My introduction to copulas

Roger Nelsen is Professor Emeritus of Mathematics at Lewis & Clark College in Portland, Oregon, USA. He studied mathematics at DePauw University (BA, 1964) and Duke University (PhD, 1969). Roger joined the faculty at Lewis & Clark in the fall of 1969, and retired in 2009. Prior to Lewis & 
Clark, Roger spent a year with the Biostatistics Unit of the Centre International de Recherche sur le Cancer in Lyon, France. He has had visiting appointments at the University of Massachusetts in Amherst and Mount Holyoke College in South Hadley, Massachusetts. In addition to his monograph 
"An Introduction to Copulas", Roger has authored or co-authored eleven books published by the Mathematical Association of America. He has served on the editorial boards of two MAA journals and several of their book series. 
The fifth interview of this series features a conversation with Roger Nelsen. His Springer book An Introduction 
to Copulas is eponymous for a mathematical precise and well written entrance into the fascinating field of copulas. Moreover, it serves as a classical reference book, a large number of citations bear witness to this. He collaborated with seminal researchers in our field, and kindly shares his view and memories with us. Roger’s second strand of research – a combination of mathematical puzzles, art, and visualization – are “Proofs Without Words.” He published several books, containing elegant visual justifications of mathematical 
statements. In the following, our questions to Roger Nelsen are typeset in bold-face.

tions for their "sabbatical lectureship" program, a program for small college faculty wishing to spend a year in a research institution. I applied. I was accepted, and drove east. When I arrived in Amherst, I went to the ofce of the Director of the Sabbatical Lectureship Program to introduce myself and o er thanks. The Director? Professor Berthold Schweizer. We talked a bit, he asked about my interests, I mentioned probability. He took a book o his shelf, handed it to me, and said that I might enjoy reading Chapter 6. The book? Probabilistic Metric Spaces by B. Schweizer and A. Sklar, published earlier that year; see [34] (reprinted in [35]). The title of Chapter 6? Copulas. I did enjoy reading Chapter 6 (and more). It changed the direction of my career.
Perhaps, nowadays, not everyone in dependence modeling has heard about Probabilistic Metric Spaces. Could you brie y explain what they are and why copulas appear in this context?
Sure. Recall that (informally) a metric space consists of a set S and a metric d that measures "distances" between points, say p and q, in S. In a probabilistic metric space, we replace the distance d(p, q) by a distribution function Fpq, whose value Fpq(x) for any real x is the probability that the distance between p and q is less than x. The rst di culty in the construction of a probabilistic metric space arises when one tries to nd a probabilistic analog of the triangle inequality d(p, r) ≤ d(p, q) + d(q, r), that is, what is the corresponding relationship among the distribution functions Fpr, Fpq , and Fqr for all p, q, and r in S? Karl Menger (Bert Schweizer's PhD advisor) proposed

