## The Annals of Statistics

### Consistent estimation of distributions with type II bias with applications in competing risks problems

#### Abstract

A random variable X is symmetric about 0 if X and -X have the same distribution. There is a large literature on the estimation of a distribution function (DF) under the symmetry restriction and tests for checking this symmetry assumption. Often the alternative describes some notion of skewness or one-sided bias. Various notions can be described by an ordering of the distributions of X and -X. One such important ordering is that $P(0<X<X\le x)-P(-x\le X<0)$ is increasing in $x<0$. The distribution of X is said to have a Type II positive bias in this case. If X has a density f, then this corresponds to the density ordering $f(-x)\le f(x)$ for $x<0$. It is known that the nonparametric maximum likelihood estimator (NPMLE) of the DF under this restriction is inconsistent. We provide a projection-type estimator that is similar to a consistent estimator of two DFs under uniform stochastic ordering, where the NPMLE also fails to be consistent. The weak convergence of the estimator has been derived which can be used for testing the null hypothesis of symmetry against this one-sided alternative. It also turns out that the same procedure can be used to estimate two cumulative incidence functions in a competing risks problem under the restriction that the cause specific hazard rates are ordered. We also provide some real life examples.

#### Article information

Source
Ann. Statist. Volume 32, Number 1 (2004), 245-267.

Dates
First available in Project Euclid: 12 March 2004

http://projecteuclid.org/euclid.aos/1079120136

Digital Object Identifier
doi:10.1214/aos/1079120136

Mathematical Reviews number (MathSciNet)
MR2051007

Zentralblatt MATH identifier
02113758

#### Citation

El Barmi, Hammou; Mukerjee, Hari. Consistent estimation of distributions with type II bias with applications in competing risks problems. Ann. Statist. 32 (2004), no. 1, 245--267. doi:10.1214/aos/1079120136. http://projecteuclid.org/euclid.aos/1079120136.

#### References

• Aly, E. A. A., Kochar, S. C. and McKeague, I. W. (1994). Some tests for comparing cumulative incidence functions and cause-specific hazard rates. J. Amer. Statist. Assoc. 89 994--999.
• Arcones, M. A. and Samaniego, F. J. (2000). On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint. Ann. Statist. 28 116--150.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Block, H. W. and Basu, A. P. (1974). A continuous bivariate exponential extension. J. Amer. Statist. Assoc. 69 1031--1037.
• Dykstra, R., Kochar, S. and Robertson, T. (1995). Likelihood ratio tests for symmetry against some one-sided alternatives. Ann. Inst. Statist. Math. 47 719--730.
• Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.
• Hall, W. J. and Wellner, J. A. (1979). Estimation of a mean residual life. Unpublished manuscript.
• Hoel, D. G. (1972). A representation of mortality data by competing risks. Biometrics 28 475--478.
• Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457--481.
• Kochar, S. C., Mukerjee, H. and Samaniego, F. J. (2000). Estimation of a monotone mean residual life. Ann. Statist. 28 905--921.
• Lévy, P. (1948). Processus stochastiques et mouvement Brownien. Gauthier-Villars, Paris.
• Lin, D. Y. (1997). Nonparametric inference for cumulative incidence functions in competing risks studies. Statistics in Medicine 16 901--910.
• Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space $D[0,\infty)$. J. Appl. Probab. 10 109--121.
• Moore, D. S. and McCabe, G. P. (1993). Introduction to the Practice of Statistics, 2nd ed. Freeman, New York.
• Mukerjee, H. (1996). Estimation of survival functions under uniform stochastic ordering. J. Amer. Statist. Assoc. 91 1684--1689.
• Rojo, J. and Samaniego, F. J. (1991). On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard. Statist. Probab. Lett. 11 267--271.
• Rojo, J. and Samaniego, F. J. (1993). On estimating a survival curve subject to a uniform stochastic ordering constraint. J. Amer. Statist. Assoc. 88 566--572.
• Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. (Corrections posted at www.stat.washington.edu/jaw/RESEARCH/ BOOKS/book1.html.)
• Stone, C. (1963). Weak convergence of stochastic processes defined on semifinite time intervals. Proc. Amer. Math. Soc. 14 694--696.
• Yanagimoto, T. and Sibuya, M. (1972). Test of symmetry of a one-dimensional distribution against positive biasedness. Ann. Inst. Statist. Math. 24 423--434.