统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Continuous Time-Parametric Inference

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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|Proceed with Caution

The first jibe thrown at the Bayesian by the Frequentist is that his analysis is not “objective.” One retort is that of course it isn’t; the clue is in the phrase subjective probabilities. The argument is that there is no such thing as objectivity or that the data can speak for themselves. Inference from data is always channelled through an interpreter, different interpreters start with different background information, and this is bound to influence the inference. The Frequentist will then say that scientific statements should not depend on the opinions of the speaker, particularly the opinions of one of those dodgy Bayesians. He will add, more seriously, that the subjective approach is a vehicle for individual decision making, but it is not appropriate for objective scientific reporting.

A more practical criticism of the Bayesian approach is the difficulty of creating a prior distribution. When the data are extensive, we know that the posterior is mainly determined by the likelihood, the prior having little impact. However, particularly in multiparameter cases, an apparently harmless prior can hide unsuspected and undesirable features. Further, even if you want your input to be negligible compared with that of the data, there is no such thing as an uninformative prior, though time and again you will see this claimed in published work. The classic example is taking a uniform prior for a probability, say $\pi$, to express no preference for any one value over any other in the range $(0,1)$. Unfortunately, the unintended consequence is that this choice expresses preference for smaller values of $\pi^{2}$ over larger ones on $(0,1)$.

Let us now turn our fire on the Frequentist approach. A hypothesis test produces a p-value: if this is very small, doubt is thrown on the hypothesis. By doubt we must mean that, in the light of the data $D$, we view the hypothesis $H$ as being dubious, unlikely, and improbable. But the $\mathrm{p}$-value arises from $p(D \mid H)$, not $p(H \mid D)$, and for ” $H$ improbable” we need the second one. So, the p-value does not do the job that we might like it to do (but see DeGroot, 1973).

Confidence intervals are open to similar criticism. They do not give probabilities: the carefully calculated interval $(0.19,0.31)$, for example, either spans $\theta$ or does not, which could have been said without getting out of bed. The argument that this interval has been randomly selected from a population of such intervals, $95 \%$ of which do $\operatorname{span} \theta$, sounds like a cunning attempt to persuade the listener that this one spans $\theta$ with probability $0.95$. But the latter statement is invalid to the Frequentist because it confers a probability on the parameter. If you want probabilities out (posterior), you have to put probabilities in (prior).

So, what do I think, you ask? Or maybe you don’t, but I will tell you anyway. Well, I cannot really say, even though this fence has rather sharp spikes. On this, I am not able to be dogmatic-I can see both sides of the argument. The Bayesian approach has some attractive features: it is logically consistent; nuisance parameters are not a nuisance, you just integrate them out; you do not have to rely on sometimes-dubious asymptotic approximations, as you do with the Frequentist approach; the computational problems, which inhibited

the application of the Bayesian approach in years gone by, are now mainly solved. On the other hand, the assignment of appropriate priors is tricky, and the Frequentist stand against subjectivism in scientific inference is eminently reasonable.

In putting this book together, as an outgrowth of Classical Competing Risks, much new material needed to be introduced. Some of the former things had to go, and one of them was the use of McMC to produce Bayesian posteriors in some of the applications. Nevertheless, all the methodology is still based on likelihood functions, which lies from the Bayesian approach but a short step or a long stretch, depending on your point of view. Also, currently the major $R$ package, survival, is mainly Frequentist, and I did feel the need to base things around it and other freely available $\mathrm{R}$ programs.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Random Samples

We have a sample $\left(t_{1}, \ldots, t_{n}\right)$ of observed lifetimes. Strictly speaking, no two observed values should be equal when they arise from a continuous distribution. In practice, though, rounding will often produce such ties.

The likelihood contributions are $f\left(t_{i} ; \theta\right)$ for an observed failure time $t_{i}$ and $F\left(t_{i} ; \theta\right)$ for one right-censored at $t_{i}$. The latter give information from events that have not yet occurred. It is sometimes not appreciated that such non-events, like unobserved failures, can provide useful information. Sheerluck Holmes was well aware of this: he gained a vital clue from the “curious incident” that the dog did not bark in Silver Blaze (Doyle, 1950). The overall likelihood function is
$$L=\prod_{\text {abs }} f\left(t_{i} ; \theta\right) \times \prod_{\text {cens }} \bar{F}\left(t_{i} ; \theta\right),$$
where $\prod_{\text {aus }}$ and $\prod_{\text {cens }}$ are the products over the observed and right-censored times, respectively. The appearance of $\bar{F}\left(t_{i} ; \theta\right)$ in the expression for $L$ assumes that the censoring tells us nothing further about the failure time than that it is beyond $t_{i}$. It is not always the case that censoring is non-informative; for example, in some circumstances censoring is associated with imminent failure.

