Bi-s*-Concave Distributions

Area of Science:

• Statistics
• Probability Theory

Background:

• Distribution functions are fundamental in statistical modeling.
• Existing shape constraints like bi-log-concavity have limitations.
• Nonparametric confidence bands are crucial for statistical inference.

Purpose of the Study:

• Introduce new shape-constrained distribution functions: bi-s*-concave classes.
• Extend existing results for bi-log-concave distributions.
• Develop novel confidence bands that incorporate bi-s*-concavity.

Main Methods:

• Define and analyze the properties of bi-s*-concave distribution functions.
• Establish the relationship between s-concave densities and bi-s*-concave distribution functions.
• Adapt and build upon existing nonparametric confidence bands.

Main Results:

• Demonstrate that every s-concave density has a bi-s*-concave distribution function for s* ≤ s/(s + 1).
• Introduce new confidence bands that account for bi-s*-concavity.
• Show that the new bands extend those for bi-log-concavity.

Conclusions:

• Bi-s*-concavity offers a valuable generalization for distribution functions.
• The developed confidence bands provide enhanced statistical inference capabilities.
• Connections are established with the Csörgő-Révész constant and quantile processes.