## 统计代写|生存模型代写survival model代考|STA628

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## 统计代写|生存模型代写survival model代考|Historical Approaches to Measuring Default Rates

The accurate measurement of the default risk is critical in the pricing of debt, measuring bond performance, and assessing bond market efficiency.

Early measurement approaches calculated the default rate in a particular year of a bond’s term by dividing the value of defaults by the original value of the bond issue, rather than by the surviving population of bonds at time of default. (The actuarial reader will recognize the former measure as an unconditional one, analogous to the life table concept denoted by ${ }_n \mid q_0$, and the latter measure as a conditional one, analogous to the life table concept denoted by $q_n$.) The former approach fails to consider that a bond can “die” in ways other than through default, such as by call redemption, sinking fund, or maturity. These annual default rates were then averaged over several years to obtain an average annual rate. For example, for the period 1978-87, the average annual default rate, measured in the traditional way, was $1.86 \%$ per year.

The traditional default rate measurement uses the par (or other) value of the defaulting bond in the numerator of the rate calculation. A more relevant measure for investors is not the rate of default, per se, but rather the rate of investment value actually lost (default loss rate measurement). Suppose an investor purchases a bond at par value. The bond defaults by failing to make a particular coupon payment, and the investor sells the bond immediately after the defaulted coupon date at $40 \%$ of its par value. Then the default loss rate calculation would use a loss of $60 \%$ of par plus one coupon in the numerator, whereas the traditional default rate calculation would use the entire par value. By taking into account that the defaulting bond can be sold, on the average, for about $40 \%$ of par, the 1978-87 average annual default loss rate was about $1.20 \%$ per year, compared to the $1.86 \%$ traditionally measured default rate.

## 统计代写|生存模型代写survival model代考|Altman ‘s BondMortality Rate Concept

As suggested in part (c) of Example 11.7, the “actuarial method” advocated by Altman tracks the survival pattern of a cohort group of bonds over time. In his model, bonds can exit from the original population by either default, calls, sinking funds, or maturities, so that we have a multiple-decrement environment. (To measure default rates, the non-default exits can be combined so that the model becomes double-decrement only.) Then if $D(t)$ denotes the value of defaulting debt in year $t$ and $P(t)$ denotes the value of the surviving population of bonds at the start of year $t$, then
$$M M R_{\ell}=\frac{D(t)}{P(t)}$$
gives the marginal mortality rate for year $t$.
Altman further defines
$$S R_t=1-M M R_t$$
to be the bond survival rate in year $t$, and
$$C M R_t=1-\prod_{j=1}^t S R_j$$
to be the cumulative mortality rate over the interval $(0, t)$.
Finally, Altman considers the mortality/survival experience of various classes of bonds over the seventeen-year sample period of 1971 through 1987. Since the values of $M M R_t$ were calculated at each duration $t$ for each separate issue year, then they must be combined to reach a rate for duration $t$ for the entire study. The combining is done on a value-weighted basis, since an unweighted average could be misleading if new issue amounts varied significantly from year to year. The reader interested in Altman’s numerical results is referred to [2].

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Altman ‘s BondMortality Rate Concept

$$M M R_{\ell}=\frac{D(t)}{P(t)}$$

Altman进一步定义
$$S R_t=1-M M R_t$$

$$C M R_t=1-\prod_{j=1}^t S R_j$$

## 统计代写|生存模型代写survival model代考|ST227

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## 统计代写|生存模型代写survival model代考|Use of Concomitant Variables

Intuitively, the distribution of strike durations may vary according to various characteristics of the labor group involved, such as (1) size of group, (2) type of industry, (3) time of year, (4) disputed issue(s), and (5) current state of the economy. As introduced in Section 8.5 , these concomitant variables can be represented in a parametric model fit to the data. In the case of the strike data, the first four of the influences listed above were taken into account by considering only data for which these variables were constant: $1000^{+}$size group, manufacturing industry, June strikes, and wage disputes.

