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# 统计代写|生存模型代考Survival Models代写|Special Case $C$ with Random Censoring

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## 统计代写|生存模型代考Survival Models代写|Special Case $C$ with Random Censoring

As an alternative to recording, and using, exact or average values of $s_i$, for the $c_x$ persons with $0<s_i<1$, it is sometimes assumed that the values of $s_i$ are randomly distributed over $(x, x+1]$. Then scheduled ending time is a random variable, $S$, and we let $g(s)$ denote its PDF. In survival data analysis, where termination of observation due to the ending of the observation period is called censoring, this structure is referred to as a random censoring mechanism.

To construct the likelihood for Special Case $C$ with partial data, we need the probability that a person in $c_x$ will die before the scheduled ending age. If a precise scheduled ending age is not used, and a random distribution of scheduled ending ages is assumed instead, then we must first calculate the marginal probability of death before scheduled ending age. If we let $\bar{q}_x$ denote this marginal probability, then
$$\bar{q}_x=1-\bar{p}_x=1-\int_0^1 g(s) \cdot{ }_s p_x d s .$$
The integral in (7.47) shows that the marginal probability $\bar{p}_x$ is a weighted average value of ${ }_s p_x$, where the sum of the weights is $\int_0^1 g(s) d s=1$.
To evaluate (7.47), we must make a distribution assumption with respect to mortality, and specify the distribution of $S$ as well. If we make the linear assumption for mortality, and the uniform distribution for $S$ (so that $g(s)=1)$, then $(7.47)$ evaluates to
$$\bar{q}_x=\frac{1}{2} \cdot q_x$$
The Special Case $\mathrm{C}$ likelihood is
$$L=\left(q_x\right)^{d^{\prime \prime}} \cdot\left(1-q_x\right)^{n-c-d^{\prime \prime}} \cdot\left(\bar{q}_x\right)^{d^{\prime}} \cdot\left(1-\bar{q}_x\right)^{c-d^{\prime}}$$
under random censoring, as opposed to (7.35) for the likelihood when an average (fixed) value of $s$ is assumed. Of course if (7.48) is used for $\bar{q}_x$, then (7.49) becomes the same as (7.35) evaluated under the uniform assumption with $s=\frac{1}{2}$, and the resulting MLE is given by Estimator (7.32) with $s=\frac{1}{2}$.
The evaluation of (7.47) assuming other mortality distributions, and the resulting MLE’s, is pursued in the exercises.

## 统计代写|生存模型代考Survival Models代写|Summary of Single-Decrement MLE ‘s

We have derived a general, full data, exponential distribution MLE, given by Estimator (7.23), which, of course, is applicable to Special Cases A, B, and $C$ as well.
Similarly, Estimator (7.27) is the general, full data, uniform distribution MLE, with Special Cases A, B, and C versions given by (7.7), (7.30) and (7.32), respectively.
For partial data, we have developed results for Special Cases A and $\mathrm{C}$ only, under each of the exponential and uniform distributions. The partial data MLE’s for Special Case B are derived analogously to those for Special Case $C$, and are left to the exercises. The general case, partial data, MLE’s are considerably more complex, and are not pursued in this text.

If both death and withdrawal are random events operating in the estimation interval $(x, x+1]$, then we seek to estimate $q_x=q_x^{\prime(d)}$ within a doubledecrement environment. The basic mathematics and notation for doubledecrement theory was presented in Section 5.3, and employed in Section 6.3 in connection with moment estimation. Now we will explore maximum likelihood estimation in a double-decrement environment.

As before, let $x+r_i$ be the age at which person $i$ enters $(x, x+1], 0 \leq r_i<1$, and let $n_x$ be the total number of persons in the sample. Let $x+t_i$ be the age at which person $i$ leaves $(x, x+1], 0<t_i \leq 1$, whether as an interval survivor $\left(t_i=1\right)$, as an observed ender $\left(t_i<1\right)$, or as a result of one of the random events death or withdrawal.

Thus each person is under observation from age $x+r_i$ to age $x+t_i$, and the probability of this is $t_i-r_i p_{x+r_i}^{(\tau)}$. For interval survivors and enders, this “total survival” (i.e., neither dying nor withdrawing) is the contribution to the likelihood. For each death and withdrawal, the respective density functions, given by $(5.11 \mathrm{a})$ and $(5.11 \mathrm{~b})$, are needed, so $t_{t,-r_i} p_{x+r_i}^{(\tau)}$ must be multiplied by the appropriate force. Thus the overall likelihood is
\begin{aligned} L & =\prod_{i=1}^n t_t-r_i p_{x+r_i}^{(\tau)} \cdot \prod_{\mathcal{D}} \mu_{x+t_i}^{(d)} \cdot \prod_W \mu_{x+t_i}^{(w)} \ & =\prod_{i=1}^n \frac{t_i-r_i}{} p_{x+r_i}^{\prime(d)} \cdot \prod_{i=1}^n t_t-r_i p_{x+r_i}^{\prime(w)} \cdot \prod_{\mathcal{D}} \mu_{x+t_i}^{(d)} \cdot \prod_W \mu_{x+t_i \cdot}^{(w)} \end{aligned}

# 生存模型代考

## 统计代写|生存模型代考Survival Models代写|Special Case $C$ with Random Censoring

$$\bar{q}_x=1-\bar{p}_x=1-\int_0^1 g(s) \cdot{ }_s p_x d s .$$

$$\bar{q}_x=\frac{1}{2} \cdot q_x$$

$$L=\left(q_x\right)^{d^{\prime \prime}} \cdot\left(1-q_x\right)^{n-c-d^{\prime \prime}} \cdot\left(\bar{q}_x\right)^{d^{\prime}} \cdot\left(1-\bar{q}_x\right)^{c-d^{\prime}}$$

(7.47)假设其他死亡率分布，以及由此产生的MLE，是在练习中进行评价的。

## 统计代写|生存模型代考Survival Models代写|Summary of Single-Decrement MLE ‘s

\begin{aligned} L & =\prod_{i=1}^n t_t-r_i p_{x+r_i}^{(\tau)} \cdot \prod_{\mathcal{D}} \mu_{x+t_i}^{(d)} \cdot \prod_W \mu_{x+t_i}^{(w)} \ & =\prod_{i=1}^n \frac{t_i-r_i}{} p_{x+r_i}^{\prime(d)} \cdot \prod_{i=1}^n t_t-r_i p_{x+r_i}^{\prime(w)} \cdot \prod_{\mathcal{D}} \mu_{x+t_i}^{(d)} \cdot \prod_W \mu_{x+t_i \cdot}^{(w)} \end{aligned}

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## MATLAB代写

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