Modified maximum likelihood estimation through artificial neural networks: A case study on estimating the rate of Rayleigh process’ occurrence

Abstract

The research develops a new Software Reliability Growth Model that utilizes the Non-Homogeneous Poisson Process (NHPP) with a Rayleigh process mean function. A final stage applies Modified Maximum Likelihood Estimator (MMLE) together with Artificial Neural Network (ANN) for better parameter value estimation to achieve improved estimation accuracy and convergence rates. The proposed model achieves a comprehensive evaluation by comparing its results against standard Inverse Rayleigh and Exponential and Half-Logistic, and Gamma distribution models. Testing occurred through the utilization of Mean Squared Error (MSE) and Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), and log-likelihood (log L). The Rayleigh-based NHPP model exhibits superior performance against competing models because it produces lower MSE results, which reflects its strong ability for software reliability assessments. The research presents a study that evaluates MMLE methods versus Bayesian methods for estimation. MMLE produces lower MSE outcomes than Bayesian estimation for every dataset in the four experimental conditions, as shown in Table 8. The MSE from MMLE amounts to 0.0200, while Data Set 1 exhibits superior performance compared to the Bayesian method’s values of 0.1120. New evidence demonstrates that MMLE gives superior performance to Bayesian estimation when estimating parameters of software failure data in Data Sets 2 through 4. The parameter uncertainty of Bayesian estimation produces MSE values higher than the values obtained from MMLE, although it incorporates prior beliefs. The research demonstrates how MMLE works well for practical reliability engineering applications, although Bayesian Variational Inference and Markov Chain Monte Carlo (MCMC) should be used for future efficiency improvements.

Journal of Statistical Research 2025, Vol. 59, No. 2, pp. 279-303