In this section we expand our parametric model to include a concomitant variable addressing the general economic condition, an index of industrial production. We choose the Cox model given by Equation (8.77), with constant underlying hazard $\lambda=e^{a_0}$, and concomitant variable $z_1$ representing a fixed value of the index of industrial production associated with each strike. Then the hazard function is
$$\lambda\left(t_j ; a_0, a_1, z_1\right)=e^{a_0+a_1 z_1},$$
and the survival function is
$$S\left(t_j ; a_0, a_1, z_1\right)=\exp \left[-t_j \cdot e^{a_0+a_1 z_1}\right]$$
so the log-likelihood function, from Equation (11.16b), is
$$\ell\left(a_0, a_1\right)=-\sum_{j=1}^n t_j \cdot e^{a_0+a_1 z_1}+\sum_{j=1}^n \delta_j\left(a_0+a_1 z_1\right) .$$

## 统计代写|生存模型代写survival model代考|MORTGAGE LOAN PREPAYMENTS

In this section we will briefly explore the survival pattern of amortized loans, specifically those secured by a mortgage on real property. We assume that the actuarial reader is familiar with the mathematics of such loans, as described, for example, in Kellison [44].

When we speak of the survival pattern of an entity, there must be some definition of what it is that constitutes survival and, consequently, what constitutes lack of survival, or failure, of the entity. Consider a conventional mortgage loan of amount $L$, to be repaid over 30 years by 360 level monthly payments. If we were to define “survival” to be the continuation of the regular monthly payments, and “failure” to be the complete repayment of the loan, then the observed survival pattern, measured in months, for a loan that made all regular payments would be $S^{\circ}(t)=1.00$ for $0 \leq t<360$, and $S^{\circ}(360)=0$.

Most mortgage loans, however, do not behave with this pattern of perfect regularity, since there are two significant events that could occur to disturb it, namely default and prepayment. These two risks are discussed in Section 11.3.1. The major use of mortgage loan survival patterns is in conjunction with mortgage-backed securities (MBS), so an introduction to this investment instrument is presented in Section 11.3.2. Finally, in Section 11.3.3 we describe various models of mortgage loan survival in light of the default and prepayment risks that have, or have had, some use in the investment arena.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Use of Concomitant Variables

$$\lambda\left(t_j ; a_0, a_1, z_1\right)=e^{a_0+a_1 z_1},$$

$$S\left(t_j ; a_0, a_1, z_1\right)=\exp \left[-t_j \cdot e^{a_0+a_1 z_1}\right]$$

$$\ell\left(a_0, a_1\right)=-\sum_{j=1}^n t_j \cdot e^{a_0+a_1 z_1}+\sum_{j=1}^n \delta_j\left(a_0+a_1 z_1\right) .$$

## 统计代写|生存模型代写survival model代考|STAT633

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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。

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

$$\mu_a^*=\mu_{a b}+\mu_{a c}+\mu_{a d} .$$

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

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

In the case of group insurance or group pension plans, a large number of individual persons are covered under a single policy or plan. There will exist a key date for such a plan, called the plan anniversary or plan valuation date. On such a date it will be necessary to calculate premium rates, actuarial present values of accrued benefits, or other financial values. For this purpose it will be convenient for all members of the plan to be an integral age on this key date. This integral age is called the fiscal age.

Note that this situation is similar to that under insuring ages, where each individual insured was an integral insuring age on that person’s policy anniversary. Here the same idea holds, with the further condition that the policy anniversary is the same date for all persons. It is traditional to refer to this date as the $\boldsymbol{T}$-date.

Historically the terms T-date and fiscal age were adopted to indicate that the T-date was the terminal date of the fiscal year of an enterprise. In this text we feel that the major application of the fiscal age concept is to studies of mortality under group insurance or pension plans. Thus the T-date is the plan anniversary. The term fiscal age is not particularly descriptive, but we will retain it for the sake of tradition.

## 统计代写|生存模型代写survival model代考|Fiscal Year of Birth

Analogous to the definitions of insuring age and valuation year of birth in Section 9.3, we assign each person in a group plan a fiscal age (FA) as of some particular T-date, say the T-date in calendar year $z$. This fiscal age would likely be the actual age nearest birthday on that date, or it could be the actual age last birthday, or the actual calendar age. Regardless of how it is assigned, we then define the fiscal year of birth (FYB) as
$$F Y B=z-F A .$$
Just as was true for $V Y B$ under insuring ages, $F Y B$ could be the same as the person’s actual $C Y B$, or it could be one year less or one year greater. Once $F Y B$ has been assigned, the T-date in the $F Y B$ is then the hypothetical date of birth for each person in the group plan.

The natural choice of an observation period is one that runs from the T-date in a certain year to the T-date in a later year. Note that the T-date is the anniversary for all members of the group plan, so a T-to-T study is both a dateto-date study and an anniversary-to-anniversary study.

