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In reliability studies, often we only have one failure data recorded in a life testing experiment. If there are two parameters in the reliability model, such as the model using Weibull distribution, then maximum likelihood estimation of parameters becomes a difficult problem. Mao and Chen published a real data set of the lifetime of a certain type of bearings which only contains one failure data. They used a Bayesian method to analyze the data and obtained some results for model parameter estimation. However, in their method the choice of prior distribution will affect heavily the final results. In this paper, we propose a Monte Carlo EM (MCEM) algorithm to estimate reliability model parameters using the Weibull distribution. Based on the same data set of Mao and Chen, we obtain some results using the MCEM algorithm. Our results do not depend on the choice of arbitrary prior distributions.

A maintained system is generally modeled using point processes. The most common processes used are the renewal process and the non homogeneous Poisson process corresponding to maximal and minimal repair situations with homogeneous Poisson process being a special case of both. A general repair formulation with a factor indicating the degree of repair is introduced into the minimal repair model to form an Arithmetic Reduction of Intensity model. These processes are generally able to model maintained systems with a fair degree of accuracy when the system is operating under stable conditions. However whenever there is a change in the environment these models which are monotonic in nature are not able to accommodate this change. Such systems operating under different environments need to be modelled by segmented models with the system domain divided into segments at the points of changes in the environment. The individual segments can then be modeled by any of the above point process models and these can be combined to form a composite model. This paper proposes a statistical model of such an operating/maintenance environment. Its purpose is to quantify the impacts of changes in the environment on the failure intensities. Field data from an industrial-setting demonstrate that appropriate parameter estimates for such phenomena can be obtained and such models are shown to more accurately describe the maintained system in a changing environment than the single point process models usually used.

The failure processes of maintained systems operating in a changing environment may be affected by the changes and exhibit different failure behaviour before and after the changes. Such processes exhibiting abrupt changes in failure intensities at specified times require segmented models with the process domain divided into segments at the points of changes in the environment to represent them. The individual segments can be modeled by any of the usual point process models and combined to form a composite segmented model with multiple change points. This paper proposes such segmented models with multiple change points to represent the failure processes of these systems and uses a hierarchical binary segmentation method to obtain the location of the changes. Its purpose is to quantify the impacts of changes in the environment on the failure intensities. These models are applied to the field data from an industrial setting; parameter estimates obtained and are shown to more accurately describe the failure processes of maintained system in a changing environment than the single point process models usually used. The interpretation and use of these models for maintained systems is also depicted.

The electric power supply cable is one of the most critical components of an electric mine loader. The effects of operating conditions such as fault types, cable types, numbers and types of repair done, the machines on which these cables are used, are analyzed using the proportional hazards model. The relatively important operating conditions influencing the life length of the cable are identified and their magnitudes are estimated. Before fitting any model to the data, simple graphical tools have been used in formulating covariates and selecting the suitable model. The proportional hazards model is found to be an effective tool for analyzing the effects of covariates. Graphical methods have been used to test the goodness-of-fit of the proportional hazards model.

This note designs a statistic model involving TM system with Gaussian noise. Based on adaptive approximation of the parameters in TM system and greedy choices for the generalized Fourier coefficients, a regression function is determined for this model from the finite observed data. Finally its statistical properties are proved.

This paper introduces a Bayesian approach to blending rainfall observations from both satellite remote sensing and ground rain gauges and to estimating errors of the blended data in terms of credible interval. The Bayesian posterior estimate (BPE) approach treats one observation as prior information and uses another observation as likelihood function information and as a correction. The BPE is an alternative to the minimum mean square estimate (MMSE) approach, also known as the least square approach. The posterior estimate outputs a probability density function (PDF), while the MMSE approach yields only an estimated mean and mean square error (MSE). When a diffusive rain rate model is assumed, the sampling errors of the Tropical Rainfall Measuring Mission (TRMM) satellite and a regular array of ground rain gauges are calculated under assumptions of idealized conditions: homogenous statistical properties of the rain rate, and flush visits of TRMM approximately twice a day. The optimal blend of the TRMM and ground gauges is determined by the optimal weight for each. The weight for the rain gauges w_g is a nonlinear function of the ratio of the space gap to the time gap. The error of the blended product increases as the gaps of time and space samplings expand, and the increase is almost linear when neither time gap nor space gap is small, but becomes strongly nonlinear when the space gap is very small. Our results are helpful in the design of appropriate sampling strategies for rainfall measurements.

We propose a new class of survival models for time-to-event data with a cure fraction. This new model is an extension of the promotion time cure rate model. Furthermore, we extend the model to the regression model to evaluate the effect of covariates in the cure fraction. An expectation-maximization algorithm is adopted for estimating the model parameters. A simulation study is conducted in order to assess the proposed model and the computation algorithm. The methodology is illustrated using a real Brazilian bank personal loan portfolio data set.

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.

Work in the last two decades on Bayesian nonparametric methods for mixture models finds that a posterior distribution is a double mixture. One first selects a partition of the objects based on a distribution on the partitions, and then performs a traditional parametric posterior analysis on the data corresponding to each cluster of the given partition. It is known that such a partition distribution favors partitions for which the clustering process is defined by predictive quantities such as predictive densities or weights. If a posterior distribution is a statistical guide to the unknown, this partition distribution could be used as a basis for a statistical model for clustering in which the partition is a parameter. The corresponding maximum likelihood estimator or posterior mode is used as an estimator of the partition. We also discuss methods to approximate these estimators based on a weighted Chinese restaurant process. A numerical example on a leukemia data set is given.