Fpr(x + y) ≥ T(Fpq(x), Fqr(y)),
where T is a triangular norm, or t-norm (see [24]). Like copulas, t-norms map [ , ] to [ , ] and some of them can be used to join distribution functions. In fact, a t-norm is a bivariate copula if and only if it is 2-increasing, and a bivariate copula is a t-norm if and only if it is commutative and associative (see, e.g., [1,23]). So, in a sense, it was inevitable that copulas would arise in the study of probabilistic metric spaces.
Do you have a scienti c role model? Without a doubt the role model and mentor in my adult life was Bert Schweizer; see Figure 2. The sabbatical year at UMass was just the beginning of a long and productive relationship. I will forever be grateful for his wise counsel and guidance.
You have worked for a long time in a college and have been clearly very active in research. Are these two activities compatible? Or was your experience quite singular?
I think the two are quite compatible, for persons who enjoy both research and teaching. When I rst started to teach at Lewis & Clark, research was encouraged but not required. The teaching load was 6 courses a year, which took up most of one's time (there were no graduate students at the College, and still aren't). But during the late 70s and 80s, that changed. The teaching load decreased to 5 courses, and the expectation shifted towards the idea that to be an e ective and inspiring teacher, one needs to be active in research. I agree; it makes one's teaching better. It is common now for the math faculty at Lewis & Clark to do publishable joint work with undergraduate students.
Being in academia comes with the bene t of traveling to exotic places and meeting interesting people. What journeys (e.g. to conferences) were particularly remarkable and what meetings with people do you remember with pleasure?
After the 1983-84 sabbatical year, I returned to Massachusetts in the fall of 1986 for a one semester visit to Mount Holyoke College in South Hadley. That semester began with the 24th International Symposium on Functional Equations (ISFE), held at Mount Holyoke. It was at that meeting where I met Claudi Alsina, Juan Quesada Molina, Carlo Sempi, Howard Sherwood, and Mike Taylor. Abe Sklar and Jerry Frank were also there; I had met them during the sabbatical year. I think it was later that fall that I met Christian Genest (see [10]) when he came to UMass to present a talk. Another memorable conference was the rst of the symposia on distributions with given marginals, held in Rome in April 1990; see the proceedings in [7]. At that conference I met Giorgio Dall'Aglio (see [11]), Ingram Olkin, Al Marshall, Carles Cuadras, Sam Kotz, Ludger Rüschendorf (see [9]), Marco Scarsini, José Antonio Rodríguez Lallena, and many others.
Many times you have collaborated with colleagues in Spain. How did this collaboration start? As I mentioned earlier, I had met Claudi Alsina, Juan Quesada Molina, and José Antonio Rodríguez Lallena at conferences in 1986 and 1990. In May 1995 the 33rd ISFE was held in Caldes de Malavella, Spain (near Barcelona), and after that conference I travelled to Granada to visit Juan. We had collaborated earlier with Bert Schweizer and Carlo Sempi on the paper [30] via email. José Antonio had completed his PhD a few years earlier under Juan's direction (see [33]), and had joined the faculty at the University of Almería. It may have been at that time (or perhaps a subsequent visit to Granada) that I met José Antonio's student Manuel (Manolo) Úbeda Flores, and accepted their invitation to be a co-advisor for Manolo's thesis [36]. I consider this a special honor, as I have never held a position in a PhD-granting institution. Su ce it to say that I think the four of us (Juan, José Antonio, Manolo, and me) have made a pretty good team.
Besides research, what are your recommendations for a trip to Spain? My recommendations for visiting Spain? First of all, do visit! Spain is a country of contrasts, be sure to visit Andalusia and Catalonia. I have yet to visit the Basque country and Galicia, but they are on my bucket list. When you go, spend some time, at least two or three months. The art, architecture, music, food, wine, all are great, but be sure to get to know the people. At Lewis & Clark we encourage our students to spend at least one semester studying abroad, and about two thirds of them do so. Many of the College's overseas programs utilize a faculty member to lead the program, and so after the visit to Spain in 1995 I studied Spanish and had the good fortune to lead our program in Sevilla three times, in 1999, 2003, and 2009. Each time I stayed in Europe after the program ended, and visited colleagues in Spain and Italy.

Can you describe a recipe for becoming a great researcher: are the ingredients technical skills, dedication, and having ideas? In which proportion? To which extent does luck or perhaps even "relational skills" play a role?
No, not really. I don't consider myself a "great researcher." But I'll give you my recipe anyway. It's simple: who you know is just as important as what you know. Attend conferences, meetings, workshops, etc. Meet people. Cultivate collaborators and coauthors. Luck does play a role -I've been blessed to be in the right place at the right time to meet my collaborators. I think of mathematical research as a team sport -and I've been fortunate to have teammates from Spain, Italy, Belgium, Mexico, Brazil, Argentina, Turkey, and of course the U.S. The work I feel the best about is the work I've done with others.
Where do you usually get your inspirations and what was the most improbable place to have a mathematical idea?
The inspirations most often come from discussions with colleagues. There is something to the adage that "the whole is greater than the sum of its parts" when it comes to mathematical research! The most improbable place for an inspiration was undoubtedly on a beach in Almería one beautiful afternoon. I was in Almería with Manolo, José Antonio, and Juan working on a paper on properties of quasi-copulas. It was our habit to work together in the morning, then go our separate ways in the afternoon. One afternoon Juan and I went to the beach. We had been searching for an example of (or a counterexample to) one suspected property of quasicopulas that morning, and were still thinking about it that afternoon. Without paper or a black board, we had only the sand to draw support diagrams -and that's the way we found the example (or counterexample) we were searching for.

