The New Mixed Generalized Erlang Distribution

In probability theory, the mixture distribution M has a density function for the collection of random variables and weighted by wi ≥ 0 and . These mixed distributions are used in various disciplines and aim to enrich the collection distribution to more parameters. A more general mixture is derived by Kadri and Halat, by proving the existence of such mixture by wi ∈ R, and maintaining . Kadri and Halat provided many examples and applications for such new mixed distributions. In this paper, we introduce a new mixed distribution of the Generalized Erlang distribution, which is derived from the Hypoexponential distribution. We characterize this new distribution by deriving simply closed expressions for the related functions of the probability density function, cumulative distribution function, moment generating function, reliability function, hazard function, and moments.

Adaptive Conditional Hazard Regression Modeling of Multiple Event Times

Recurrent event time data and more general multiple event time data are commonly analyzed using extensions of Cox regression, or proportional hazards regression, as used with single event time data. These methods treat covariates, either time-invariant or time-varying, as having multiplicative effects while general dependence on time is left un-estimated. An adaptive approach is formulated for analyzing multiple event time data. Conditional hazard rates are modeled in terms of dependence on both time and covariates using fractional polynomials restricted so that the conditional hazard rates are positive-valued and so that excess time probability functions (generalizing survival functions for single event times) are decreasing. Maximum likelihood is used to estimate parameters adjusting for right censored event times. Likelihood cross-validation (LCV) scores are used to compare models. Adaptive searches through alternate conditional hazard rate models are controlled by LCV scores combined with tolerance parameters. These searches identify effective models for the underlying multiple event time data. Conditional hazard regression is demonstrated using data on times between tumor recurrence for bladder cancer patients. Analyses of theory-based models for these data using extensions of Cox regression provide conflicting results on effects to treatment group and the initial number of tumors. On the other hand, fractional polynomial analyses of these theory-based models provide consistent results identifying significant effects to treatment group and initial number of tumors using both model-based and robust empirical tests. Adaptive analyses further identify distinct moderation by group of the effect of tumor order and an additive effect to group after controlling for nonlinear effects to initial number of tumors and tumor order. Results of example analyses indicate that adaptive conditional hazard rate modeling can generate useful insights into multiple event time data.

An Adaptive Approach for Hazard Regression Modeling

Regression models for survival time data involve estimation of the hazard rate as a function of predictor variables and associated slope parameters. An adaptive approach is formulated for such hazard regression modeling. The hazard rate is modeled using fractional polynomials, that is, linear combinations of products of power transforms of time together with other available predictors. These fractional polynomial models are restricted to generating positive-valued hazard rates and decreasing survival times. Exponentially distributed survival times are a special case. Parameters are estimated using maximum likelihood estimation allowing for right censored survival times. Models are evaluated and compared using likelihood cross-validation (LCV) scores. LCV scores and tolerance parameters are used to control an adaptive search through alternative fractional polynomial hazard rate models to identify effective models for the underlying survival time data. These methods are demonstrated using two different survival time data sets including survival times for lung cancer patients and for multiple myeloma patients. For the lung cancer data, the hazard rate depends distinctly on time. However, controlling for cell type provides a distinct improvement while the hazard rate depends only on cell type and no longer on time. Furthermore, Cox regression is unable to identify a cell type effect. For the multiple myeloma data, the hazard rate also depends distinctly on time. Moreover, consideration of hemoglobin at diagnosis provides a distinct improvement, the hazard rate still depends distinctly on time, and hemoglobin distinctly moderates the effect of time on the hazard rate. These results indicate that adaptive hazard rate modeling can provide unique insights into survival time data.

Adaptive Conditional Hazard Regression Modeling of Multiple Event Times

Recurrent event time data and more general multiple event time data are commonly analyzed using extensions of Cox regression, or proportional hazards regression, as used with single event time data. These methods treat covariates, either time-invariant or time-varying, as having multiplicative effects while general dependence on time is left un-estimated. An adaptive approach is formulated for analyzing multiple event time data. Conditional hazard rates are modeled in terms of dependence on both time and covariates using fractional polynomials restricted so that the conditional hazard rates are positive-valued and so that excess time probability functions (generalizing survival functions for single event times) are decreasing. Maximum likelihood is used to estimate parameters adjusting for right censored event times. Likelihood cross-validation (LCV) scores are used to compare models. Adaptive searches through alternate conditional hazard rate models are controlled by LCV scores combined with tolerance parameters. These searches identify effective models for the underlying multiple event time data. Conditional hazard regression is demonstrated using data on times between tumor recurrence for bladder cancer patients. Analyses of theory-based models for these data using extensions of Cox regression provide conflicting results on effects to treatment group and the initial number of tumors. On the other hand, fractional polynomial analyses of these theory-based models provide consistent results identifying significant effects to treatment group and initial number of tumors using both model-based and robust empirical tests. Adaptive analyses further identify distinct moderation by group of the effect of tumor order and an additive effect to group after controlling for nonlinear effects to initial number of tumors and tumor order. Results of example analyses indicate that adaptive conditional hazard rate modeling can generate useful insights into multiple event time data.

