Inside Interesting Integrals

In some of my research - notably that relating to statistical distribution theory, and that in Bayesian econometrics - I spend quite a bit of time dealing with integration problems. As I noted in this recent post, integration is something that we really can't avoid in econometrics - even if it's effectively just "lurking behind the scenes", and not right in our face.

Contrary to what you might think, this can be rather interesting!

We can use software, such as Maple, or Mathematica, to help us to evaluate many complicated integrals. Of course, that wasn't always so, and in any case it's a pity to let your computer have all the fun when you could get in there and get your hands dirty with some hands-on work. Is there anything more thrilling than "cracking" a nasty looking integral?

I rely a lot on the classic book, Table of Integrals, Series and Products, by Gradshteyn Ryzhik. It provides a systematic tabulation of thousands of integrals and other functions. I know that there are zillions of books that discuss various standard methods (and non-standard tricks) to help us evaluate integrals. I'm not qualified to judge which ones are the best, but here's one that caught my attention some time back and which I've enjoyed delving into in recent months.

It's written by an electrical engineer, Paul J. Nahin, and it's called Inside Interesting Integrals.

I just love Paul's style, and I think that you will too. For instance, he describes his book in the following way -

"A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and Devilishly Seductive Maneuvers for Computing Nearly 200 Perplexing Definite Integrals From Physics, Engineering, and Mathematics. (Plus 60 Challenge Problems with Complete, Detailed Solutions.)"

Well, that certainly got my attention!

And then there's the book's "dedication":

"This book is dedicated to all who, when they read the following line from John le Carre´’s 1989 Cold War spy novel The Russia House, immediately know they have encountered a most interesting character:

'Even when he didn’t follow what he was looking at, he could relish a good page of mathematics all day long.'

as well as to all who understand how frustrating is the lament in Anthony Zee’s book Quantum Field Theory in a Nutshell:

'Ah, if we could only do the integral … . But we can’t.' "

What's not to love about that?

Take a look at Inside Interesting Integrals - it's a gem.

© 2016, David E. Giles

My Favourite Book

Well, perhaps it's not really my favourite book, but it's certainly right up there with the most heavily thumbed tomes on my office bookshelf.

I'm referring to Tables of Integrals, Series and Products, by Gradshteyn and Ryzhik. I picked up a used copy of the 4^th ed. (1965) for about $5 some years ago at Powell's bookstore in Portland, and it's saved me more anguish and time than I can possibly estimate.

For example (click for LARGER version):

(Sample page)

The book is now in its 8^th edition (2014). You can download the 7th ed. (2007) on a pay-by-the-chapter basis from here, and you should be aware of the associated errata document.

I also came across this link.

© 2015, David E. Giles

Econometricians' Debt to Alan Turing

The other day, Carol and I went with friends to see the movie, The Imitation Game. I definitely recommend it.

I was previously aware of many of Alan Turing's contributions, especially in relation to the Turing Machine, cryptography, computing, and artificial intelligence. However, I hadn't realized the extent of Turing's use of, and contributions to, a range of important statistical tools. Some of these tools have a direct bearing on Econometrics.

For example:

• (HT to Lief Bluck for this one.) In 1935, at the tender age of 22, Turing was appointed a Fellow at King's College, Cambridge, on the basis of his 1934 (undergraduate) thesis in which he proved the Central Limit Theorem. More specifically, he derived a proof of what we now call the Lindeberg-Lévy Central Limit Theorem.  He was not aware of Lindeberg's earlier work (1920-1922) on this problem. Lindeberg, in turn, was unaware of Lyapunov's earlier results. (Hint: there was no internet back then!). How many times has your econometrics instructor waved her/his arms and muttered ".......as a result of the central limit theorem....."?
• In 1939, Turing developed what Wald and his collaborators would later call "sequential analysis". Yes, that's Abraham Wald who's associated with the Wald tests that you use all of the time.Turing's wartime work on this subject remained classified until the 1980's. Wald's work became well-established in the literature by the late 1940's, and was included in the statistics courses that I took as a student in the 1960's. Did I mention that Wald's wartime associates included some familiar names from economics? Namely, Trygve Haavelmo, Harold Hotelling, Jacob Marschak, Milton Friedman, W. Allen Wallis, and Kenneth Arrow.
• The mathematician/statistician I. J. ("Jack") Good was a member of Turing's team at Bletchley Park that cracked the Enigma code. Good was hugely influential in the development of modern Bayesian methods, many of which have found their way into econometrics. He described the use of Bayesian inference in the Enigma project in his "conversation" with Banks (1996). (This work also gave us the Good-Turing estimator - e.g., see Good, 1953.)
• Turing (1948) devised the LU ("Lower and Upper") Decomposition that is widely used for matrix inversion and for solving systems of linear equations. Just think how many times you invert matrices when you're doing your econometrics, and how important it is that the calculations are both fast and accurate!

