### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Some Discrete Survival Distributions

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• 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|多元统计分析代写Multivariate Statistical Analysis代考|Mixed Discrete-Continuous Survival Distributions

Suppose that $T$ has a mixed discrete-continuous distribution, with atoms of probability at points $0=\tau_{0}<\tau_{1}<\tau_{2}<\ldots$, together with a density $f^{c}(t)$ on $(0, \infty)$. An all-too-familiar example is the waiting time in a queue: $T$ is then either zero (rarely) or capped at closing time $\tau$ (when the shutter comes down just as you reach the counter), or continuous on $(0, \tau)$. (My wife always seems to beat the queue, although she maintains that when she first met me there was no queue to beat.)

If $\bar{F}$ is continuous at $t, \bar{F}(t-)=\bar{F}(t)$, whereas if $t=\tau_{l}, \bar{F}\left(\tau_{l}-\right)=\bar{F}\left(\tau_{l}\right)+$ $\mathrm{P}\left(T=\tau_{l}\right)$. Then,
$$\bar{F}(t)=\mathrm{P}(T>t)=\sum_{\mathrm{r}{l}>t} \mathrm{P}\left(T=\tau{l}\right)+\int_{t}^{\infty} f^{c}(s) d s$$
the density component $f^{c}(t)$ is defined as $-d \bar{F}(t) / d t$ at points between the r . Now,
$$\bar{F}(t)=\bar{F}(0) \prod_{l=1}^{l(t)}\left[\left{\bar{F}\left(\tau_{l}\right) / \bar{F}\left(\tau_{l}-\right)\right}\left{\bar{F}\left(\tau_{l}-\right) / \bar{F}\left(\tau_{l-1}\right)\right}\right]\left{\bar{F}(t) / \bar{F}\left(\tau_{l(t)}\right)\right}$$
where $l(t)=\max \left{l: \tau_{l} \leq t\right}$. But $F(0)=1-h_{0}$, where $h_{0}=\mathrm{P}(T=0)$, and $\bar{F}\left(\tau_{l}\right) / \bar{F}\left(\tau_{l}-\right)=\mathrm{P}\left(T>\tau_{l}\right) / \mathrm{P}\left(T>\tau_{l}-\right)=1-\mathrm{P}\left(T \leq \tau_{l} \mid T>\tau_{l}-\right)=1-h_{l}$
say. Also,
$$\bar{F}\left(\tau_{l}-\right) / \bar{F}\left(\tau_{l-1}\right)=\exp \left{-\int_{\tau_{l-1}}^{\tau_{l}-} h^{c}(s) d s\right}$$
where
$$h^{c}(s)=f^{c}(s) / \bar{F}(s)=-d \log \bar{F}(s) / d s .$$
The $h_{l}$ are the discrete hazard contributions at the discontinuities, and $h^{c}$ is the continuous component of the hazard function. Last,
$$F(t) / F\left(\tau_{l(t)}\right)=\exp \left{-\int_{\tau_{l(t)}}^{t} h^{c}(s) d s\right}$$
which equals 1 if $t=\tau_{l(t)}$. Substituting into the expression given for $F(t)$ a few lines above yields the well-known formula (e.g., Cox, 1972, Section 1 ):
$$\bar{F}(t)=\left{\prod_{s=1}^{l(t)}\left(1-h_{s}\right)\right} \exp \left{-\int_{0}^{t} h^{c}(s) d s\right}$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|From Discrete to Continuous

Consider now a purely discrete distribution in which the density component is absent, so that
$$h_{l}=\mathrm{P}\left(\tau_{l-1}\tau_{l-1}\right)$$