Nonparametric estimation for hazard rate monotonously decreasing system

Estimation of density and hazard rate is very important to the reliability analysis of a system. In order to estimate the density and hazard rate of a hazard rate monotonously decreasing system, a new nonparametric estimator is put forward. The estimator is based on the kernel function method and optimum algorithm. Numerical experiment shows that the method is accurate enough and can be used in many cases.

Landslide Hazard Zonation Mapping in Ghat Road Section of Kolli Hills, India

Landslides are the most common natural disaster in hilly terrain which causes changes in landscape and damage to life and property. The main objective of the present study was to carry out landslide hazard zonation mapping on 1:50,000 scale along ghat road section of Kolli hills using a Landslide Hazard Evaluation Factor(LHEF) rating scheme. The landslide hazard zonation map has been prepared by overlaying the terrain evaluation maps with facet map of the study area. The terrain evaluation maps include lithology, structure, slope morphometry, relative relief, land use and land cover and hydrogeological condition. The LHEF rating scheme and the Total Estimated Hazard(TEHD) were calculated as per the Bureau of Indian Standard(BIS) guidelines(IS: 14496(Part-2) 1998) for the purpose of preparation of Landslide Hazard Zonation(LHZ) map in mountainous terrains. The correction due to triggering factors such as seismicity, rainfall and anthropogenic activities were also incorporated with Total Estimated Hazard to get final corrected TEHD. The landslide hazard zonation map was classified as the high, moderate and low hazard zones along the ghat road section based on corrected TEHD.

Parameter Estimation Based on Censored Data under Partially Accelerated Life Testing for Hybrid Systems due to Unknown Failure Causes

In general,simple subsystems like series or parallel are integrated to produce a complex hybrid system.The reliability of a system is determined by the reliability of its constituent components.It is often extremely difficult or impossible to get specific information about the component that caused the system to fail.Unknown failure causes are instances in which the actual cause of systemfailure is unknown.On the other side,thanks to current advanced technology based on computers,automation,and simulation,products have become incredibly dependable and trustworthy,and as a result,obtaining failure data for testing such exceptionally reliable items have become a very costly and time-consuming procedure.Therefore,because of its capacity to produce rapid and adequate failure data in a short period of time,accelerated life testing(ALT)is the most utilized approach in the field of product reliability and life testing.Based on progressively hybrid censored(PrHC)data froma three-component parallel series hybrid system that failed to owe to unknown causes,this paper investigates a challenging problem of parameter estimation and reliability assessment under a step stress partially accelerated life-test(SSPALT).Failures of components are considered to follow a power linear hazard rate(PLHR),which can be used when the failure rate displays linear,decreasing,increasing or bathtub failure patterns.The Tempered random variable(TRV)model is considered to reflect the effect of the high stress level used to induce early failure data.The maximum likelihood estimation(MLE)approach is used to estimate the parameters of the PLHR distribution and the acceleration factor.A variance covariance matrix(VCM)is then obtained to construct the approximate confidence intervals(ACIs).In addition,studentized bootstrap confidence intervals(ST-B CIs)are also constructed and compared with ACIs in terms of their respective interval lengths(ILs).Moreover,a simulation study is conducted to demonstrate the performance of the estimation procedures and the methodology discussed in this paper.Finally,real failure data from the air conditioning systems of an airplane is used to illustrate further the performance of the suggested estimation technique.

