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Bounds for a joint distribution function with fixed sub-distribution functions: Application to competing risks.

A V Peterson1

  • 1Department of Statistics, Stanford University, Stanford, California 94305.

Proceedings of the National Academy of Sciences of the United States of America
|January 1, 1976
PubMed
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This study establishes precise bounds for joint and marginal survival functions, crucial for analyzing risks when sub-survival functions are known. These findings offer valuable tools for statistical modeling and risk assessment.

Area of Science:

  • Statistics
  • Survival Analysis
  • Risk Theory

Background:

  • Survival functions are fundamental in statistical analysis for modeling time-to-event data.
  • Joint and marginal survival functions describe the probability of multiple events occurring or not occurring over time.
  • Understanding these functions is critical in fields like reliability engineering, medical research, and finance.

Purpose of the Study:

  • To derive sharp bounds for joint survival functions G(t1, t2,...,tr) and marginal survival functions S(j)(t).
  • To establish these bounds under the condition that sub-survival functions S(j)*(t) are fixed.
  • To apply these bounds to competing risks problems and develop empirical bounds.

Main Methods:

  • Development of theoretical bounds for survival functions using mathematical inequalities.

Related Experiment Videos

  • Formulation of specific theorems for the cases of r=2 and general r.
  • Application of derived bounds to the context of competing risks analysis.

    • Main Results:

    • Sharp bounds were derived for joint survival functions P(X(1) > t1, ..., X(r) > tr).
    • Bounds were also established for marginal survival functions P(X(j) > t).
    • Empirical bounds were presented for competing risks scenarios based on observed data.

    Conclusions:

    • The study provides a rigorous mathematical framework for bounding survival functions.
    • The derived bounds are applicable to complex scenarios, including competing risks.
    • The results offer practical tools for estimating survival probabilities and assessing risks in various applications.