Let $\delta_{l}=\tau_{l}-\tau_{l-1}$ and $g\left(\tau_{l}\right)=h_{l} / \delta_{l}$, so that, in the limit $\delta_{l} \downarrow 0, g\left(\tau_{l}\right)$ is defined as the hazard function at $\tau_{l}$ of a continuous variate. Now,
\begin{aligned} \log \bar{F}(t) &=\log \prod_{n_{l} \leq t}\left(1-h_{l}\right)=\sum_{\tau_{l} \leq t} \log \left{1-g\left(\tau_{l}\right) \delta_{l}\right} \ &=-\sum_{\tau_{i} \leq t} g\left(\tau_{l}\right) \delta_{l}+O\left{\sum_{\eta_{l} \leq t} g\left(\tau_{l}\right)^{2} \delta_{l}^{2}\right} \end{aligned}
In the limit $\max \left(\delta_{l}\right) \rightarrow 0$ we obtain
$$F(t)=\exp \left{-\int_{0}^{t} g(s) d s\right}$$
This illustrates the transition from an increasingly dense discrete distribution to a continuous one. This is just an informal sketch of material dealt with in much greater depth by Gill and Johansen $(1990$, Section 4.1). The reverse transition, obtained by dividing up the continuous time scale into intervals $\left(\tau_{l-1}, \tau_{l}\right)$, is accomplished simply by defining
$$h_{l}=1-\exp \left{-\int_{t_{l-1}}^{\tau_{l}} g(s) d s\right}$$
Then,
$$\bar{F}\left(\tau_{k}\right)=\exp \left{-\int_{0}^{\tau_{k}} g(s) d s\right}=\exp \left{-\sum_{l=1}^{k} \int_{\tau_{l-1}}^{\tau_{l}} g(s) d s\right}=\prod_{s=1}^{k}\left(1-h_{s}\right)$$

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Rieman–Stieltjes Integrals

We describe here a convenient notation, which can be used for discrete, continuous, and mixed distributions alike. Suppose first, that $T$ is continuous with distribution function $F(t)=\mathrm{P}(T \leq t)$. Its mean is then calculated as $\mathrm{E}(T)=\int_{0}^{\infty} t f(t) d t$, where $f$ is its density function. But $f(t)=d F(t) / d t$, so we can write $\mathrm{E}(T)=\int_{0}^{\infty} t d F(t)$. Now suppose that $T$ is discrete, taking values $t_{j}$ with probabilities $p_{j}(j=1,2, \ldots)$ : then $\mathrm{E}(T)=\sum_{j} t_{j} p_{j}$. But $d F(t)=F(t+d t)-F(t), \operatorname{so} d F(t)=0$ if the interval $(t, t+d t]$ does not include one of the $t_{j}$, and $d F(t)=p_{j}$ if $t_{j} \in(t, t+d t]$. In that case, $\int_{0}^{\infty} t d F(t)$ reduces to $\sum_{j} t_{j} p_{j}$ since $d F(t)$ is only non-zero at the $t_{j}$. In either case, continuous or discrete, and also when $T$ has a mixed discrete-continuous distribution, the form $\int_{0}^{\infty} t d F(t)$ serves to define $E(T)$. More generally, we can define $\int_{0}^{\infty} g(t) d F(t)$ in the same way, where $g$ is some function of $t$. This style of integral is known as Rieman-Stieltjes. (Of course, there is a more formal argument for all this, but here is not the place to be pedantic. It is sufficient that $g$ be continuous and $F$ of bounded variation-look it up if you feel the need.)

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Mixed Discrete-Continuous Survival Distributions

F¯(吨)=磷(吨>吨)=∑rl>吨磷(吨=τl)+∫吨∞FC(s)ds

\bar{F}(t)=\bar{F}(0) \prod_{l=1}^{l(t)}\left[\left{\bar{F}\left(\tau_{l} \right) / \bar{F}\left(\tau_{l}-\right)\right}\left{\bar{F}\left(\tau_{l}-\right) / \bar{F} \left(\tau_{l-1}\right)\right}\right]\left{\bar{F}(t) / \bar{F}\left(\tau_{l(t)}\right) \对}\bar{F}(t)=\bar{F}(0) \prod_{l=1}^{l(t)}\left[\left{\bar{F}\left(\tau_{l} \right) / \bar{F}\left(\tau_{l}-\right)\right}\left{\bar{F}\left(\tau_{l}-\right) / \bar{F} \left(\tau_{l-1}\right)\right}\right]\left{\bar{F}(t) / \bar{F}\left(\tau_{l(t)}\right) \对}