Added, 20 February, 2015: I have recently become aware of Good (1979)

References

Banks, D. L., 1996. A conversation with I. J. Good. Statistical Science, 11, 1-19.

Good, I. J., 1953.The population frequencies of species and the estimation of population parameters. Biometrika, 40, 237-264.

Good, I. J., 1979. A. M. Turing's statistical work in World War II. Biometrika, 66, 393-396.

Turing, A. M., 1948. Rounding-off errors in matrix processes. Quarterly Journal of Mechanics and Applied Mathematics, 1, 287-308.

© 2014, David E. Giles

Mathgen

H/T to my colleague, Martin Farnham, for drawing my attention to Mathgen.

 

Thanks to Nate Eldridge, a mathematician at Cornell University, who blogs at That's Mathematics!, you can randomly generate your own mathematics research paper!

 

In fact, a Mathgen-generated was recently accepted for publication at one of those pseudo-journals that seem to have sprouted with a vengeance of late. If you weren't convinced already that these publishing outlets should be avoided like the plague, this ought to do it for you!

 

Just for funzies, I decided to solicit Mathgen's assistance in writing my own paper. It took just a few seconds, and you can read it here. Constructive comments are welcomed, of course. Just don't ask me what the title means.

I have a feeling that this is going to be a particularly productive weekend!

 

(As Martin suggested to me, this is every journal editor's new nightmare!)

 

© 2012, David E. Giles

Mathematics, Economics, & the Nobel Prize

With the announcement of this year's Nobel Prize in Economic Science less than a week away, here's a recent working paper that you'll surely enjoy: "The use of mathematics in economics and its effect on a scholar's academic career", by Miguel Espinosa, Carlos Rondon, and Mauricio Romero. (Be sure that you download the latest version - dated September 2012.)

Here's the abstract:

"There has been so much debate on the increasing use of formal mathematical methods in Economics. Although there are some studies tackling these issues, those use either a little amount of papers, a small amount of scholars or cover a short period of time. We try to overcome these challenges constructing a database characterizing the main socio-demographic and academic output of a survey of 438 scholars divided into three groups: Economics Nobel Prize winners; scholars awarded with at least one of six prestigious recognitions in Economics; and academic faculty randomly selected from the top twenty Economics departments worldwide. Our results provide concrete measures of mathematization in Economics by giving statistical evidence on the increasing trend of number of equations and econometric outputs per article. We also show that for each of these variables there have been four structural breaks and three of them have been increasing ones. Furthermore, we found that the training and use of mathematics has a positive correlation with the probability of winning a Nobel Prize in certain cases. It also appears that being an empirical researcher as measured by the average number of econometrics outputs per paper has a negative correlation with someone's academic career success." (Emphasis added; DG)

The first of the highlighted conclusions doesn't surprise me. I'm not sure that I like the second one, though!

© 2012, David E. Giles

Solving Mathematical Problems - the Tricki

Hat-tip to Sean Brocklebank (Economics, University of Edinburgh), through the Subgame Equilibrium blog, for pointing us to The Tricki. 

This is a Wiki site devoted to discussing and explaining the methods of proof that are used in various areas of mathematics. Probability and Statistics are among the fields covered, although as yet there are no entries for the second of these two particular sections.

In the Elementary Probability section I especially liked the entry on "Bounding Probabilities by Expectations", where there are some examples of using Markov's Inequality to good effect.

In short, there's a wealth of great information for students and teachers of Econometrics alike. I'll certainly be using it, and I'll be looking forward to seeing some entries in the Statistics section.

© 2012, David E. Giles

Close Encounters of the Math Kind

Alert readers of this blog may have noticed (front page) that I have an Erdös Number of 4. That's to say, I've published (several) papers co-authored with someone, who co-authored a paper with someone, who co-authored a paper with the mathematician Paul Erdös.