The principal benefit of using a T-to-T observation period is that all members in the plan when the O.P. opens enter the study at an integral age $y_i$. Similarly, all members in the study sample will have an integral scheduled exit age $z_i$. In turn, we know that integral $y_i$ and $z_i$ imply $r_i=0$ and $s_i=1$ for any estimation interval $(x, x+1]$, and a Special Case A estimation problem.

Although dates other than T-dates can be used in date-to-date fiscal age studies, there is no particular advantage to this, and only T-to-T studies will be considered in this chapter.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Fiscal Year of Birth

＄$F Y B=z F A。 ＄$

## 统计代写|生存模型代写survival model代考|Calculation of Exposure

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## 统计代写|生存模型代写survival model代考|Calculation of Exposure

Estimators of this form are particularly easy to apply when the basic data is represented by $\mathbf{u}{i, x}$ for person $i$ within interval $(x, x+1]$. We first note that the number of observed deaths in $(x, x+1]$ is simply the number of $\mathbf{u}{i, x}$ vectors for which $\iota_i \neq 0$. Similarly the number of observed withdrawals in $(x, x+1]$ is the number of $u{i, x}$ vectors for which $\kappa_i \neq 0$.

The exact exposure over $(x, x+1]$ contributed by person $i$ is simply
$$(\text { Exact Exposure }){i, x}=\left[\begin{array}{c} s_i \ \iota_i \ \kappa_i \end{array}\right]-r_i,$$ where $\left[\begin{array}{l}s_i \ \iota_i \ \kappa_i\end{array}\right]$ represents the minimum of $s_i, \iota_i, \kappa_i$ that exceed zero. In other words, if $\iota_i=\kappa_i=0$, so that person $i$ neither dies nor withdraws in $(x, x+1]$, then we have (Exact Exposure) $i{i, x}=s_i-r_i$. But if person $i$ dies in $(x, x+1]$, so that $\iota_i<s_i$ and $\kappa_i=0$, then (Exact Exposure) $i_{i, x}=\iota_i-r_i$. Finally, if person $i$ withdraws in $(x, x+1)$, so that $\kappa_i<s_i$ and $\iota_i=0$, then we have (Exact Exposure $)_{i, x}=\kappa_i-r_i$.

To find the scheduled exposure for estimating the mortality probability $q_x^{\prime(d)}$ under Hoem’s moment approach, exact exposure is still used for those who withdraw and for those who neither die nor withdraw in $(x, x+1]$, but those who die are exposed to age $x+s_i$. Thus we have
$$(\text { Scheduled Exposure })_{i, x}=\left[\begin{array}{c} s_i \ \kappa_i \end{array}\right]-r_i,$$
where $\kappa_i$ is used if person $i$ withdraws in $(x, x+1]$, and $s_i$ is used otherwise.

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

In many cases an average age at event might be substituted for the exact age at event for all persons whose exact age at event falls within a certain age range. For example, consider all persons whose exact $y_i$ ‘s fall between the integers $x$ and $x+1$. We might substitute a common age $y^{\prime}$ for all such $y_i$, frequently using $y^{\prime}=x+\frac{1}{2}$ as the assumed average value of the $y_i$ ‘s. When we do this the entrants to the study have been grouped by age last birthday, since it is those with a common age last birthday $(x)$ that are being considered together.

A second type of grouping is one that is done by calendar age. Calendar age at event is defined to be the integral age $y$ obtained on the birthday in the same calendar year in which the event takes place. For example, if a person’s date of birth is September 14, 1960, and date of withdrawal is June 26,1994 , then the calendar age at withdrawal is 34 , since this person would be integral age 34 on the birthday in the calendar year of withdrawal. Calendar ages at event are easily found by
$$C A=C Y E-C Y B,$$
where $C Y E$ is the calendar year of the event and $C Y B$ is the calendar year of birth.