APPROACH OF FIVE-YEAR-AVERAGE HAZARD RATES FOR THE BREAST CANCER PATIENTS AND ANALYSES OF PROGNOSTIC FACTORS-AN APPLICATION OF COX REGRESSION MODEL

Objective: To compare with fiveyear survival after surgery for the 116 breast cancer patients treated at the First Teaching Hospital (FTH) and the 866 breast cancer patients at Hpital du SaintSacrement (HSS). Methods: Using Cox regression model, after eliminating the confounders, to develop the comparison of the fiveyear average hazard rates between two hospitals and among the levels of prognostic factors. Results: It has significant difference for the old patients (50 years old or more) between the two hospitals. Conclusion: Tumor size at pathology and involvement of lymph nodes were important prognostic factors.

Propulsion Shaft and Gearbox Failure in Marine Vessels: A Duration Model Analysis

The need to mitigate downtime in marine vessels arising from propulsion system failures has led ship operating companies to devote enormous resources for research based solutions. This paper applied duration models to determine failure probabilities of shaft and gearbox systems in service boats. Using dockyard’s event history data on boat repairs and maintenance, we applied Kaplan Meier hazard and survival curves to analyse probability of failure of shaft and gearbox systems in supply, crew and tug boats. We found that average time to shaft and gearbox failure was 8.33, 5.23 and 5.21 months for tug, supply and crew boats respectively. The hazard plots however, showed that supply boats had higher probability of failure than crew boats and then tug boats in that order. Further analysis using Cox regression model showed that the boats’ shaft and gearbox system failures were significantly affected by level of lubrication oil, stress corrosion cracking and impacts on the propulsion system’s components. The paper proposes that design of maintenance schedules for service boats should take the following into consideration: 1) estimated survival limits or failure times of propulsion system’s shaft and gearboxes, 2) significant risk factors that affect failure mode of the propulsion system components.

Power Hamza Distribution with Application to Lifetime Data

In this paper, a three-parameter lifetime distribution named power Hamza distribution (PH) is proposed. The PH distribution is a useful generalization of the Hamza distribution which accommodates heavy-tailed, upside-down bathtub and J-shaped hazard rates making it more flexible than the Hamza distribution for modelling various kinds of lifetime data. A comprehensive account of the properties of this distribution is presented. The maximum likelihood estimators of the unknown model parameters are discussed. Finally, a real-life data is analyzed for illustrative purpose proving that the PH outperforms the Hamza distribution and several other lifetime distributions.

Quantitative Structural Models to Assess Credit Risk on Individuals

Default Probabilities quantitatively measures the credit risk that a borrower will be unable or unwilling to repay its debt. An accurate model to estimate, as a function of time, these default probabilities is of crucial importance in the credit derivatives market. In this work, we adapt Merton’s [1] original works on credit risk, consumption and portfolio rules to model an individual wealth scenario, and apply it to compute this individual default probabilities. Using our model, we also compute the time depending individual default intensities, recovery rates, hazard rate and risk premiums. Hence, as a straight-forward application, our model can be used as novel way to measure the credit risk of individuals.

Applications of Dynamic-Equilibrium Continuous Markov Stochastic Processes to Elements of Survival Analysis

In this article, we summarize some results on invariant non-homogeneous and dynamic-equilibrium (DE) continuous Markov stochastic processes. Moreover, we discuss a few examples and consider a new application of DE processes to elements of survival analysis. These elements concern the stochastic quadratic-hazard-rate model, for which our work 1) generalizes the reading of its It? stochastic ordinary differential equation (ISODE) for the hazard-rate-driving independent (HRDI) variables, 2) specifies key properties of the hazard-rate function, and in particular, reveals that the baseline value of the HRDI variables is the expectation of the DE solution of the ISODE, 3) suggests practical settings for obtaining multi-dimensional probability densities necessary for consistent and systematic reconstruction of missing data by Gibbs sampling and 4) further develops the corresponding line of modeling. The resulting advantages are emphasized in connection with the framework of clinical trials of chronic obstructive pulmonary disease (COPD) where we propose the use of an endpoint reflecting the narrowing of airways. This endpoint is based on a fairly compact geometric model that quantifies the course of the obstruction, shows how it is associated with the hazard rate, and clarifies why it is life-threatening. The work also suggests a few directions for future research.

Flexible Reduced Logarithmic-Inverse Lomax Distribution with Application for Bladder Cancer

In this paper, a new method for adding parameters to a well-established distribution to obtain more flexible new families of distributions is applied to the inverse Lomax distribution (IFD). This method is known by the flexible reduced logarithmic-X family of distribution (FRL-X). The proposed distribution can be called a flexible reduced logarithmic-inverse Lomax distribution (FRL-IL). The statistical and reliability properties of the proposed models are studied including moments, order statistics, moment generating function, and quantile function. The estimation of the model parameters by maximum likelihood and the observed information matrix are also discussed. In order to assess the potential of the newly created distribution. The extended model is applied to real data and the results are given and compared to other models.

