Function Based Nonlinear Least Squares and Application to Jelinski--Moranda Software Reliability Model

A function based nonlinear least squares estimation (FNLSE) method is proposed and investigated in parameter estimation of Jelinski-Moranda software reliability model. FNLSE extends the potential fitting functions of traditional least squares estimation (LSE), and takes the logarithm transformed nonlinear least squares estimation (LogLSE) as a special case. A novel power transformation function based nonlinear least squares estimation (powLSE) is proposed and applied to the parameter estimation of Jelinski-Moranda model. Solved with Newton-Raphson method, Both LogLSE and powLSE of Jelinski-Moranda models are applied to the mean time between failures (MTBF) predications on six standard software failure time data sets. The experimental results demonstrate the effectiveness of powLSE with optimal power index compared to the classical least–squares estimation (LSE), maximum likelihood estimation (MLE) and LogLSE in terms of recursively relative error (RE) index and Braun statistic index.

💡 Research Summary

The paper addresses a fundamental limitation in software reliability modeling, namely the sensitivity of parameter estimation to heteroscedasticity and non‑normal error structures. Traditional approaches for the Jelinski‑Moranda (JM) model—Maximum Likelihood Estimation (MLE) and Ordinary Least Squares (LSE)—assume homoscedastic, Gaussian residuals. Real failure‑time data, however, often violate these assumptions, leading to biased estimates and poor mean time between failures (MTBF) predictions.

To overcome this, the authors introduce a Function‑Based Nonlinear Least Squares Estimation (FNLSE) framework. The core idea is to apply a monotonic, differentiable transformation ( H(\cdot) ) to both the observed inter‑failure times ( y_i ) and the model‑predicted values ( f(x_i,\beta) ). The transformed residual sum of squares, \