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Survival analysis plays a crucial role in understanding time-to-event data and modeling the underlying hazard rates. In this study, we propose a novel Bayesian survival analysis approach by incorporating the Type II half-logistic Weibull Cox-proportional hazard (Cox-PH) model. This model combines the advantages of the half-logistic Weibull distribution with the flexibility of the Cox-PH framework, allowing for efficient estimation of survival probabilities and hazard ratios. This study aims to apply Bayesian Analysis of Type II half-logistic Weibull Cox-PH model and compare it with Weibull Cox-PH, and Weibull models. These distributions are analyzed and fitted to censored survival data using the probabilistic programming language Stan. To incorporate censored mechanisms within the Stan framework, customized codes are developed. The models are then compared using information criteria under Bayesian paradigm.

The recent rapid spread of deadly epidemics have precipitated an urgent need to speed up the development of different treatments, as well as methods of evaluating their efficacy. The first step towards this is the collection of data relating to the cure rate in groups of patients who have had different treatments applied to them. As most of the available data in these cases is random, it is now the role of statisticians to provide efficient statistical tests to assess the treatment methods through the data. This research aims to provide a new statistical test with high efficiency to reach the right decision with accurate results as quickly as possible using parallel computing algorithms through Wolfram Mathematica software.

A total of nine methods are compared and tables are presented for selecting the best performing estimator for a two-parameter Weibull distribution using randomly right censored data. The results indicate that a best performing method can be selected given a certain sample size, censoring level, and shape parameter range. Such a comparison of estimators is necessary as the best performing method is shown to vary across these values. The estimation methods tested include the Maximum Likelihood Estimator, Kaplan-Meier Estimator, Piecewise Exponential Estimator, Földes, Rejt, and Winter Estimator, Klein, Lee, and Moeschberger Partially Parametric Estimator, Ross Estimator, White Estimator, Bain and Engelhardt Estimator, and the Modified Profile Maximum Likelihood Estimator. Recommendations are provided for applying the studied approach to other types of data and distributions.

In this paper, we consider an interval estimation of the age replacement problem, where the underlying failure time probability is given by the two-parameter Weibull distribution. The parametric bootstrap is used to obtain the probability distribution of an estimator for the optimal age replacement time minimizing the expected cost per unit time in the steady state. We focus on both cases with complete and incomplete samples of failure time data, and calculate not only the higher moments of an estimator for the optimal age replacement time but also the confidence interval.

The purpose of this paper is to propose an approach for testing multivariate uniformity. The proposed test procedure works for both complete sample case and incomplete sample case. What multivariate uniformity test does is to check whether the population distribution, from which a random sample is drawn is a multivariate uniform distribution. Particularly, it is desired to check whether the population distribution is a uniform distribution on a $p$-dimensional unit cube. There are some existing research papers in the literature. The proposed method works when only some arbitrarily selected order statistics are available for the test.

A new three-parameter compounded extended-exponential distribution “Poisson Nadarajah–Haghighi” is introduced and studied, which is quite flexible and can be used effectively in modeling survival data. It can have increasing, decreasing, upside-down bathtub and bathtub-shaped failure rate. A comprehensive account of the mathematical properties of the model is presented. We discuss maximum likelihood estimation for complete and censored data. The suitability of the maximum likelihood method to estimate its parameters is assessed by a Monte Carlo simulation study. Four empirical illustrations of the new model are presented to real data and the results are quite satisfactory.

A simulation study is presented where the Piecewise Exponential Estimator is compared with the Kaplan-Meier Estimator when the TTT-plot is used for identification of failure distribution models and also to determine the replacement time for a repairable system. We can conclude that the Piecewise Exponential Estimator seems to be better than the Kaplan-Meier Estimator, especially when we have heavy censoring and/or small sample sizes.

The applicability of the fatigue-life distribution of Birnbaum–Saunders for modeling the components of partially prestressed concrete girder bridges is argued by the agreement of the calculated distribution with a large data set comprised of many small censored samples, each put into standardized form, of the fatigue-lives of bridge components. Also, the Weibull and lognormal distributions are compared by the appropriateness of the assumptions in their derivation to the cumulative damage to bridge components.