What inspired you to write An Introduction to Copulas ([28], second edition in [29])?
I was in Amherst in November 1995 to attend a symposium at UMass celebrating Bert Schweizer's retirement. At a co ee break after one session with several talks about copulas, someone asked several of us what one should read to learn more about them. We mentioned several research papers and conference proceedings. I think I then mentioned that perhaps the time had come for "someone" to write an introductory monograph on copulas. A colleague, I forget whom, responded "Good idea Roger, why don't you write it?" I let the comment lie dormant until the following September when I was in Prague for the third conference on distributions with given marginals; see [3]. I asked Abe Sklar, Giorgio Dall'Aglio, and Ingram Olkin if they thought there might be interest among statisticians for such a book. They responded positively, and knowing that I was eligible for a sabbatical in 1997-1998, I began to give the idea of writing the book some serious thought.
Your book contains a long, analytical proof of Sklar's theorem. Where does the proof come from? Was Abe involved in this part of the book?
The outline for the proof comes from the book Probabilistic Metric Spaces by Bert and Abe. There it is given for the n-dimensional case, but it is incomplete. I re-worked it for bivariate copulas, and Bert and Abe helped me ll in some of the gaps.
Which researchers inspired and supported you during this project? I wrote most of the book during my sabbatical leave in 1997-1998. I am grateful to my colleagues at Mount Holyoke College for graciously inviting me to spend the year with them. One advantage to working at Mount Holyoke was that it is only about ten miles south of UMass -and Bert Schweizer.
Was it di cult to nd a publishing house?
No. At the time I was working on the book Ingram Olkin was one of the editors of Springer's Lecture Notes in Statistics Series, and he encouraged me to submit the manuscript to Springer.
Many notations and names of copula families have been introduced in your book and became standard in the community. Why did you use the symbols W, Π, and M for the three basic copulas?
Most of the names come from the literature, either from the authors (e.g., Ali-Mikhail-Haq), or the name used by the authors for the functions (e.g., ordinal sums and shu es of M). About the symbols M, W, and Π for copulas: they are the symbols used in the book Probabilistic Metric Spaces (see [34]). The symbols Π and M obviously stand for product and minimum. I think W is an upside-down M, since as a bivariate copula the graph of W is a twisted re ection in Π of the graph of M (see eq. 5.6.5 in [34]).
In your book, you presented a long list of bivariate copula families, some of them named after one or more colleagues who have worked on these. What is your "favorite" family?
It's probably the family known as Shu es of M (see [25]). I've found it to be quite useful for nding counterexamples to certain implications concerning dependence properties, and to show that certain inequalities are best-possible. It's also easy to draw diagrams of their support.
Would you share some anecdotes about your book? I can't tell you how many times I've been told that it's not a statistics book (i.e., no data), in spite of its appearance in the Springer Series in Statistics! How many copies of your book have you signed? Not too many. Maybe a dozen -maybe two dozen, no more. What was your worst experience with the review process of any of your papers? Did you experience that it became easier to get your papers published after your book on copulas became a standard reference?
At rst it was di cult to nd the right journal for early papers on copulas with Bert. When we submitted the paper to a statistics journal, the rejection letter often stated that the paper was "too mathematical"; when submitted to a mathematics journal, "too statistical." This was not surprising, as the copula concept does not appear to have been well known in the statistics community at that time. The rst appearance of "copula" as a keyword in the Current Index to Statistics was in 1981, and in the rst 18 volumes of the Index, there were only 11 references (total, not per year) to papers about copulas. That has clearly changed, and the book may have played a role. According to the Google Scholar website, the book in its two editions has been cited 10,000 times (as of 2017). That's hard to believe -makes me think that one should not trust data one nds on the Web.

Copulas
When did you realize that copulas have become so popular?
About the time when I began to see papers with applications in nance, actuarial science, hydrology, etc.; one of the early papers concerning applications is [4]. An even earlier paper is [22]. I discussed copulas with Bob Clemen when he was at the University of Oregon; Jouini was his PhD student at the time.

How was your collaboration with Abe Sklar and Bert Schweizer in the early age of copulas?
Most of the early collaboration was with Bert and those who had worked with Bert such as Jerry Frank and Claudi Alsina. Although Abe and I often discussed issues related to copulas, we don't have a joint paper, something I really regret.

