\mu_3}
$$
和
$$
\operatorname{Var}(R)=\operatorname{Var}\left(T_{2 b}\right)+\operatorname{Var}\left(T_3\right)=\frac{1}{\mu_{2 b}^2}+\frac{1}{\mu_3^2},
$$

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