\bar{F}\left(\tau_{l}-\right) / \bar{F}\left(\tau_{l-1}\right)=\exp \left{-\int_{\tau_{l -1}}^{\tau_{l}-} h^{c}(s) d s\right}\bar{F}\left(\tau_{l}-\right) / \bar{F}\left(\tau_{l-1}\right)=\exp \left{-\int_{\tau_{l -1}}^{\tau_{l}-} h^{c}(s) d s\right}

HC(s)=FC(s)/F¯(s)=−d日志⁡F¯(s)/ds.

F(t) / F\left(\tau_{l(t)}\right)=\exp \left{-\int_{\tau_{l(t)}}^{t} h^{c}(s ) d s\right}F(t) / F\left(\tau_{l(t)}\right)=\exp \left{-\int_{\tau_{l(t)}}^{t} h^{c}(s ) d s\right}

\bar{F}(t)=\left{\prod_{s=1}^{l(t)}\left(1-h_{s}\right)\right} \exp \left{-\int_{ 0}^{t} h^{c}(s) d s\right}\bar{F}(t)=\left{\prod_{s=1}^{l(t)}\left(1-h_{s}\right)\right} \exp \left{-\int_{ 0}^{t} h^{c}(s) d s\right}

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|From Discrete to Continuous

Hl=磷(τl−1τl−1)

\begin{对齐} \log \bar{F}(t) &=\log \prod_{n_{l} \leq t}\left(1-h_{l}\right)=\sum_{\tau_{l } \leq t} \log \left{1-g\left(\tau_{l}\right) \delta_{l}\right} \ &=-\sum_{\tau_{i} \leq t} g\左(\tau_{l}\right) \delta_{l}+O\left{\sum_{\eta_{l} \leq t} g\left(\tau_{l}\right)^{2} \delta_ {l}^{2}\right} \end{对齐}\begin{对齐} \log \bar{F}(t) &=\log \prod_{n_{l} \leq t}\left(1-h_{l}\right)=\sum_{\tau_{l } \leq t} \log \left{1-g\left(\tau_{l}\right) \delta_{l}\right} \ &=-\sum_{\tau_{i} \leq t} g\左(\tau_{l}\right) \delta_{l}+O\left{\sum_{\eta_{l} \leq t} g\left(\tau_{l}\right)^{2} \delta_ {l}^{2}\right} \end{对齐}

F(t)=\exp \left{-\int_{0}^{t} g(s) d s\right}F(t)=\exp \left{-\int_{0}^{t} g(s) d s\right}

h_{l}=1-\exp \left{-\int_{t_{l-1}}^{\tau_{l}} g(s) d s\right}h_{l}=1-\exp \left{-\int_{t_{l-1}}^{\tau_{l}} g(s) d s\right}

\bar{F}\left(\tau_{k}\right)=\exp \left{-\int_{0}^{\tau_{k}} g(s) d s\right}=\exp \left{ -\sum_{l=1}^{k} \int_{\tau_{l-1}}^{\tau_{l}} g(s) d s\right}=\prod_{s=1}^{k }\left(1-h_{s}\right)\bar{F}\left(\tau_{k}\right)=\exp \left{-\int_{0}^{\tau_{k}} g(s) d s\right}=\exp \left{ -\sum_{l=1}^{k} \int_{\tau_{l-1}}^{\tau_{l}} g(s) d s\right}=\prod_{s=1}^{k }\left(1-h_{s}\right)

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