Most young scientists have not had the chance to meet Abe Sklar. What kind of scientist is he? Is he surprised/happy that "his theorem" has become so famous?
Abe is a remarkable person. He is not only a ne mathematician, he is a warm, friendly, and kind person. He has a dry but keen sense of humor. His writing is precise and to the point. I suspect that he is pleased about the theorem -not so much that it is "famous," but that it has found so many applications.
Is it correct to say that you started with an analytical view on copulas? During your career, has this ever changed to a probabilistic or statistical perspective?
The analytic view is no doubt due to the fact that my rst exposure to copulas was via Bert and Abe's book Probabilistic Metric Spaces. As I've learned more about copulas, my view has become more multidisciplinary. I don't see the three perspectives you mention as being mutually exclusive.
Why did it take many years before copulas were appreciated in statistics and applications? What was the triggering event (if any) that has changed the development of this concept?
As I mentioned earlier, it was di cult at rst to nd journals interested in publishing papers about copulas. The copula concept was not new. It is present in the work of Maurice Fréchet [15], Giorgio Dall'Aglio [6], Robert Féron [12], Wassily Hoe ding [19,20], and later Paul Deheuvels [8], Janos Galambos [17], and others. I suspect that the sequence of conferences on distributions with xed or given marginal, beginning with the one in Rome in 1990, played a role in bringing together researchers from many countries.
It seems that you like to represent graphically mathematical concepts (see the PWW section below). In fact, some of your works are related to "geometric" constructions of copulas. When did you start considering such constructions? What was the main motivation behind?
You are correct; I like to illustrate mathematical concepts geometrically when it's possible. The geometric interpretation of positive quadrant dependence for two random variables -the graph of their copula lies on or above the graph of Π -led me ask if there are similar geometric interpretations of other dependence properties. The motivation for me was simple: if I can visualize something I'm more likely to understand it.
In one of your articles with Frank and Schweizer you presented a solution of the best-possible bounds for the sum of two random variables; see [14]. How did you arrive at the problem? Have you ever suspected that the problem would become so popular in risk management years later?
When I arrived in Amherst in the fall of 1983 Bert and Jerry had already started on that paper. After I learned a bit about copulas and triangle functions (Chapters 6 and 7 in Probabilistic Metric Spaces), they invited me to join them on the paper, part of my training in working with copulas. At the time we had no idea that copulas would nd all their current applications.

In one of your articles with Alsina and Schweizer, you introduced the concept of quasi-copulas. What was your original motivation?
The article to which you refer is the 1993 Statistics & Probability Letters paper characterizing a class of certain binary operations on distribution functions derivable from functions on random variables; see [2]. We discovered that the operation in the characterization was somewhat more general than a bivariate copula. The operation is "almost" a copula, hence our choice of the name quasi-copula. Our de nition in that paper is cumbersome, and it was soon replaced by a much better one in a 1999 paper by Christian Genest, Juan Quesada Molina, José Antonio Rodríguez Lallena, and Carlo Sempi; see [18].
You contributed to popularizing Archimedean copulas. Why are they so popular in your opinion? I think Archimedean copulas are "popular" for several reasons. First, they are rather easy to construct from univariate functions (the generators). Secondly, members of the Archimedean family exhibit a wide variety of dependence structures. Although the basic Archimedean copulas are bivariate with a single real parameter, many can be extended to yield multivariate copulas, and it is often possible to add parameters for more exibility.
In your book you listed 22 families of Archimedean copulas. How did you select them? Why not 23 or 31?
When I was working on the rst edition of the copulas book, Claudi Alsina, Jerry Frank, and Bert Schweizer were working on their book about associative functions on intervals (see [1]). They kindly shared a draft of that book with me, which included a table of 25 families of one parameter triangular norms. Many (but not all) triangular norms are copulas, and Claudi, Jerry, and Bert permitted me to adapt their table, using the families in their list that included copulas. I did not include two of their families in which there were no copulas and one family with very few copulas.
You "translated" many dependence concepts into the language of copulas. Now, while independence is clear, dependence has many facets. How would you explain "dependence" ?
For me it's like the "negative" de nition of an irrational number -it's a real number that is not rational. Similarly dependence has a "negative " de nition -it is any relation between random variables other than independence. Since (in the continuous setting) each pair of independent random variables has the copula Π, dependent pairs are those with any copula other than Π.
Is there any problem related to dependence/association where copulas should not be used? I must admit I'm not a fan of the normal copula. It seems to me that many of the important properties of the bivariate normal distribution are lost when the margins are replaced by non-normal distribution functions. But I'm not a statistician, and I'm sure many researchers would disagree with me.
What is the mathematical result/scienti c paper you are most proud of? As I recall, pride is one of the seven deadly sins, so I hesitate to answer this question! However, there are a couple of results in my papers that I rather like. One is in the 2005 Comptes Rendus paper [31] with Manolo Úbeda Flores, showing that the set of bivariate quasi-copulas is the lattice-theoretic completion of the set of bivariate copulas. This paper grew out of a speculation Bert Schweizer made at Manolo's thesis defense in 2001. Another is the result in the 2007 JSPI paper [16] with Greg Fredricks, where we found su cient conditions on the copula for the values of the population version of Spearman's rho to be about 50% larger than those for Kendall's tau for distributions near independence.
What is the copula or dependence result you would have expected to be very di cult but has been solved? What is the copula or dependence related result you would have expected to be very easy but still is open?
Excellent questions, but hard to answer. What you nd to be easy, I probably nd to be di cult. But rarely the reverse.

