ESTIMATION OF A MONOTONE DENSITY IN S-SAMPLE BIASED SAMPLING MODELS.

Area of Science:

• Statistics
• Nonparametric Inference
• Biased Sampling Models

Background:

• The estimation of decreasing density functions in biased sampling models is crucial for various statistical applications.
• Existing literature lacks methods for determining the monotone maximum likelihood estimator and its asymptotic distribution for s > 1 samples due to complex likelihood structures.

Purpose of the Study:

• To develop and validate a monotone maximum likelihood estimator (ĝ) for decreasing density functions in general s-sample biased sampling models.
• To establish the existence, uniqueness, consistency, and asymptotic distribution of the proposed estimator.

Main Methods:

• Developed a novel approach to address the non-standard structures of the likelihood function, including non-separability.
• Employed purely analytic arguments to demonstrate the tightness of the estimator, avoiding traditional geometric methods.
• Proposed an indirect strategy to achieve the convergence rate for linear functionals involving weight functions.

Main Results:

• Successfully established the existence, uniqueness, self-characterization, and consistency of the monotone maximum likelihood estimator (ĝ).
• Determined the asymptotic distribution of ĝ at a fixed point, resolving a long-standing problem for s > 1.
• Provided a method to achieve the convergence rate for the linear functional ∫ w.

Conclusions:

• The study successfully addresses the challenges in nonparametric estimation of decreasing densities under biased sampling.
• The established theoretical properties of the monotone maximum likelihood estimator provide a robust tool for statistical analysis.
• This work significantly advances the understanding and application of biased sampling models in statistical inference.