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Abstract

A new three-sample nonparametric estimation of ordered survival functions is proposed based on three independent samples from distribution functions F(x), G(x) and H(x), which satisfy the stochastic order constraints $\bar{F} (x) \leq \bar{G} (x) \leq \bar{H} (x)$ for all $x \geq 0$ . The proposed estimators are computationally simpler than the other existing estimators and exhibit comparable mean squared errors. Such an estimation procedure is expected to have potential applications in connection with quality improvement plans (involving estimation of survival probabilities of initial, improved and ideal versions of engineered products) and clinical trials (involving estimation of survival functions of controlled group, low-dose and high-dose groups). The strong uniform consistency of the proposed estimators is established, and related weak convergence results are also derived. A simulation study is conducted for the performance evaluation of the proposed estimators. Illustrative examples involving real-life datasets are also presented. Extensions to the case of randomly right-censored observations and to the general k-ordered populations have also been highlighted.

AMS Subject Classification: 62G05, 62G30, 62N02

Keywords

Survival functions stochastic orders nonparametric estimation strong uniform consistency weak convergence

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References

Lehmann

Ordered families of distributions. Ann Math Stat 1955; 26(3): 399–419.

Müller

and Stoyan

Comparison methods for stochastic models and risks . Chichester: John Wiley; 2002

Marshall

and Olkin

Life distributions . New York: Springer; 2007.

Shaked

and Shanthikumar

Stochastic orders . New York: Springer; 2007.

Brunk

, Franck

, Hanson

and Hogg

Maximum likelihood estimation of the distributions of two stochastically ordered random variables. J Am Stat Assoc 1966; 61(316): 1067–1080.

Dykstra

Maximum likelihood estimation of the survival functions of stochastically ordered random variables. J Am Stat Assoc 1982; 77(379): 621–628.

S-H

Estimation of distribution functions under order restrictions. Stat Risk Model 1987; 5(3-4): 251–262.

Feltz

and Dykstra

Maximum likelihood estimation of the survival functions of n stochastically ordered random variables. J Am Stat Assoc 1985; 80(392): 1012–1019.

Dykstra

, Kochar

and Robertson

Statistical inference for uniform stochastic ordering in several populations. Ann of Stat 1991; 19(2):870–888.

10.

Rojo

and Ma

On the estimation of stochastically ordered survival functions. J Stat Comput Simul 1996; 55(1-2): 1–21.

11.

Puri

and Singh

Estimation of a distribution function dominating stochastically a known distribution function. Aust J Stat 1992; 34(1): 31–38.

12.

Rojo

On the estimation of survival functions under a stochastic order constraint. Lect Notes 2004; 44: 37–61.

13.

Barmi

HEI

and Mukerjee

Inferences under a stochastic ordering constraint: the k-sample case. J Am Stat Assoc 2005; 100(469): 252–261.

14.

Park

, Taylor

JMG

, and Kalbfleisch

Pointwise nonparametric maximum likelihood estimator of stochastically ordered survivor functions. Biometrika 2012; 99(2): 327–343.

15.

Kaur

and Singh

Estimation of a distribution function bounded by two known distribution functions. Commun Stat-Theory Meth 1990; 19(9): 3343–3360.

16.

Kaplan

and Meier

Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958; 53(282): 457–481.

17.

Stute

and Wang

J-L

The strong law under random censorship. Ann Stat 1993; 21(3): 1591–1607.

18.

Breslow

and Crowley

A large sample study of the life table and product limit estimates under random censorship. Ann Stat 1974; 2(3): 437–453.

19.

Földes

and RejtöL

LIL type result for the product limit estimator. Z Wahrscheinlichkeitstheorie Geb 1981; 56(1): 75–86.

20.

Lee

PSC-7-mouse changeover experiment A.1.5.6. Statistical analysis of visible skin tumor data (Tobacco Advisory Council Document TA 51) . London: Tobacco Research Council; 1979.

21.

Billingsley

Convergence of probability measures . John Wiley & Sons; 2013.

22.

Rojo

On the weak convergence of certain estimators of stochastically ordered survival functions. J Nonparam Stat 1995; 4(4): 349–363.

A Simple Three-sample Estimation of Survival Functions Under Order Restrictions

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