It is easy to see that a person with calendar age $w$ at event has an exact age at event that falls within $(w-1, w+1)$. If the $w^{\text {th }}$ birthday is January 1 , and the event is December 31 of the same year, then the calendar age at event is $w$ but the exact age is practically $w+1$. Conversely, if the $w^{t h}$ birthday is December 31, and the event is January 1 of the same year, then the calendar age at event is still $w$, but the exact age is virtually $w-1$.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Calculation of Exposure

$$(\text { Exact Exposure }){i, x}=\left[\begin{array}{c} s_i \ \iota_i \ \kappa_i \end{array}\right]-r_i,$$其中$\left[\begin{array}{l}s_i \ \iota_i \ \kappa_i\end{array}\right]$表示$s_i, \iota_i, \kappa_i$大于零的最小值。换句话说，如果$\iota_i=\kappa_i=0$，那么那个人$i$既没有死亡也没有退出$(x, x+1]$，那么我们有(精确暴露)$i{i, x}=s_i-r_i$。但如果某人$i$死于$(x, x+1]$，那么$\iota_i<s_i$和$\kappa_i=0$，那么(确切暴露)$i_{i, x}=\iota_i-r_i$。最后，如果人$i$退出$(x, x+1)$，那么$\kappa_i<s_i$和$\iota_i=0$，然后我们有(确切曝光$)_{i, x}=\kappa_i-r_i$。

$$(\text { Scheduled Exposure })_{i, x}=\left[\begin{array}{c} s_i \ \kappa_i \end{array}\right]-r_i,$$

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

$$C A=C Y E-C Y B,$$

## 统计代写|生存模型代写survival model代考|Grouped Times of Death

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## 统计代写|生存模型代写survival model代考|Grouped Times of Death

This study design was described in Section 4.4, along with techniques for estimating a tabular survival model from such data. We are now interested in the estimation of parametric models, and we will consider the methods of maximum likelihood and least squares for this purpose.
Suppose $n$ persons are alive at $t=0$, and we observe their deaths in $k$ non-overlapping intervals of equal length. Let $d_i$ be the number of deaths observed in $(i, i+1]$. The probability of death in $(i, i+1]$ for a person alive at $t=0$ is $i \mid q_0=S(i)-S(i+1)$, so the contribution of the $(i+1)^{s t}$ interval to the likelihood is
$$L_i=[S(i)-S(i+1)]^{d_i}$$
The overall likelihood is
$$L=\prod_{i=0}^{k-1}[S(i)-S(i+1)]^{d_i} .$$
Then $S(i)-S(i+1)$ is written in terms of the unknown parameters of the chosen parametric model, and the parameters are found by maximizing $L$.
If the chosen $S(t)$ is sufficiently simple, such as a one-parameter uniform or exponential model, then convenient expressions for the MLE’s of the parameters can be found. Otherwise, as frequently occurs, the likelihood equations must be solved numerically, or the likelihood is maximized numerically without taking derivatives.

## 统计代写|生存模型代写survival model代考|Maximum Likelihood Approaches

As a first example, consider a sample of $n$ laboratory mice all alive at $t=0$. We observe the exact time of each death up to time $t=r$, and cease

observation at that time with some of the mice still alive. Since not all have died we have an incomplete data situation. Note that the study design is longitudinal (not cross-sectional), and those that are not observed to die are enders, not random withdrawals. In a clinical setting this type of study is said to be truncated.

The contribution of each death to the likelihood is the PDF for death at the time that death actually occurs. The contribution for each survivor at $t=r$ is simply the probability of living to $t=r$, namely $S(r)$. If there are $d$ deaths in total, out of the original sample of size $n$, then we have
$$L=[S(r)]^{n-d} \cdot \prod_{i=1}^d f\left(t_i\right),$$
where $t_i$ is the time of the $i^{t h}$ death.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Grouped Times of Death

$$L_i=[S(i)-S(i+1)]^{d_i}$$

$$L=\prod_{i=0}^{k-1}[S(i)-S(i+1)]^{d_i} .$$

## 统计代写|生存模型代写survival model代考|Maximum Likelihood Approaches

$$L=[S(r)]^{n-d} \cdot \prod_{i=1}^d f\left(t_i\right),$$

## 统计代写|生存模型代写survival model代考|Full Data, Uniform Distribution for Mortality

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## 统计代写|生存模型代写survival model代考|Full Data, Uniform Distribution for Mortality

