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Given X1, ..,Xn iid from some univariate for now distribution, in this exercise we consider a seemingly trivial goal

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I expect you to upload your solutions on Moodle as a single running  R Markdown file (.rmd) + its .html output,  named with your surnames.

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Exercise: Estimating a population mean. . . in 2020. . .

1.   Introduction

Given {X1, . . . , Xn} iid from some (univariate for now) distribution, in this exercise we consider a seemingly trivial goal:

 estimate the population mean µ = E(X)  

An obvious choice would be the plug-in estimator, the empirical mean that you all know and love. . .

This estimator is computationally attractive, requires no prior knowledge, and automatically scales with the population variance

σ. In addition, tweaking a bit the Central Limit Theorem, we also know that result that also holds non-asymptotically under some suitable technical conditions. If these conditions are not met, we still have Chebyshev’s inequality, which says that with a probability of at least 1 − α an exponentially weaker bound that will especially hurt in modern applications where many means have to be estimated simultaneusly (e.g. empirical risk minimization methods).

One may in fact try to extend the MoM estimator to the multivariate case, only one problem: what is a median in the multivariate case?  Given n points   x1, . . . , xn    in Rd, the center of the smallest ball that contains at least  half of the points may be considered as a notion of a multivariate median. Computing such a median is totally a nontrivial problem! The multivariate MoM estimator may be defined as the geometric median of the sample means of the k blocks defined before. As in the univariate case, the theoretical optimal block number is k٨ = 8 log(1/α) . As a quick final remark, let me notice that, for the purpose of experimenting a bit, we could replace this last summary with any robust multivariate location estimator. 

1.On robust procedures and heavy-tailed distribution in Data Science

As mentioned, MoM estimators should have an edge on the beloved sample average in multivariate, heavy-tailed cases. At this point in time, heavy-tailed distributions have been accepted as realistic models for various phenomena:

•  www-session characteristics (e.g. sizes and durations of sub-sessions; sizes of responses inter-response time intervals)

•  on/off-periods of packet traffic

•  file sizes

•  service-time in queueing model

•  flood levels of rivers

•  major insurance claims

•  extreme levels of ozon concentrations

•  high wind-speed values

•  wave heights during a storm

•  low and high temperature

But there’s more. As you probably know, recent technological developments have allowed companies and state organizations to collect and store huge datasets. Big datasets have also challenged scientists in statistics and computer science to develop new methods. In fact, because of the very “unstructured” way in which these datasets are collected, oftentimes they tend to be corrupted by nasty outliers and/or exhibit heavy tails.

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