Assessing Conformance with Benford's Law: Goodness-Of-Fit Tests and Simultaneous Confidence Intervals.

Area of Science:

• Data analysis and statistics
• Scientific integrity and fraud detection

Background:

• Benford's Law describes the distribution of first significant digits in numerical datasets.
• It is widely applied in scientific fields including biology, genomics, neuroscience, and image analysis.
• Applications extend to detecting fraud and validating mathematical models.

Purpose of the Study:

• To statistically compare methods for assessing data conformance with Benford's Law.
• To identify reliable statistical tests and confidence intervals for Benford's Law analysis.
• To address the over-rejection issue with common binomial confidence intervals.

Main Methods:

• Implementation and comparison of discrete versions of Cramér-von Mises (CvM) statistical tests.
• Evaluation of simultaneous confidence intervals for assessing conformance.
• Statistical analysis comparing traditional methods with proposed alternatives.

Main Results:

• Many commonly used binomial confidence intervals incorrectly reject data that actually follows Benford's Law.
• The CvM statistic Ud(2) and Pearson's chi-square statistic are effective for conformance testing.
• Goodman's simultaneous confidence intervals provide reliable assessment, especially when visualized.

Conclusions:

• Recommends using CvM statistic Ud(2), Pearson's chi-square, and Goodman's simultaneous confidence intervals for robust Benford's Law analysis.
• Highlights the importance of appropriate statistical methods to avoid false rejections.
• Emphasizes the utility of visual inspection with confidence intervals for understanding data deviations.