A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics

Leonid Hanin () and Lyudmila Pavlova
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Leonid Hanin: Department of Mathematics and Statistics, Idaho State University, 921 S. 8th Avenue, Stop 8085, Pocatello, ID 83209-8085, USA
Lyudmila Pavlova: School of Applied Mathematics and Computational Physics, Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya ul. 29, 195251 St. Petersburg, Russia

Mathematics, 2023, vol. 11, issue 20, 1-19

Abstract: We study classification of random sequences of characters selected from a given alphabet into two classes characterized by distinct character selection probabilities and length distributions. The classification is based on the sign of the log-likelihood score (LLS) consisting of a random sum and a random term depending on the length distributions for the two classes. For long sequences selected from a large alphabet, computing misclassification error rates is not feasible either theoretically or computationally. To mitigate this problem, we computed limiting distributions for two versions of the normalized LLS applicable to long sequences whose class-specific length follows a translated negative binomial distribution (TNBD). The two limiting distributions turned out to be plain or transformed Erlang distributions. This allowed us to establish the asymptotic accuracy of the likelihood-based classification of random sequences with TNBD length distributions. Our limit theorem generalizes a classic theorem on geometric random sums due to Rényi and is closely related to the published results of V. Korolev and coworkers on negative binomial random sums. As an illustration, we applied our limit theorem to the classification of DNA sequences contained in the genome of the bacterium Bacillus subtilis into two classes: protein-coding genes and standard noncoding open reading frames. We found that TNBDs provide an excellent fit to the length distributions for both classes and that the limiting distributions capture essential features of the normalized empirical LLS fairly well.

Keywords: Rényi theorem; sequence classification; classification accuracy; random sum; translated negative binomial distribution; Kullback–Leibler distance; Erlang distribution; protein-coding gene; open reading frame (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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