Blog Archives

Inference for AR(p) Time Series

January 28, 2014
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Inference for AR(p) Time Series

Consider a (stationary) autoregressive process, say of order 2, for some white noise with variance . Here is a code to generate such a process, > phi1=.25 > phi2=.7 > n=1000 > set.seed(1) > e=rnorm(n) > Z=rep(0,n) > for(t in 3:n) Z=phi1*Z+phi2*Z+e > Z=Z > n=length(Z) > plot(Z,type="l") Here, we have to estimate two sets of parameters: the autoregressive...

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Bias of Hill Estimators

January 28, 2014
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Bias of Hill Estimators

In the MAT8595 course, we’ve seen yesterday Hill estimator of the tail index. To be more specific, we did see see that if , with , then Hill estimators for are given by for . Then we did say that satisfies some consistency in the sense that if , but not too fast, i.e. (under additional assumptions on the...

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Causal Autoregressive Time Series

January 21, 2014
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Causal Autoregressive Time Series

In the MAT8181 graduate course on Time Series, we will discuss (almost) only causal models. For instance, with , with some white noise , those models are obtained when . In that case, we’ve seen that was actually the innovation process, and we can write which is actually a mean-square convergent series (using simple Analysis arguments on series). From that...

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Visualizing Autoregressive Time Series

January 21, 2014
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Visualizing Autoregressive Time Series

In the MAT8181 graduate course on Time Series, we started discussing autoregressive models. Just to illustrate, here is some code to plot  – causal – process, > graphar1=function(phi){ + nf <- layout(matrix(c(1,1,1,1,2,3,4,5), 2, 4, byrow=TRUE), respect=TRUE) + e=rnorm(n) + X=rep(0,n) + for(t in 2:n) X=phi*X+e + plot(X,type="l",ylab="") + abline(h=mean(X),lwd=2,col="red") + abline(h=mean(X)+2*sd(X),lty=2,col="red") + abline(h=mean(X)-2*sd(X),lty=2,col="red") + u=seq(-1,1,by=.001) + plot(0:1,0:1,col="white",xlab="",ylab="",axes=FALSE,ylim=c(-2,2),xlim=c(-2.5,2.5)) + polygon(c(u,rev(u)),c(sqrt(1-u^2),rev(-sqrt(1-u^2))),col="light yellow")...

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Statistical Interests in Large Cities

January 10, 2014
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Statistical Interests in Large Cities

I always thought that there were some kind of schools in statistics, areas (not to say universities or laboratories) where people had common interest in term of statistical methodology. Like people with strong interest in extreme values, or in Lévy Processes. I wanted to check this point so I did extract information about articles puslished in about 35 journals...

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Sequences defined using a Linear Recurrence

January 6, 2014
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Sequences defined using a Linear Recurrence

In the introduction to the time series course (MAT8181) this morning, we did spend some time on the expression of (deterministic) sequences defined using a linear recurence (we will need that later on, so I wanted to make sure that those results were familiar to everyone). First order recurence The most simple case is the first order recurence, where...

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Random points on some hemisphere

December 18, 2013
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Random points on some hemisphere

In my previous post, I tried to answer the following question Consider  points uniformly distributed on a sphere. What is the probability that the  points lie on a same hemisphere, for some hemisphere (there is no south or north here) ? If I have been able to use Monte Carlo simulations in dimension 2 (on a circle, not on a sphere), I could...

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Conditional dependence measures

December 17, 2013
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Conditional dependence measures

This week, I spend some time at the Workshop on Nonparametric Curve Smoothing conference at Concordia. Yesterday afternoon, Noël Veraverbeke show an interesting graph, to illustrate conditional copulas (and the derivation of conditional dependence measures, such as Kendall’s tau, or Spearman’s rho). A long time ago, in my PhD thesis (mainly on conditional copulas) I did try to derive conditional...

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On Wigner’s law (and the semi-circle)

December 16, 2013
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On Wigner’s law (and the semi-circle)

There is something that I love about mathematics: sometimes, you discover – by chance – a law. It has always been there, it might have been well known by some people (specialized in some given field), but you did not know it. And then, you discover it, and you start wondering how comes you never heard about it before… I...

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Random points on the Earth

December 7, 2013
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Random points on the Earth

The problem with puzzles is that you keep it in your head for days, until you find an answer. Or at least some ideas about a possible answer. This is what happened to me a few weeks ago, when a colleague of mine asked me the following question : Consider points uniformly distributed on a sphere. What is the...

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