Under the uniform distribution, (7.52), being the same as (7.20), is maximized by the value of $q_x$ which satisfies
$$\frac{d_x}{q_x}+\sum_{i=1}^n \frac{r_i}{1-r_i \cdot q_x}-\sum_{\overline{\mathcal{D}}} \frac{t_i}{1-t_i \cdot q_x}=0$$
where the last summation is taken over all persons who do not die. Equation (7.53), like its single-decrement counterpart (7.27), must, in general. be solved by iteration.
A quadratic solution is found for (7.53) under Special Case A, with $r_i=0$ for all $i, t_i=1$ for all enders, and $t_i=t$, a constant, for all withdrawals. (The derivation of this result is left as an exercise.) For all other situations, we must solve a higher order polynomial equation to obtain $\hat{q}_x=\hat{q}_x^{\prime}(d)$, with the attendant possibility of multiple roots. Under the exponential assumption, on the other hand, all cases have a unique solution for $\hat{\mu}$ given by (7.23), and thus a unique solution for $\hat{\boldsymbol{g}}_x$.
Estimation of $q_x^{\prime(d)}$ from partial data in the presence of random withdrawals is more complex than with full data. We consider only Special Case A, where $r_i=0$ for all $i$ and $s_i=1$ for all $i$ who do not die or withdraw. Suppose our only information is that from a sample of $n_x$ persons at exact age $x, d_x$ died and $w_x$ withdrew in $(x, x+1]$, so that $n_x-d_x-w_x$ survived to age $x+1$. Exact ages at death and withdrawal are not available.
The likelihood of this sample result is
$$L=\left[q_x^{(d)}\right]^{d_x} \cdot\left[q_x^{(w)}\right]^{w_x} \cdot\left[1-q_x^{(d)}-q_x^{(w)}\right]^{n_x-d_x-w_x},$$
where $q_x^{(d)}$ and $q_{\mathrm{x}}^{(w)}$ are defined by (5.12a) and (5.12b), respectively. To find the MLE’s of $q_x^{(d)}$ and $q_x^{(w)}$, we first write
$$\ln L=d_x \cdot \ln q_x^{(d)}+w_x \cdot \ln q_x^{(w)}+\left(n_x-d_x-w_x\right) \cdot \ln \left(1-q_x^{(d)}-q_x^{(w)}\right) .$$
Then we find
$$\frac{\partial \ln L}{\partial q_x^{(d)}}=\frac{d_x}{q_x^{(d)}}-\frac{n_x-d_x-w_x}{1-q_x^{(d)}-q_x^{(w)}}=0$$
and
$$\frac{\partial \ln L}{\partial q_x^{(i x)}}=\frac{w_x}{q_x^{(w)}}-\frac{n_x-d_x-w_x}{1-q_x^{(d)}-q_x^{(w)}}=0,$$
which solve simultaneously for the expected results
$$\hat{q}_x^{(d)}=\frac{d_x}{n_x}$$
and
$$\stackrel{\wedge}{q}_x^{(w)}=\frac{w_x}{n_x} .$$

## 统计代写|生存模型代写survival model代考|Partial Data (Special Case A), Exponential Distributions

From Equations (5.28a) and (5.28b), with $q_x^{(d)}$ and $q_x^{(w)}$ replaced by their estimators given by (7.57a) and (7.57b), we directly have
$$\widehat{q}_x^{\prime(d)}=1-\left(\frac{n-d-w}{n}\right)^{d /(d+w)}$$
and
$${\mathcal{q}_x^{(w)}}^{(w)}=1-\left(\frac{n-d-w}{n}\right)^{w /(d+w)}$$
Note that MLE’s (7.58a) and (7.58b) are the same as the moment estimators (6.37a) and (6.37b).

It should be recognized that it was the independence assumption for the random events death and withdrawal which allowed us to reach the general solutions for $\hat{\mu}=\hat{\mu}^{(d)}$ under the exponential distribution, given by (7.23), and $\hat{q}_x=\hat{q}_x^{\prime}(d)$ under the uniform distribution, given by (7.53), for the full data case. If the independence assumption is not valid, and a dependent model is assumed, then the estimation of $q_x$ is more complex. For a discussion of this, the interested reader is referred to Robinson [63].

We chose to develop the partial data situation only for Special Case A, since it is the only one of the special cases which has convenient solutions for $\hat{q}_x^{\prime(d)}$ and $\hat{q}_x^{\prime(w)}$. The more general partial data situation for the random withdrawal model, which embraces all of our special cases, is given by Broffitt [14].

