统计代写|生存模型代写survival model代考|STAT633
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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|Description of the Model
The CCRC is made up of $m$ independent living units (ILU) and a skilled nursing facility (SNF), to which residents can transfer on either a temporary or permanent basis. Transfer to the SNF is considered temporary if the resident is expected to recover and return to his or her ILU, which will remain vacant during the temporary stay at the SNF. If a return to independent living is not expected, then the transfer to the SNF is considered permanent. Residents may leave the community, either by death or voluntary withdrawal, at any time while residing in their ILU’s or in the SNF (either temporarily or permanently).
Thus we see that there are four states to the model:
State (a): Residence in the ILU.
State (b): Temporary residence in the SNF.
State (c): Permanent residence in the SNF.
State (d): Departed from the CCRC, either by death or voluntary withdrawal.
Persons in State (a) can transfer to any of States (b), (c), or (d). Persons in State (b) can transfer back to State (a), or to States (c) or (d). Persons in State (c) can transfer only to State (d), not back to States (a) or (b). Persons in State (d) are departed from the CCRC, and cannot transfer to any of the other states. The model is illustrated in Figure 10.4.
统计代写|生存模型代写survival model代考|Assumptions and Properties of the Model
It is assumed that there is an adequate waiting list for residence in the CCRC, such that any vacancy occurring in an ILU as a result of transfer to either State (c) or State (d) is immediately filled by a new resident. As stated above, transfer to State (b) keeps the ILU vacant pending transfer back to State (a) upon recovery. However, if the person considered to be temporarily in the SNF (i.e., in State (b)) is reclassified as a permanent resident of the SNF or dies or otherwise withdraws (i.e., transfers to State (c) or State (d)), then the vacancy in that person’s ILU is immediately filled.
This “high demand” assumption has two important consequences The first is that the total number of ILU residents plus temporary SNF residents is $m$ at all times. The second is that the $m$ ILU’s can be assumed to operate independently of each other. This, in turn, allows us to analyze the model by first analyzing a single unit.
As with the Panjer AIDS model of Section 10.2, we assume here that the force of transition from State $(i)$ to State $(j)$, denoted $\mu_{i j}$, is constant. Thus $\mu_{a b}$ denotes the force of transition for a resident in an ILU to make a temporary transfer to the SNF. Similarly, $\mu_{b a}$ denotes the “force of recovery” for a resident in the SNF to return to an active status in his or her ILU.
As explained in Section 10.3.1, and illustrated in Figure 10.4, the non-zero forces of transition for a person in State (a) are $\mu_{a b}, \mu_{a c}$, and $\mu_{a d}$. The non-zero forces for a person in State (b) are $\mu_{b a}, \mu_{b c}$, and $\mu_{b d}$. For a person in State (c), the only non-zero force of transition is $\mu_{c d}$. It should be clear that no forces of transition apply to persons in State (d), since all such persons have departed from the CCRC.
From these observations it is clear that a person in State (a) is facing a triple-decrement environment, so that various probabilities can be determined from the three forces of transition using familiar actuarial concepts and notation. In particular, the total force of transition is given by
$$
\mu_a^*=\mu_{a b}+\mu_{a c}+\mu_{a d} .
$$
生存模型代考
统计代写|生存模型代写survival model代考|Description of the Model
CCRC由$m$独立生活单位(ILU)和一个熟练护理设施(SNF)组成，居民可以临时或永久转移到这些设施。如果居民预计将恢复并返回他或她的ILU，则被认为是临时转移到SNF，该ILU在SNF临时停留期间将保持空缺。如果不期望恢复独立生活，则认为转移到SNF是永久性的。居民在居住在其临时单位或SNF期间(临时或永久)可随时因死亡或自愿撤离离开社区。
因此，我们看到模型有四种状态:
国家(a):在国际劳工组织居住。
状态(b):在国家安全区内的临时住所。
国家(c):在瑞士永久居住。
国家(d):因死亡或自愿退出而退出《儿童权利公约》。
在(a)国的人员可以转移到(b)、(c)或(d)国中的任何国家。在(b)国的人员可以转移回(a)国或(c)国或(d)国。在(c)国的人员只能转移到(d)国，不能转移回(a)或(b)国。在(d)国的人员脱离了CCRC，不能转移到任何其他国家。该模型如图10.4所示。
统计代写|生存模型代写survival model代考|Assumptions and Properties of the Model
假定有足够的等待在中心居住的名单，因此，由于转移到(c)国或(d)国而在国际劳工单位出现的任何空缺都立即由新居民填补。如上所述，转移到状态(b)使ILU空置，等待恢复后转移回状态(a)。但是，如果被认为暂时在SNF(即在(b)国)的人被重新分类为SNF的永久居民，或死亡或以其他方式退出(即转移到(c)国或(d)国)，则该人员的ILU空缺将立即填补。
这种“高需求”的假设有两个重要的结果:第一，ILU居民加上临时SNF居民的总人数在任何时候都是$m$。第二点是$m$逻辑单元可以被假定为相互独立地运行。反过来，这允许我们通过首先分析单个单元来分析模型。
与第10.2节的Panjer艾滋病模型一样，我们假设从状态$(i)$到状态$(j)$的过渡力(表示为$\mu_{i j}$)是恒定的。因此$\mu_{a b}$表示在一个ILU的居民临时转移到SNF的过渡力。类似地，$\mu_{b a}$表示SNF中的居民回到其ILU中活动状态的“恢复力”。
如第10.3.1节所述，如图10.4所示，状态(a)的人的非零过渡力为$\mu_{a b}, \mu_{a c}$和$\mu_{a d}$。处于状态(b)的人的非零力为$\mu_{b a}, \mu_{b c}$和$\mu_{b d}$。对于处于(c)状态的人，惟一的非零过渡力是$\mu_{c d}$。应该清楚的是，过渡力量不适用于(d)国的人员，因为所有这些人都已离开《儿童权利公约》。
从这些观察中可以清楚地看出，处于状态(a)的人面临着三重递减环境，因此可以使用熟悉的精算概念和符号从三种过渡力量中确定各种概率。特别地，总过渡力由
$$
\mu_a^*=\mu_{a b}+\mu_{a c}+\mu_{a d} .
$$
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