In our previous work, we proposed wavelet shrinkage estimation (WSE) for nonhomogeneous Poisson process (NHPP)-based software reliability models (SRMs), where WSE is a data-transform-based nonparametric estimation method. Among many variance-stabilizing data transformations, the Anscombe transform and the Fisz transform were employed. We have shown that it could provide higher goodness-of-fit performance than the conventional maximum likelihood estimation (MLE) and the least squares estimation (LSE) in many cases, in spite of its non-parametric nature, through numerical experiments with real software-fault count data. With the aim of improving the estimation accuracy of WSE, in this paper we introduce other three data transformations to preprocess the software-fault count data and investigate the influence of different data transformations to the estimation accuracy of WSE through goodness-of-fit test. 1. Introduction In the field of software reliability engineering, the quantitative assessment of software reliability has become one of the main issues of this area. Especially, people are interested in finding several software intensity functions from the software-fault count data observed in the software testing phases, since the software intensity function in discrete time denotes the number of software faults detected per unit time. This directly makes it possible to estimate the number of remaining software faults and the quantitative software reliability, which is defined as the probability that software system does not fail during a specified time period under a specified operational environment. Moreover, these evaluation measures can be used in the decision making such as allocation of development resources and software release scheduling. Therefore, we are interested in developing a high-accuracy estimation method for the software intensity function. Among over hundreds of software reliability models (SRMs) [1–3], nonhomogeneous Poisson process (NHPP)-based SRMs have gained much popularity in actual software testing phases. In many cases, the NHPP-based SRM is formulated as a parametric model, where the mean value function or its difference in discrete time or derivative in continuous time, called “software intensity function,” can be considered as a unique parameter to govern the probabilistic property. One class of parametric NHPP-based SRMs is concerned with modeling the number of software faults detected in testing phases, initiated by Goel and Okumoto [4]. Afterwards, many parametric NHPP-based SRMs were proposed in the literatures [5–8]
References
[1]
M. R. Lyu, Ed., Handbook of Software Reliability Engineering, McGraw-Hill, New York, NY, USA, 1996.
[2]
J. D. Musa, A. Iannino, and K. Okumoto, Software Reliability, Measurement, Prediction, Application, McGraw-Hill, New York, NY, USA, 1987.
[3]
H. Pham, Software Reliability, Springer, Singapore, Singapore, 2000.
[4]
A. L. Goel and K. Okumoto, “Time-dependent error-detection rate model for software reliability and other performance measures,” IEEE Transactions on Reliability, vol. R-28, no. 3, pp. 206–211, 1979.
[5]
A. L. Goel, “Software reliability models: assumptions, limitations and applicability,” IEEE Transactions on Software Engineering, vol. SE-11, no. 12, pp. 1411–1423, 1985.
[6]
X. Xiao and T. Dohi, “Estimating software reliability using extreme value distribution,” in Proceedings of the International Conference on Advances in Software Engineering and Its Applications (ASEA '11), vol. CCIS 257, pp. 399–406, Springer, 2011.
[7]
X. Xiao, H. Okamura, and T. Dohi, “NHPP-based software reliability models using equilibrium distribution,” IEICE Transactions of the Fundamentals A, vol. E95-A, no. 5, pp. 894–902, 2012.
[8]
S. Yamada, M. Ohba, and S. Osaki, “S-shaped reliability growth modeling for software error detection,” IEEE Transactions on Reliability, vol. R-32, no. 5, pp. 475–484, 1983.
[9]
A. Sofer and D. R. Miller, “A non-parametric software reliability growth model,” IEEE Transactions on Reliability, vol. R-40, no. 3, pp. 329–337, 1991.
[10]
A. Gandy and U. Jensen, “A non-parametric approach to software reliability,” Applied Stochastic Models in Business and Industry, vol. 20, no. 1, pp. 3–15, 2004.
[11]
M. Barghout, B. Littlewood, and A. Abdel-Ghaly, “A non-parametric order statistics software reliability model,” Software Testing Verification and Reliability, vol. 8, no. 3, pp. 113–132, 1998.
[12]
Z. Wang, J. Wang, and X. Liang, “Non-parametric estimation for NHPP software reliability models,” Journal of Applied Statistics, vol. 34, no. 1, pp. 107–119, 2007.
[13]
M. A. El-Aroui and J. L. Soler, “A bayes nonparametric framework for software-reliability analysis,” IEEE Transactions on Reliability, vol. 45, no. 4, pp. 652–660, 1996.
[14]
S. P. Wilson and F. J. Samaniego, “Nonparametric analysis of the order-statistic model in software reliability,” IEEE Transactions on Software Engineering, vol. 33, no. 3, pp. 198–208, 2007.
[15]
X. Xiao and T. Dohi, “Wavelet-based approach for estimating software reliability,” in Proceedings of the 20th International Symposium on Software Reliability Engineering (ISSRE '09), pp. 11–20, IEEE CS Press, November 2009.
[16]
F. J. Anscombe, “The transformation of Poisson, binomial and negative binomial data,” Biometrika, vol. 35, no. 3-4, pp. 246–254, 1948.
[17]
M. Fisz, “The limiting distribution of a function of two independent random variables and its statistical application,” Colloquium Mathematicum, vol. 3, pp. 138–146, 1955.
[18]
M. S. Bartlett, “The square root transformation in the analysis of variance,” Journal of the Royal Statistical Society, vol. 3, no. 1, pp. 68–78, 1936.
[19]
M. F. Freeman and J. W. Tukey, “Transformations related to the angular and the square root,” The Annals of Mathematical Statistics, vol. 21, no. 4, pp. 607–611, 1950.
[20]
H. Ishii, T. Dohi, and H. Okamura, “Software reliability prediction based on least squares estimation,” Quality Technology and Quantitative Management Journal. In press.
[21]
D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, 1994.
[22]
G. P. Nason, “Wavelet shrinkage using cross-validation,” Journal of the Royal Statistical Society B, vol. 58, no. 2, pp. 463–479, 1996.