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Full Data, Uniform Distribution for Mortality

$$\frac{d_x}{q_x}+\sum_{i=1}^n \frac{r_i}{1-r_i \cdot q_x}-\sum_{\overline{\mathcal{D}}} \frac{t_i}{1-t_i \cdot q_x}=0$$

$$\ln L=d_x \cdot \ln q_x^{(d)}+w_x \cdot \ln q_x^{(w)}+\left(n_x-d_x-w_x\right) \cdot \ln \left(1-q_x^{(d)}-q_x^{(w)}\right) .$$

$$\frac{\partial \ln L}{\partial q_x^{(d)}}=\frac{d_x}{q_x^{(d)}}-\frac{n_x-d_x-w_x}{1-q_x^{(d)}-q_x^{(w)}}=0$$

$$\frac{\partial \ln L}{\partial q_x^{(i x)}}=\frac{w_x}{q_x^{(w)}}-\frac{n_x-d_x-w_x}{1-q_x^{(d)}-q_x^{(w)}}=0,$$

$$\hat{q}_x^{(d)}=\frac{d_x}{n_x}$$

$$\stackrel{\wedge}{q}_x^{(w)}=\frac{w_x}{n_x} .$$

## 统计代写|生存模型代写survival model代考|Partial Data (Special Case A), Exponential Distributions

$$\widehat{q}_x^{\prime(d)}=1-\left(\frac{n-d-w}{n}\right)^{d /(d+w)}$$

$${\mathcal{q}_x^{(w)}}^{(w)}=1-\left(\frac{n-d-w}{n}\right)^{w /(d+w)}$$

## 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|General Form for Full Data

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。

## 统计代写|生存模型代写survival model代考|General Form for Full Data

We recall that $n_x$ is the total number of persons in the sample who contribute to the estimation interval $(x, x+1]$, and $x+r_i$ is the age at which the $i^{\text {th }}$ person enters $(x, x+1], 0 \leq r_i<1$. Let $x+t_i$ be the age at which the $i^{\text {th }}$ person leaves $(x, x+1], 0<t_i \leq 1$, whether as a scheduled, and observed, ender, as an interval survivor, or as a death. Let $\delta_i$ be the indicator variable for person $i$ defined by (7.17). Then if $t_i=1$ and $\delta_i=0$, person $i$ is a survivor; if $t_i<1$ and $\delta_i=0$, person $i$ is an observed ender; if $t_i \leq 1$ and $\delta_i=1$, person $i$ is a death.

As of age $x+r_i$, person $i$ is scheduled to leave observation at some age not later than $x+1$, but the possibility of prior death implies that the age at which observation actually does cease is a random result. Thus $t_i$ is the realization of a random variable $T_i$. On the other hand, $r_i$, the time of entry, is not a random variable.

We can now see that if $r_i=0$ for all $i$, we have either Special Case A or Special Case C. Similarly, if $t_i=1$ for all $i$ for which $\delta_i=0$ (i.e., for all $i$ who do not die), we have either Special Case A or Special Case B. This is summarized in Table 7.1 .

The general form of the contribution to $L$ by person $i$ is
$$L_i={ }{t_i-r_i} p{x+r_i}\left(\mu_{x+t_i}\right)^{\delta_1},$$
since, if $\delta_i=0$, the likelihood is merely the probability of survival from $x+r_i$ to $x+t_i$, and if $\delta_i=1$, it is the density function for death at $x+t_i$ given alive at $x+r_i$. The overall likelihood is
$$L=\prod_{i=1}^n t_i-r_i p_{x+r_i}\left(\mu_{x+t_i}\right)^{\delta_i} .$$
To evaluate (7.20) we must make a distribution assumption. We will again consider two cases.

## 统计代写|生存模型代写survival model代考|Full Data, Exponential Distribution

Under the exponential (constant force) assumption, we find
$$L=(\mu)^d \cdot \prod_{i=1}^n e^{-\left(t_i-r_i\right) \mu}$$
$\mathrm{SO}$
$$\ell=\ln L=d \cdot \ln \mu-\mu \cdot \sum_{i=1}^n\left(t_i-r_i\right),$$
and it is easy to see that $\frac{d \ell}{d \mu}=0$ produces
$$\hat{\mu}=\frac{d_x}{\sum_{i=1}^n\left(t_i-r_i\right)}$$
Estimator (7.23) is of the same form as Estimator (7.15), namely the ratio of $d_x$ to the exact exposure of the sample, which is the sample central rate. (This will be the result for all cases of full data evaluated under the exponential assumption.) Thus (7.23) is the general full data MLE, and Special Cases $A, B$, and $C$ are all contained within it.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|General Form for Full Data

