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Multivariate Inferential Statistics
Published in Jhareswar Maiti, Multivariate Statistical Modeling in Engineering and Management, 2023
The confidence region is an ellipsoid centered at x¯ and axes are defined by χp2(α), where α is the probability level of significance. For 95 % confidence region α=0.05.
Estimation of P (X < Y) for Gompertz distribution based on upper records
Published in International Journal of Modelling and Simulation, 2022
Ayush Tripathi, Umesh Singh, Sanjay Kumar Singh
This distribution has been considered by many authors to study its statistical properties and applications in the field of reliability, survival and actuarial data. Pollard & Valkovics discussed the properties and applications of the Gompertz distribution [4]. Z. Chen estimated the unknown parameters of the Gompertz distribution [5]. Jaheen obtained Bayes estimator under LINEX and squared error loss function (SELF) based on upper record values [6]. Jaheen again considered the problem of estimation for the parameter of Gompertz distribution based on progressive Type-II censored data [7]. Chang and Tsai obtained maximum likelihood (ML) estimator with their exact confidence interval and exact joint confidence region for the parameters of the Gompertz distribution based on progressive Type-II-censored samples [8]. Moments of the Gompertz distribution are discussed by Lenart [9]. He also showed superiority of maximum likelihood estimator over moment estimator for this distribution. Wang et al. discussed parameter estimation based on record values under the classical and Bayesian paradigm [10]. El-Din et al. conducted a comparative study between ML and Bayes estimator for Gompertz distribution based on Type-II progressively hybrid-censored data [11]. Recently, Dey et al. provided a detail study of the Gompertz distribution based on a complete sample [12].
Adaptive Minimum Confidence Region Rule for Multivariate Initialization Bias Truncation in Discrete-Event Simulations
Published in Technometrics, 2020
Jianguo Wu, Honglun Xu, Feng Ju, Tzu-Liang (Bill) Tseng
where is the chi-squared distribution with p degree of freedom. A joint confidence region for the mean satisfies where is the upper th percentile of distribution. Clearly, the confidence region is a p-dimensinal ellipsoid (ellipse for p = 2) with semi-axes where are eigenvalues of the covariance matrix Σ satisfying . The volume of the confidence region can be calculated as (Wilson 2010)
Data-driven software reliability evaluation under incomplete knowledge on fault count distribution
Published in Quality Engineering, 2020
Tadashi Dohi, Junjun Zheng, Hiroyuki Okamura
Next, we concern the interval estimation of the software intensity function and its related software reliability measures such as the mean value function and quantitative software reliability. Yamada and Osaki (1985), Joe (1989), Reese and van Pul (1995), Zhao and Xie (1996) gave asymptotically approximate confidence intervals of model parameters for specific NHPP-based SRMs. Yin and Trivedi (1999) obtained exact confidence intervals of model parameters for the exponential NHPP-based SRM (Goel and Okumoto 1979). Okamura et al. (2007) developed a variational approach to approximate the confidence interval, of model parameters from both viewpoints of frequentist and Bayesian approaches. In this way, considerable attentions have been paid to the interval estimation of model parameters. However, this approach is rather limited if one is interested in the confidence region of arbitrary software reliability measures. For instance, when we are interested in the confidence interval of the mean value function, it does not mean the confidence region of NHPP itself. It is worth mentioning that the probability distribution of an arbitrary complex estimator has to be evaluated based on the underlying software fault count data. That is to say, it is impossible to derive analytically the confidence regions of software reliability measures themselves, such as software intensity function, mean value function and quantitative software reliability function, without employing approximate techniques.