Let $c_{i}=I\left(t_{i}\right.$ observed $)$, in terms of the indicator function. So, $c_{i}$ is the censoring indicator, $c_{i}=1$ if $t_{i}$ is observed and $c_{i}=0$ if $t_{i}$ is right-censored. The likelihood function can then be written as
$$L=\prod_{i=1}^{n}\left{f\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right)^{1-c_{i}}\right}=\prod_{i=1}^{n}\left{h\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right)\right}$$
Different symbols are used for the censoring indicator by different authors. Some use $\delta_{i}$, but we will mostly reserve Greek letters for parameters here; further, I prefer to spell censoring with a $c$. Perhaps we should use $C_{i}$ instead of $c_{i}$, adhering to the convention that capitals are used for random variables. However, that looks a bit odd, against most usage. Finally, the term censoring indicator should, strictly speaking, be replaced by non-censoring, or observation, indicator; but let’s not be too fussy.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Type-I Censoring

Consider a random sample from an exponential distribution with mean $\xi$. The observations are right-censored at fixed time $a>0$, that is, we only observe $t_{a}=\min (a, t)$ : this is known as Type-I censoring. Thus,
$$\mathrm{E}\left(t_{a}\right)=\int_{0}^{a} t\left(\xi^{-1} \mathrm{e}^{-t / \xi}\right) d t+a \mathrm{P}(t>a)=\xi\left(1-\mathrm{e}^{-a / \xi}\right) .$$
Suppose that the data comprise $t_{1}, \ldots, t_{r}$ (observed values, all $\leq a$ ) and $t_{r+1}, \ldots, t_{n}$ (right-censored, all $\left.=a\right)$. Then,
$$\mathrm{E}(r)=n \mathrm{P}(t \leq a)=n\left(1-\mathrm{e}^{-a / \xi}\right)$$
The log-likelihood function is given by
$$I(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{i} / \xi}\right) \times \prod_{i=r+1}^{n} \mathrm{e}^{-a / \xi}\right}=-r \log \xi-\xi^{-1} t_{+}$$
where $t_{+}=\sum_{i=1}^{r} t_{i}+(n-r) a$ is the Total Time on Test, a term from reliability engineering. The score function is $l^{\prime}(\xi)=-r \xi^{-1}+t_{+} \xi^{-2}$, and the information function is $-l^{\prime \prime}(\xi)=-r \xi^{-2}+2 t_{+} \xi^{-3}$. The mle, obtained as the solution of $l^{\prime}(\xi)=0$, is $\xi=t_{+} / r$, and its variance is approximated by $-l^{\prime \prime}(\xi)^{-1}=\xi^{2} / r$ (Appendix B).

Consider a random sample from an exponential distribution with mean $\xi$. However, this time we observe only the $r$ smallest $t_{i}$ s, where $r$ is a predetermined number: this is known as Type-II censoring. Let $t_{(1)}, \ldots, t_{(n)}$ be the sample order statistics (the $t_{i} s$ rearranged in ascending order). To calculate the likelihood function we use (a) the density $\xi^{-1} \mathrm{e}^{-\mathrm{f} / \xi}$ for $t_{(1)}, \ldots, t_{(r)}$ (since their values are observed) and (b) the survivor function $e^{-t / \xi}$ evaluated at $t=t_{(r)}$ for $t_{(r+1), \ldots,} t_{(n)}$ (since we know only that their values exceed $t_{(r)}$ ). The log-likelihood function is now
$$l(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{(i)} / \xi}\right) \times \prod_{i=r+1}^{n}\left(\mathrm{e}^{-t_{(r)} / \xi}\right)\right}=-r \log \xi-\xi^{-1} t_{+}$$
which looks much the same as for Type-I censoring though now $r$ is nonrandom and $t_{+}=\sum_{i=1}^{r} t_{(i)}+(n-r) t_{(r)}$. The score function is $l^{\prime}(\xi)=-r \xi^{-1}+$ $t_{+} \xi^{-2}$ and the information function is $-l^{\prime \prime}(\xi)=-r \xi^{-2}+2 t_{+} \xi^{-3}$. The mle is $\hat{\xi}=t_{+} / r$, and its variance is approximated by $-l^{\prime \prime}(\hat{\xi})^{-1}=\hat{\xi}^{2} / r$.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Random Samples

L=\prod_{i=1}^{n}\left{f\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right )^{1-c_{i}}\right}=\prod_{i=1}^{n}\left{h\left(t_{i} ; \theta\right)^{c_{i}} F \left(t_{i} ; \theta\right)\right}L=\prod_{i=1}^{n}\left{f\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right )^{1-c_{i}}\right}=\prod_{i=1}^{n}\left{h\left(t_{i} ; \theta\right)^{c_{i}} F \left(t_{i} ; \theta\right)\right}

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Type-I Censoring

I(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{i} / \xi}\right ) \times \prod_{i=r+1}^{n} \mathrm{e}^{-a / \xi}\right}=-r \log \xi-\xi^{-1} t_{+ }I(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{i} / \xi}\right ) \times \prod_{i=r+1}^{n} \mathrm{e}^{-a / \xi}\right}=-r \log \xi-\xi^{-1} t_{+ }

l(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{(i)} / \xi} \right) \times \prod_{i=r+1}^{n}\left(\mathrm{e}^{-t_{(r)} / \xi}\right)\right}=-r \log \xi-\xi^{-1} t_{+}l(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{(i)} / \xi} \right) \times \prod_{i=r+1}^{n}\left(\mathrm{e}^{-t_{(r)} / \xi}\right)\right}=-r \log \xi-\xi^{-1} t_{+}

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