### 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Some Continuous 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|The Exponential Distribution

The distribution can be defined by its survivor function $F(t)=\exp (-t / \xi)$; the scale parameter $\xi>0$ is actually the mean lifetime $\xi=\mathrm{E}(T)$. The primary distinguishing feature of the distribution is arguably its constant hazard function: $h(t)=1 / \xi$. It is the only distribution with such a constant hazard (subject to the usual mathematical provisos). So, as time proceeds, there is no recognition of age: the probability of imminent failure remains at the same level throughout. This makes the distribution both discreditable (in most applications) and compelling (as a prototype model). The lack-of-memory property is expressed, if I recall, as
$$\mathrm{P}(T>a+b \mid T>a)=\mathrm{P}(T>b)$$
your mission, if you decide to accept it, is to prove this-see the Exercises.

A natural generalisation of the exponential survivor function is $F(t)=$ $\exp \left{-(t / \xi)^{v}\right}$, raising $t / \xi$ to a power $v>0$; the exponential distribution is regained when $v=1$. The corresponding hazard function is $h(t)=(v / \xi)(t / \xi)^{v-1}$, an increasing function of $t$ when the shape parameter $v>1$ and decreasing when $v<1$. The mean and variance are expressed in terms of the gamma function (see the Exercises), but with an explicit survivor function, quantiles are more accessible. Thus, the upper $q$ th quantile $t_{q}$, for which $\mathrm{P}\left(T>t_{q}\right)=q$, is given by $t_{q}=\xi(-\log q)^{1 / v}$.

The Weibull distribution is fairly ubiquitous in reliability, even boasting a must-have handbook, commonly referred to as the Weibull Bible. The reasons for this popularity probably have to do with its being an extreme-value distribution with the associated weakest-link interpretation, the variety of shapes that can be accommodated by the hazard function, and the simple form of $\bar{F}(t)$, which facilitates data plotting.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Pareto Distribution

The Pareto distribution can arise naturally in the survival context as follows. Begin with an exponential distribution for $T: F(t)=e^{-t / \xi}$. Suppose that, due to circumstances beyond our control (as they say, whenever I go by train), $\xi$ varies randomly over individual units. Specifically, say $\lambda=1 / \xi$ has a gamma distribution with density
$$f(\lambda)=\Gamma(\gamma)^{-1} \alpha^{\gamma} \lambda^{\gamma-1} \mathrm{e}^{-\alpha \lambda}$$
(This somewhat artificial distributional assumption oils the wheels: a choice always has to be made between hair-shirt realism and mathematical tractability.) Thus, the conditional survivor function of $T$ is $\bar{F}(t \mid \lambda)=\mathrm{e}^{-\lambda t}$ and the unconditional one is obtained as
$$\bar{F}(t)=\int_{0}^{\infty} \bar{F}(t \mid \lambda) f(\lambda) d \lambda=\Gamma(\gamma)^{-1} \alpha^{\gamma} \int_{0}^{\infty} \lambda^{\gamma-1} \mathrm{e}^{-\lambda(t+\alpha)} d \lambda=(1+t / \alpha)^{-\gamma} .$$
The parameters of this Pareto survivor function are $\alpha>0$ (scale) and $\gamma>0$ (shape).

The hazard function, $h(t)=\gamma /(\alpha+t)$, is decreasing in $t$ and tends to zero as $t \rightarrow \infty$. Such behaviour is fairly atypical but not altogether unknown in practice. It would be appropriate for units that become less failure-prone with age, ones having ever-decreasing probability of imminent failure, settling in as time goes on. (One might cite as an example humans, who become less gaffe-prone with age-if only that were true!) However, there will certainly be failure at some finite time since $\bar{F}(\infty)=0$, a reflection of the fact that $h(t)$ here does not decrease fast enough for $\int_{0}^{\infty} h(s) d s$ to be finite.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Shape of Hazard

A variety of alternative distributions has been applied in survival analysis and reliability. Essentially, any distribution on $(0, \infty)$ will serve. Thus, one can contemplate distributions such as the gamma, Gompertz, Burr, and inverse Gaussian. And then, when one has finished contemplating them, one can note that distributions on $(-\infty, \infty)$ can be converted, usually via a log-transform: this yields the log-normal and the log-logistic, for example. (And, yes, you can start a sentence with and: and one of the best-loved hymns in the English language starts with And.) Some properties of these will be set as exercises below.

