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# 统计代写|生存模型代考Survival Models代写|Anniversary-to-Anniversary Studies

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## 统计代写|生存模型代考Survival Models代写|Anniversary-to-Anniversary Studies

When insuring ages are used, a natural choice for the observation period is one that opens on the policy anniversary in a designated year for each insured person involved in the study. Similarly, the observation period would close on the policy anniversary in a later year. For example, an observation period might be defined as running from policy anniversaries in 1994 to those in 1998. Note that each person involved in the study has his or her own observation period.

Observation periods that run from a fixed date to a later fixed date can be used with insuring ages, but there are definite advantages to the anniversary-to-anniversary study when insuring ages are used. In this text we emphasize anniversary-to-anniversary studies with insuring ages.

The major convenient consequence of anniversary-to-anniversary studies with insuring ages is that all persons enter the study at an integral age $y_i$. This is true whether the person enters the study by joining the group via policy issue during the O.P. (at an integral insuring age), or by already being in the group when the O.P. opens. In the latter case, entry is at a policy anniversary which is always the attainment of an integral (insuring) age. Since $y_i$ is an integer, it follows that $r_i=0$ for any estimation interval $(x, x+1)$, since $x$ is integral.

Similarly, with the O.P. ending on a policy anniversary, all scheduled ending ages $z_i$ are integers, from which it follows that $s_i=1$ for all estimation intervals $(x, x+1]$. Hence all vectors $\boldsymbol{u}_{i, x}$ are of the convenient form $\left[0,1, \iota_i, \kappa_i\right]$, and all anniversary-to-anniversary insuring age studies belong to our Special Case A.

If there are no withdrawals, then the estimate $\hat{q}x$ is found from (6.7) simply by counting the number of vectors $\mathbf{u}{i, x}$ with $r_i=0$, which gives $n_x$, and the number with $\iota_i \neq 0$, which gives $d_x$, for $(x, x+1]$. If there are withdrawals, the number of them in $(x, x+1], w_x$, is the number of vectors $u_{i, x}$ with $\kappa_i \neq 0$. With $n_x, d_x$ and $w_x$ available, $q_x^{\prime(d)}$ and $q_x^{\prime(w)}$ can be estimated by (6.34) or (6.37). Of course any of the exposure-based estimators could be used as well. Note that since all $s_i=1$, the moment estimator and the actuarial estimator are the same.

## 统计代写|生存模型代考Survival Models代写|Select Studies

For the purpose of estimating $S(t ; x)$, as defined in Section 1.2, we consider only those policies issued at $I A=x$. Again assuming an anniversary-toanniversary observation period, the data processing is quite similar to that described for insuring age studies, with the following parallel features:
(1) The vector $\boldsymbol{v}i$ now represents the policy durations at entry, scheduled exit, death or withdrawal, rather than the insuring ages of the insured at such events. Note that for policies issued during the O.P., the duration at entry to the study is 0 . (2) The vector $\mathbf{u}{i, t}$ represents the location of these events within estimation interval $(t, t+1]$.
(3) Withdrawals are grouped by calendar duration, instead of calendar insuring age. Note that calendar duration is defined by
$$C D=C Y W-C Y I .$$
An example will show the similarity of select studies to insuring age studies.

# 生存模型代考

## 统计代写|生存模型代考Survival Models代写|Select Studies

(1)向量$\boldsymbol{v}i$现在表示进入、计划退出、死亡或退出时的保单期限，而不是被保险人在此类事件中的保险年龄。注意，对于在op期间发布的策略，进入研究的持续时间为0。(2)向量$\mathbf{u}{i, t}$表示这些事件在估计区间$(t, t+1]$内的位置。
(3)提款按日历期限分组，而不是按日历投保年龄分组。注意，日历持续时间由
$$C D=C Y W-C Y I .$$

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