Nonparametric estimation of a distribution with type I bias with applications to competing risks

Abstract

A random variable X has a symmetric distribution about a if and only if X − a and −X + a are identically distributed. By considering various types of partial orderings between the distributions of X − a and −X + a, one obtains various types of partial skewness or one-sided bias. For example, F has Type I bias about a if F¯(a + x) ≥ F((a − x)−) for all x > 0; here F¯ = 1 − F. In this article we assume that a = 0, and propose a nonparametric estimator of a continuous distribution function F under the restriction that it has Type I bias. We derive the weak convergence of the resulting process which is used to test for symmetry against that type of bias. The new estimator is then compared with the nonparametric likelihood estimator (NPMLE), [Fcirc] _n , of F in terms of mean squared error. A simulation study seems to indicate that the new estimator outperforms the NPMLE uniformly at all the quantiles of the distributions that we have investigated.

It turns out that the results developed here could be used to compare two risks in a competing risks problem. We show how this can be done and illustrate the theory with a real life example.

Keywords:

• Type I bias
• Competing risks
• Nonparametric maximum likelihood
• Weak convergence

Acknowledgement

The authors thank the editor, a referee, and Professor Hari Mukerjee for helpful comments and suggestions that led to a much improved paper. The authors are especially grateful to the referee for his painstaking scrutiny that found several errors and many typos.