For most systems, hazard functions increase with age as the system becomes increasingly prone to crack-up. (No laughing at the back-your professor isn’t gaga yet.) The trade description for this is IFR (increasing failure rate). Likewise, DFR stands for decreasing failure rate: this would apply to systems that wear in or learn from experience (of others, preferably), becoming less liable to failure with age. The bog-standard example is the Weibull hazard, IFR for $v>1$ and DFR for $v<1$.

In some cases the legendary bathtub hazard shape (down-along-up) crops up. In animal lifetimes this can reflect high infant mortality, followed by a lower and fairly constant hazard, eventually ending up with the ravages of old age. To model such a shape we might simply add together three Weibull possibilities, that is, take the hazard function as
$$h(t)=\left(v_{0} / \xi_{0}\right)+\left(v_{1} / \xi_{1}\right)\left(t / \xi_{1}\right)^{v_{1}-1}+\left(v_{2} / \xi_{2}\right)\left(t / \xi_{2}\right)^{v_{2}-1} .$$
The corresponding survivor function is
\begin{aligned} \bar{F}(t) &=\exp \left{-\int_{0}^{t} h(s) d s\right}=\exp \left{-t / \xi_{0}-\left(t / \xi_{1}\right)^{v_{1}}-\left(t / \xi_{2}\right)^{v_{2}}\right} \ &=G_{0}(t) G_{1}(t) G_{2}(t), \end{aligned}
where the $G_{j}(j=0,1,2)$ are the survivor functions of the three contributing distributions. The representation in terms of random variables is $T=$ $\min \left(T_{0}, T_{1}, T_{2}\right)$, where the $T_{j}$ are independent with survivor functions $G_{j}$.
In reliability applications the bathtub hazard is relevant for manufactured units when there is relatively high risk of early failure (wear in), a high risk of late failure (wear out), and a lower hazard in between (where it is constant). Unfortunately, in my experience, it is often only the first two characteristics that are apparent in practice. In medical applications the bathtub hazard can reflect treatment with non-negligible operative mortality that is otherwise life preserving; as Boag (1949) put it, it is not the malady but the remedy that can prove fatal. The construction, whereby a specified hazard form is constructed by combining several contributions, is generally applicable, not just confined to the minimum of three independent Weibull variates.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Exponential Distribution

Weibull 分布在可靠性方面相当普遍，甚至拥有一本必备手册，通常称为 Weibull 圣经。这种流行的原因可能与它是具有相关最弱链接解释的极值分布，风险函数可以容纳的各种形状以及简单的形式有关F¯(吨)，这有助于数据绘图。

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Pareto Distribution

F(λ)=Γ(C)−1一种CλC−1和−一种λ
（这个有点人为的分布假设为车轮加油：总是必须在毛衫现实主义和数学易处理性之间做出选择。）因此，条件幸存者函数吨是F¯(吨∣λ)=和−λ吨并且无条件的获得为
F¯(吨)=∫0∞F¯(吨∣λ)F(λ)dλ=Γ(C)−1一种C∫0∞λC−1和−λ(吨+一种)dλ=(1+吨/一种)−C.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Shape of Hazard

H(吨)=(在0/X0)+(在1/X1)(吨/X1)在1−1+(在2/X2)(吨/X2)在2−1.

\begin{对齐} \bar{F}(t) &=\exp \left{-\int_{0}^{t} h(s) d s\right}=\exp \left{-t / \xi_{ 0}-\left(t / \xi_{1}\right)^{v_{1}}-\left(t / \xi_{2}\right)^{v_{2}}\right} \ &= G_{0}(t) G_{1}(t) G_{2}(t), \end{aligned}\begin{对齐} \bar{F}(t) &=\exp \left{-\int_{0}^{t} h(s) d s\right}=\exp \left{-t / \xi_{ 0}-\left(t / \xi_{1}\right)^{v_{1}}-\left(t / \xi_{2}\right)^{v_{2}}\right} \ &= G_{0}(t) G_{1}(t) G_{2}(t), \end{aligned}

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