While researching a topic in probability, I encountered a special mathematical function that I had not previously known about. The function is called Owen's T function (Owen, 1956, Ann. Math. Stat.). It is useful for computing multivariate probabilities, for defining the skew-normal distribution, and for computing probabilities associated with certain
I write a lot of Monte Carlo simulations. This article discusses how to assess whether two independent Monte Carlo estimates are close to each other. This result can be used to test the correctness of a Monte Carlo simulation. It can also be used to compare different Monte Carlo algorithms
SAS has more than 25 common probability distributions that are supported in the PDF, CDF, QUANTILE, and RAND functions. If you want to work with a less common distribution, you can implement these functions yourself. For example, I previously showed how to use PROC FCMP in Base SAS to implement
The classical multivariate normal (MVN) distribution is a standard model for correlated data. It is a simple model, it is easy to fit the MVN model to data, and the parameters in the model (locations and correlations) are intuitive. Of course, normality is a strong assumption that is not always
In a previous article, I implemented an algorithm due to Niels Waller (TAS, 2020) that uses the method of alternating projections (MAP) to generate random correlation matrices that have a specified set of eigenvalues. The algorithm is iterative, and the MAP method is not guaranteed to converge, although Waller claims
One of the great algorithms of linear algebra is the Gram-Schmidt orthogonalization process, which enables you to construct an orthogonal basis for a linear subspace from any set of linearly independent vectors that span the subspace. The Gram-Schmidt process is the basis for the QR decomposition in numerical linear algebra,
A previous article discusses the problem of fitting a kernel density estimate (KDE) to data that are strictly positive. If you use a standard KDE, the resulting density curve might estimate non-zero probability for negative values. This is unsatisfactory because quantities like lengths and mass cannot be negative. The previous
The kernel density estimate (KDE) is a powerful tool for estimating the density of univariate data. The KDE is a flexible model. For example, it can fit data distributions that are multimodal or have long tails. It is nonparametric, which means that you do not need to assume any form
Pareto charts are used by quality engineers to visually display a set of causes that produce defects in a manufacturing process. The chart is based on the famous Pareto Principle (the 80/20 rule), which states that 20% of the causes often produce 80% of the defects. If you focus on
A recent article discussed Pareto charts and how to create them in SAS. A Pareto chart is used in quality control to identify the most likely reasons that a manufacturing process produces defective items. Suppose there are k different reasons why a process can fail, and you intend to randomly