The maximum likelihood estimates of the parameters of the fatigue lives of the prestressing steel, mild steel, and concrete are found from randomly censored samples, for several censored data-sets under various stress regimes. The parameters are postulated to be log-linear functions of the maximum stress and stress regimes. Weighted least-square regression is applied to determine the unknown coefficients. To compensate for differences in the precision of the estimates obtained under each stress regime the variances are calculated by the bootstrap statistical method. This procedure gives an estimated distribution of the fatigue life with its parameters functions of the stress regime which is applicable under virtually any realistic condition.

Conditional probability distribution models have been widely used in economics and finance. In this chapter, we introduce two closely related popular methods to estimate conditional distribution models—Maximum Likelihood Estimation (MLE) and Quasi-MLE (QMLE). MLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is correctly specified, and QMLE is a parameter estimator that maximizes the model likelihood function of the random sample when the conditional distribution model is misspecified. Because the score function is an MDS and the dynamic Information Matrix (IM) equality holds when a conditional distribution model is correctly specified, the asymptotic properties of MLE is analogous to those of the OLS estimator when the regression disturbance is an MDS with conditional homoskedasticity, and we can use the Wald test, LM test and Likelihood Ratio (LR) test for hypothesis testing, where the LR test is analogous to the J · F test statistic. On the other hand, when the conditional distribution model is misspecified, the score function has mean zero, but it may no longer be an MDS and the dynamic IM equality may fail. As a result, the asymptotic properties of QMLE are analogous to those of the OLS estimator when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. Robust Wald tests and LM tests can be constructed for hypothesis testing, but the LR test can no longer be used, for a reason similar to the failure of the F-test statistic when the regression disturbance displays serial correlation and/or conditional heteroskedasticity. We discuss methods to test the MDS property of the score function, and the dynamic IM equality, and correct specification of a conditional distribution model.

Since the pioneer work of Thomas & Grunkemeier (1975) and Owen (1988), empirical likelihood has been developed as a powerful nonparametric inference approach and become popular in statistical literature. There are many applications of empirical likelihood in survival analysis. In this paper, we present an overview of recent developments of empirical likelihood methods for survival data. In particular, we discuss empirical likelihood results for a general mean functional of the distribution function, a functional of the hazard function, the Cox proportional hazards model, and a semiparametric accelerated failure time model.

Based on an existing kernel survival function estimate that admits censored data, we develop confidence intervals to help assess the validity of the estimate. Practical issues of estimation are discussed and then the developments are applied to a real data set. The results are analyzed and discussed further.

Xia, Tong, Li and Zhu (2002) proposed a general estimation method termed minimum average variance estimation (MAVE) for semiparametric models. The method has been found very useful in estimating complicated semiparametric models (Xia, Zhang and Tong, 2004; Xia and Härdle, 2006) and general dimension reduction (Xia, 2008; Wang and Xia, 2008). The method is also convenient to combine with other methods in order to incorporate additional statistical requirements (Wang and Yin, 2007). In this paper, we give a general review on the method and discuss some issues arising in estimating semiparametric models and dimension reduction (Li, 1991 and Cook, 1998) when complicated statistical requirements are imposed, including quantile regression, sparsity of variables and censored data.

The ultimate solution to the high volume of excess foundry sands generated in metal casting facilities is to beneficially reuse them. A variety of possible reuse programs suit the utilization of foundry sands. These programs mainly relate to civil and agricultural applications, e.g., highway embankment, concrete, backfills, topsoil, and growing amendments. Foundry sand exhibits qualified physical, mechanical, or chemical properties for these end-use products and is acknowledged as a marketable resource. Most often, foundry sand is characterized to match a target material, which has been adopted as a standardized component in an end-use product, e.g. fine aggregate in concrete. If the match gives favorable outcomes, the foundary sand is introduced fully or partially replacing the target material to form a technically competitive end-use product, e.g. foundry sand used in concrete. In addition to its engineering qualifications, foundry sand's environmental safety is also ensured to comply with regulations. The chemical characterization of foundry sand needs to be accomplished to demonstrate that it will not pose a thread to environments and human health during reuse programs. This chapter is organized to present the system (foundry) sand operation, foundry sand generation and management, chemical characterization of foundry sand, reuse applications of foundry sand.