The Triangular Distribution

One of our goals in h s book is to "dig out" suitable substitutes of the beta distribution. Only recently (less than 10 years ago) has the triangular distribution spenfcalh been investigated by D. Johnson (1 997) as a proy for the beta distn'btltion, even though its origins can be traced back to Thomas Simpson (1755) (about one century after the discovery of the beta distribution in a letter from Sir Isaac Newton to Henry Oldenberg). Very recently a "Handbook of Beta Distributions1' edited by Gupta and Nadarajah (2004) has appeared (providing and emphasizing in a single monograph the attention that the beta distribution has attracted by both statistical theoreticians and practitioners over the last century, or so). On the other hand it appears that, in our opinion, the triangular distribution has been somewhat neglected in the statistical literature (perhaps even due to its simplicity whch may discourage research efforts). In this chapter, we shall attempt to provide some chronology regarding the history of t h s distribution, state some of its properties and describe methods for estimating its parameters. Although the exposition is certainly not complete, we hope that it becomes apparent that the triangular distributions' "simplicity" is to a certain extent wrongly perceived and these distributions and their extensions are certainly worthy of further investigations. Written records on the triangular distribution seem to originate in the middle of the 18-th century when problems of combinatorial probabhty were at their peak. A historically inclined reader may wish to consult the classical book by F.N. David (1962). One of the earliest mentions of the