This week, in the MAT8181 Time Series course, we’ve discussed unit root tests. According to Wold’s theorem, if is  (weakly) stationnary then where is the innovation process, and where  is some deterministic series (just to get a result as general as possible). Observe that as discussed in a ... [Read more...]

I am currently working on a paper on Academic Blogging, from my own experience. And I wanted to do something similar to Stephen Wolfram’s personal analytics of my life. More specifically, I wanted to understand when I do post my blog entries. If I post more entries during office ... [Read more...]

As we mentioned in our previous post, as soon as we have a moving average part, inference becomes more complicated. Again, to illustrate, we do not need a two general model. Consider, here, some  process, where  is some white noise, and assume further that . __ theta=.7 __ phi=.5 __ n=1000 __ Z=rep(0,n) __ ... [Read more...]

Yesterday, we’ve seen how inference for time series was possible.  I started  with that one because it is actually the simple case. For instance, we can use ordinary least squares. There might be some possible bias (see e.g. White (1961)), but asymptotically, estimators are fine (consistent, with asymptotic normality). ... [Read more...]

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[t]=phi1*Z[t-1]+phi2*Z[t-2]+e[t] __ Z=Z[800:1000] __ ... [Read more...]

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, ... [Read more...]

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 (... [Read more...]

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 graphar1(.8) or __ graphar1(-.7) (with, on the […] [Read more...]

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 ... [Read more...]

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 ... [Read more...]

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 ... [Read more...]

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 ... [Read more...]

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 ... [Read more...]

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 ... [Read more...]

With Stéphane Tufféry we’ve been working on credit scoring1 and we’ve been using the popular german credit dataset, __ myVariableNames credit = read.table( + "http://archive.ics.uci.edu/ml/machine-learning-databases/statlog/german/german.data", + header=FALSE,col.names=myVariableNames) __ credit$class library(RColorBrewer) __ CL=brewer.pal(6, "RdBu") __ ... [Read more...]

Most of the time, when we introduce binomial models, such as the logistic or probit models, we discuss only Bernoulli variables, . This year (actually also the year before), I discuss extensions to multinomial regressions, where  is a function on some simplex. The multinomial logistic model was mention here. The idea ... [Read more...]

Thursday, I got an interesting question from a colleague of mine (JP). I mean, the way I understood the question turned out to be a nice puzzle (but I have to confess I might have misunderstood). The question is the following : consider a i.i.d. sample of continuous variables. ... [Read more...]

Today, I wanted to publish a post on generating functions, based on discussions I had with Jean-Francois while having our coffee after lunch a couple of times already. The other reason is that I publish my post while my student just finished their Probability exam (and there were a few ... [Read more...]

This morning, I was working with Julie, a student of mine, coming from Rennes, on mortality tables. Actually, we work on genealogical datasets from a small region in Québec, and we can observe a lot of volatiliy. If I borrow one of her graph, we get something like Since ... [Read more...]

This year, for Halloween, a post on candies (I promise, next year I will write another post on zombies). But I don’t want to focus on the kids problems (last year, we tried to minimize their walking distance to collect as much candies as possible, with part 1 and part 2), ... [Read more...]