Proofs Without Words
One of your favorite activities is to create proofs that explain mathematical statements without words. In short: "Proofs Without Words" (PWWs). How did this interest arise?
In the fall of 1975 the rst "proofs without words" appeared in Mathematics Magazine, see [21], and soon PWWs were appearing regularly as end-of-article ller. Like lots of readers, I enjoyed the mental challenge of guring out how a PWW illustrates the truth of a statement and hints at its more formal proof. I thought that perhaps I could create PWWs, and after a number of disappointing e orts, I succeeded in having my rst PWW appear in the Magazine in 1987; see [26] and [27]. Collecting and creating PWWs continues to be an enjoyable hobby.
Is creating a PWW more art or science? Probably a combination of the two. For me it is like solving a puzzle. Here's an example: Theorem: The area of a regular dodecagon inscribed in a unit circle is 3. The puzzle? Dissect the dodecagon into a nite number of pieces and reassemble them to form three unit squares. And do it with not too many pieces (that's the science). Some symmetry in the solution would look nice, too (that's the art).

What is your favorite PWW?
Currently it is the one I've just described. My favorites are those that are totally visual, with no labels or equations. Here's the PWW (see Figure 6) for the puzzle in the answer to the preceding question (which appeared in the January 2015 issue of the College Mathematics Journal). I hope readers noticed that the reassembly of the pieces into squares requires only translations; no rotations or re ections are needed.
What do you think about the argument that Archimedes -on purpose! -did not draw geometric gures in the correct scale and reduced his diagrams to their topological features, in order to avoid drawing the wrong conclusions from a speci c gure (See Ch. 4

in [32])?
I think George Pólya would agree, when he wrote "Geometry is the art of correct reasoning on incorrect gures." And maybe so did Jean Dieudonné when he wrote "I have decided to introduce not a single gure in the text." But yet Pólya's advice in his classic book How to Prove It is "Draw a gure." As I wrote in the introduction to the rst collection of PWW, if PWW are not proofs, what are they? They are pictures or diagrams that help one see why a particular statement is true, and how one might go about proving it is true.
Is there any PWW in the eld of copulas/dependence modeling? I haven't seen one yet. But I haven't given up looking for one, either.

What activities do you enjoy after work?
Being retired I don't consider anything I currently do "work." I enjoy travel, especially visiting my sister in Colorado and her sons and grandchildren, and visiting my brother and his wife and their sons and grand-children in Texas. I really enjoy going to conferences overseas, not just for the mathematics, but also to visit good friends, and make new ones. Here in Portland, I enjoy outdoor activities, visits to the Oregon Coast, the Columbia Gorge, and to the Cascades. For many years I had a small sailboat (named the Aftermath) on the Willamette River, and enjoyed sailing with friends until the boat disintegrated recently. Maybe I should get a new one.
A nal question: Will you miss academia and what do you wish the copula community for the (near and far) future?
I am very grateful to Lewis & Clark for letting me continue to be a member of the math department with an o ce, library privileges, internet, etc. So in a sense I don't miss academia-not yet. My wish for the copula community is simple: I wish each member will enjoy the collegiality of this community and its many friendships as I have, and have as much pleasure working with others in the community as I have enjoyed the past 30-plus years. It is a remarkable community, and I am honoured to be part of it.