$$L_i={ }{t_i-r_i} p{x+r_i}\left(\mu_{x+t_i}\right)^{\delta_1},$$

$$L=\prod_{i=1}^n t_i-r_i p_{x+r_i}\left(\mu_{x+t_i}\right)^{\delta_i} .$$

## 统计代写|生存模型代写survival model代考|Full Data, Exponential Distribution

$$L=(\mu)^d \cdot \prod_{i=1}^n e^{-\left(t_i-r_i\right) \mu}$$
$\mathrm{SO}$
$$\ell=\ln L=d \cdot \ln \mu-\mu \cdot \sum_{i=1}^n\left(t_i-r_i\right),$$

$$\hat{\mu}=\frac{d_x}{\sum_{i=1}^n\left(t_i-r_i\right)}$$
Estimator(7.23)与Estimator(7.15)的形式相同，即$d_x$与样本的确切暴露的比率，即样本中心率。(这将是在指数假设下评估完整数据的所有情况的结果。)因此(7.23)是一般的全数据MLE，特殊情况$A, B$和$C$都包含在其中。

## 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|THE ACTUARIAL APPROACH TO MOMENT ESTIMATION

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。

## 统计代写|生存模型代写survival model代考|THE ACTUARIAL APPROACH TO MOMENT ESTIMATION

Of the various approaches to the estimation of $q_x$ over the interval $(x, x+1]$ described in this text, the actuarial approach was the first to be developed, dating from the middle of the nineteenth century. There are two significant implications of this early origin:
(1) The method predates the formal, scientific development of statistical estimation theory.
(2) The method predates any kind of mechanical or electronic calculating equipment.

It will be easy to see the reflection of these two observations in the features of the actuarial method. The method was developed primarily in an intuitive manner, in the absence of a guiding statistical theory, and a key theoretical concession was made in the interest of simpler record-keeping and calculation. The traditional actuarial approach remained in use well into the second half of the twentieth century, even after the availability of both high-speed data processing capabilities and modern statistical theory had made the process nearly obsolete. Over the past ten years, however, we have seen mounting evidence to suggest that this pioneering method of survival model estimation should become, and is becoming, of historical significance only. Theref ore the presentation of the method here is in the nature of a historical summary. The reader interested in a fuller history can pursue this through the many references given here.

We will present the actuarial method assuming a double-decrement environment. The simplification that results if the environment is single-decrement will be easily seen.

## 统计代写|生存模型代写survival model代考|The Concept of Actuarial Exposure

The key difference between the traditional actuarial approach to estimation and the moment approaches presented thus far in this chapter is that the actuarial approach has not generally made use of the concept of scheduled exposure, as defined in Section 6.2.3. Rather it made use of a type of observed exposure, which we will call actuarial exposure, to distinguish it from the two other types of exposure (scheduled and exact) described earlier.
Recall that person $i$ is scheduled to be an ender to the study at age $z_i$, and this is known at entry to the study. Recall also that if $x<z_i<x+1$, then we say that person $i$ is scheduled to exit $(x, x+1]$ at age $z_i=x+s_i$, where $0<s_i<1$. On the other hand, if $z_i \geq x+1$, we say that person $i$ is scheduled to survive $(x, x+1]$, which is the same as to say scheduled to exit at $x+s_i$, where $s_i=1$.

Suppose person $i$ enters $(x, x+1]$ at age $x+r_i, 0 \leq r_i<1$. Under the actuarial method, if person $i$ is an observed ender at $x+s_i$, he contributes exposure of $\left(s_i-r_i\right)$. If he is an observed withdrawal at $x+t_i$, where, necessarily, $t_i \leq s_i$, he contributes exposure of $\left(t_i-r_i\right)$. But what if person $i$ is an observed death in $(x, x+1]$ ? We have seen that a proper moment estimation procedure would have person $i$ contribute exposure to age $x+s_i$, the scheduled exit age, and early in the historical development of the actuarial method this was intuitively recognized. However, the identification of scheduled exit age, for a person who died before reaching it, required more extensive data analysis than the early manual procedures allowed. To resolve the problem of an unknown $x+s_i$ for an observed death, it was simply assumed that $s_i=1$ for all deaths.

