After my recent articles on simulating data by using copulas, many readers commented about the power of copulas. Yes, they are powerful, and the geometry of copulas is beautiful. However, it is important to be aware of the limitations of copulas. This article creates a bizarre example of bivariate data,
Do you know what a copula is? It is a popular way to simulate multivariate correlated data. The literature for copulas is mathematically formidable, but this article provides an intuitive introduction to copulas by describing the geometry of the transformations that are involved in the simulation process. Although there are
A recent article about how to estimate a two-dimensional distribution function in SAS inspired me to think about a related computation: a 2-D cumulative sum. Suppose you have numbers in a matrix, X. A 2-D cumulative sum is a second matrix, C, such that the C[p,q] gives the sum of
This article shows how to estimate and visualize a two-dimensional cumulative distribution function (CDF) in SAS. SAS has built-in support for this computation. Although the bivariate CDF is not used as much as the univariate CDF, the bivariate version is still a useful tool in understanding the probable values of
This article uses simulation to demonstrate the fact that any continuous distribution can be transformed into the uniform distribution on (0,1). The function that performs this transformation is a familiar one: it is the cumulative distribution function (CDF). A continuous CDF is defined as an integral, so the transformation is
A previous article showed how to simulate multivariate correlated data by using the Iman-Conover transformation (Iman and Conover, 1982). The transformation preserves the marginal distributions of the original data but permutes the values (columnwise) to induce a new correlation among the variables. When I first read about the Iman-Conover transformation,
Simulating univariate data is relatively easy. Simulating multivariate data is much harder. The main difficulty is to generate variables that have given univariate distributions but also are correlated with each other according to a specified correlation matrix. However, Iman and Conover (1982, "A distribution-free approach to inducing rank correlation among
Many nonparametric statistical methods use the ranks of observations to compute distribution-free statistics. In SAS, two procedures that use ranks are PROC NPAR1WAY and PROC CORR. Whereas the SPEARMAN option in PROC CORR (which computes rank correlation) uses only the "raw" tied ranks, PROC NPAR1WAY uses transformations of the ranks,
For many univariate statistics (mean, median, standard deviation, etc.), the order of the data is unimportant. If you sort univariate data, the mean and standard deviation do not change. However, you cannot sort an individual variable (independently) if you want to preserve its relationship with other variables. This statement is
When data contain outliers, medians estimate the center of the data better than means do. In general, robust estimates of location and sale are preferred over classical moment-based estimates when the data contain outliers or are from a heavy-tailed distribution. Thus, instead of using the mean and standard deviation of
