zbMATH — the first resource for mathematics

Consistent estimation of distributions with type II bias with applications in competing risks problems. (English) Zbl 1105.62326
Summary: 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 $$P(0<X<X\leq x)-P(-x\leq X<0)$$ is increasing in $$x<0$$. The distribution of $$X$$ is said to have aType II positive bias in this case. If $$X$$ has a density $$f$$, then this corresponds to the density ordering $$f(-x)\leq f(x)$$ 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.

MSC:
 62G05 Nonparametric estimation 60E15 Inequalities; stochastic orderings 62G20 Asymptotic properties of nonparametric inference 62G10 Nonparametric hypothesis testing
Full Text:
References:
 [1] 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. · Zbl 0804.62091 [2] 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. · Zbl 1106.62332 [3] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201 [4] Block, H. W. and Basu, A. P. (1974). A continuous bivariate exponential extension. J. Amer. Statist. Assoc. 69 1031–1037. · Zbl 0299.62027 [5] 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. · Zbl 0843.62066 [6] Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York. · Zbl 0727.62096 [7] Hall, W. J. and Wellner, J. A. (1979). Estimation of a mean residual life. Unpublished manuscript. · Zbl 0417.62013 [8] Hoel, D. G. (1972). A representation of mortality data by competing risks. Biometrics 28 475–478. [9] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457–481. · Zbl 0089.14801 [10] Kochar, S. C., Mukerjee, H. and Samaniego, F. J. (2000). Estimation of a monotone mean residual life. Ann. Statist. 28 905–921. · Zbl 1105.62379 [11] Lévy, P. (1948). Processus stochastiques et mouvement Brownien. Gauthier-Villars, Paris. · Zbl 0034.22603 [12] Lin, D. Y. (1997). Nonparametric inference for cumulative incidence functions in competing risks studies. Statistics in Medicine 16 901–910. [13] Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space $$D[0,\infty)$$. J. Appl. Probab. 10 109–121. · Zbl 0258.60008 [14] Moore, D. S. and McCabe, G. P. (1993). Introduction to the Practice of Statistics , 2nd ed. Freeman, New York. · Zbl 0701.62002 [15] Mukerjee, H. (1996). Estimation of survival functions under uniform stochastic ordering. J. Amer. Statist. Assoc. 91 1684–1689. · Zbl 0885.62115 [16] 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. · Zbl 0712.62032 [17] 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. · Zbl 0773.62030 [18] 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.) · Zbl 1170.62365 [19] Stone, C. (1963). Weak convergence of stochastic processes defined on semifinite time intervals. Proc. Amer. Math. Soc. 14 694–696. · Zbl 0116.35602 [20] Yanagimoto, T. and Sibuya, M. (1972). Test of symmetry of a one-dimensional distribution against positive biasedness. Ann. Inst. Statist. Math. 24 423–434. · Zbl 0347.62023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.