The important point here is that the traditional actuarial approach to handling the data, as thoroughly described by Batten [6] and Gershenson [26], simply did not identify $x+s_i$ a priori. Rather $x+s_i$ became known $a$ posteriori only if person $i$ was, in fact, an ender, just as Hoem’s “scheduled” withdrawal age $x+t_i$ becomes known, a posteriori, only if person $i$ is an observed withdrawal. Thus, in the actuarial approach, $x+s_i$ did not become known for a death, so it was assumed to be $x+1$, just as, in Hoem’s approach, a death that might otherwise have been a withdrawal has an unknown $x+t_i$, and is therefore assumed to be $x+s_i$.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|THE ACTUARIAL APPROACH TO MOMENT ESTIMATION

(1)该方法早于统计估计理论的正式、科学发展。
(2)该方法早于任何一种机械或电子计算设备。

## 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|The Basic Moment Relationship

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|The Basic Moment Relationship

If $n_x$ is the total number of persons who contribute to $(x, x+1]$, then the total number of expected deaths is $\sum_{i=1}^{n_x} s_i-r_i q_{x+r_i}$. (For convenience we will use $n$ for $n_x$ in our summations.) When equated to the actual number of observed deaths, we obtain the moment equation
$$E\left[D_x\right]=\sum_{i=1}^n s_{i-r,} q_{x+r_i}=d_x,$$
where $D_x$ is the random variable for deaths in $(x, x+1]$, and $d_x$ is the observed number.

To solve (6.2) for our estimate of $q_x$, we will use the approximation
$$s_i-r_i q_{x+r_i} \approx\left(s_i-r_i\right) \cdot q_x$$
Then (6.2) becomes
$$E\left[D_x\right]=q_x \cdot \sum_{i=1}^n\left(s_i-r_i\right)=d_x$$
from which we easily obtain
$$\hat{q}x=\frac{d_x}{\sum{i=1}^n\left(s_i-r_i\right)}$$
the general form of the moment estimator in a single-decrement environment.

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

If $r_i=0$ and $s_i=1$ for all $n_x$ persons who contribute to $(x, x+1]$, then we have $s_{s_i-r_i} q_{x+r_i}=q_x$, and (6.2) becomes
$$E\left[D_x\right] \neq n_x \cdot q_x=d_x$$
so that (6.5) becomes
$$\hat{q}_x=\frac{d_x}{n_x}$$
Recall that this is Special Case A, as defined in Section 5.2.

We recognize (6.7) as the binomial proportion estimator already encountered in Chapter 4. We also recognize it as the maximum likelihoo estimator of the conditional mortality probability $q_x$, where the model for the likelihood is a simple binomial model. Thus the number of persons in the sample, $n_x$, can be thought of as a number of binomial trials. The standard characteristics of a binomial model are assumed to apply. Thus each trial is considered to be independent, and the probability of death on a single trial $\left(q_x\right)$ is assumed constant for all trials. In such a situation, the sample proportion of deaths, which is given by (6.7), is a natural estimator for this parameter $q_x$.

If $s_i=1$ for all, but $r_i>0$ for some of the $n_x$ persons who contribute to $(x, x+1]$, then we have $s_i-r_i q_{x+r_i}=1-r_i q_{x+r_i}$, and (6.2) becomes
$$E\left[D_x\right]=\sum_{i=1}^n{ }{1-r_i} q{x+r_i}=d_x$$
The general approximation given by (6.3) then becomes
$$1-r_i q_{x+r_i} \approx\left(1-r_i\right) \cdot q_x$$
which is the same as (3.77). Substituting (6.9) into (6.8), we obtain the result
$$\hat{q}x=\frac{d_x}{\sum{i=1}^n\left(1-r_i\right)},$$
the moment estimator for Special Case B.

# 生存模型代考

## 统计代写|生存模型代写survival model代考|The Basic Moment Relationship

$$E\left[D_x\right]=\sum_{i=1}^n s_{i-r,} q_{x+r_i}=d_x,$$

$$s_i-r_i q_{x+r_i} \approx\left(s_i-r_i\right) \cdot q_x$$

$$E\left[D_x\right]=q_x \cdot \sum_{i=1}^n\left(s_i-r_i\right)=d_x$$

$$\hat{q}x=\frac{d_x}{\sum{i=1}^n\left(s_i-r_i\right)}$$

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

$$E\left[D_x\right] \neq n_x \cdot q_x=d_x$$

$$\hat{q}_x=\frac{d_x}{n_x}$$

$$E\left[D_x\right]=\sum_{i=1}^n{ }{1-r_i} q{x+r_i}=d_x$$

$$1-r_i q_{x+r_i} \approx\left(1-r_i\right) \cdot q_x$$

$$\hat{q}x=\frac{d_x}{\sum{i=1}^n\left(1-r_i\right)},$$

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## 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。