Overall mean estimation of trace evidence in a two-level normal-normal model

Area of Science:

• Forensic Science
• Statistical Modeling
• Measurement Analysis

Background:

• Aitken and Lucy (2004) proposed a two-level model for forensic trace evidence, using likelihood ratios and assuming normality.
• This model requires estimates of within-group variation, between-group variation, and the overall mean.
• Two common estimators for the overall mean in forensic science are the weighted and unweighted means.

Purpose of the Study:

• To discuss and compare three estimators for the overall mean in the context of forensic trace evidence.
• To introduce a generalized weighted mean estimator and an optimal "plug-in" version.
• To evaluate the performance of these estimators under various conditions.

Main Methods:

• The study discusses weighted, unweighted, and a novel generalized weighted mean estimator.
• A simulation study was conducted to compare the estimators.
• The generalized weighted mean incorporates optimal weights that minimize variance, leading to a "toy estimator" and an "optimal plug-in" estimator.

Main Results:

• Under typical conditions, the unweighted mean is often preferable to the weighted mean.
• The unweighted mean and the optimal generalized weighted mean show similar performance in many practical situations.
• An artificial parameter choice demonstrated a scenario where the optimal generalized weighted mean outperformed both the weighted and unweighted means.

Conclusions:

• The choice of overall mean estimator can significantly impact forensic trace evidence analysis.
• The unweighted mean and optimal generalized weighted mean are robust choices for many practical forensic applications.
• Further investigation into the generalized weighted mean estimator is warranted for forensic applications.