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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mi>F</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mover accent="true"> <mml:mi>G</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤</mml:mo> <mml:mover accent="true"> <mml:mi>H</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . 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