An Efficient Risk Estimator with External Information Under Additive Hazards Model

Rare event data is encountered when the events of interest occur with low frequency, and the estimators based on the cohort data only may be inefficient. However, when external information is available for the estimation, the estimators utilizing external information can be more efficient. In this paper, we propose a method to incorporate external information into the estimation of the baseline hazard function and improve efficiency for estimating the absolute risk under the additive hazards model. The resulting estimators are shown to be uniformly consistent and converge weakly to Gaussian processes. Simulation studies demonstrate that the proposed method is much more efficient. An application to a bone marrow transplant data set is provided.

Characterizations on Heavy-tailed Distributions by Means of Hazard Rate

Abstract Let F(x) be a distribution function supported on [0,X), with an equilibrium distribution function Fe(x). In this paper we shall study the function $r_e(x)( - {\rm ln}{\overline F}_e ( x ))\prime = {\overline F}( x )/\int_x^\infty {\overline F}( u )du $, which is called the equilibrium hazard rate of F. By the limiting behavior of re(x) we give a criterion to identify F to be heavy-tailed or light-tailed. Two broad classes of heavy-tailed distributions are also introduced and studied.

Exponentiated Generalized Inverse Flexible Weibull Distribution:Bayesian and Non-Bayesian Estimation Under Complete and TypeⅡCensored Samples with Applications

In this paper,a new 4-parameter exponentiated generalized inverse flexible Weibull distribution is proposed.Some of its statistical properties are studied.The aim of this paper is to estimate the model parameters via several approaches,namely,maximum likelihood,maximum product spacing and Bayesian.According to Bayesian approach,several techniques are used to get the Bayesian estimators,namely,standard error function,Linex loss function and entropy loss function.The estimation herein is based on complete and censored samples.Markov Chain Monte Carlo simulation is used to discuss the behavior of the estimators for each approach.Finally,two real data sets are analyzed to obtain the flexibility of the proposed model.

AGING PROPERTIES OF THE LIFETIME IN SIMPLE ADDITIVE DEGRADATION MODELS

This paper deals with the simple additive degradation models with single random effect. The authors further study the link between the aging property of the implied lifetime and that of the random variation. It is found that both the aging property of the random variation and the analytical behavior of the mean degradation path influence the aging behavior of the implied lifetime.

Reliability Analysis of Symmetrical Columns with Eccentric Loading from Lindley Distribution

This paper shows the reliability of the symmetrical columns with eccentric loading about one and two axes due to the maximum intensity stress and minimum intensity stress.In this paper,a new lifetime distribution is introduced which is obtained by compounding exponential and gamma distributions(named as Lindley distribution).Hazard rates,mean time to failure and estimation of single parameter Lindley distribution by maximum likelihood estimator have been discussed.It is observed that when the load and the area of the cross section increase,failure of the column also increases at two intensity stresses.It is observed from the results that reliability decreases when scale parameter increases.

An Extended Weibull Model with Variable Periodicity

A new distribution for the fluctuation of materials＇ lifetime cumulative hazard rate is firstly proposed. The new distribution is extended from the Weibull distribution by adding a sine function. After that, the properties of its hazard rate function, cumulative hazard rate function, probability density function and cumulative distribution function are studied. The analysis result shows this distribution can well model the lifetime with variable and periodic hazard rate. Finally, the new distribution is verified with two real data sets as examples to demonstrate its capability.

An Imperfect Grouping Maintenance Strategy for Multi-Component Systems Using the Survival Signature

This paper presents an imperfect maintenance strategy for the multi-component systems.The proposed maintenance strategy takes into account two types of maintenance actions,namely preventive maintenance(PM)and corrective maintenance(CM).The imperfect effect of PM is modeled on the basis of the hybrid hazard rate model.Meanwhile,a new structure importance measure based on the survival signature is presented.Using this new importance measure method,an adjustment function is designed to update the PM maintenance threshold.For CM actions,a decision rule relying on the criticality level of components is introduced.In order to judge the criticality level of components,a novel structure updating method based on the survival signature is proposed.Moreover,a maintenance model considering economic dependence among components is developed.A 10-component system is finally introduced to illustrate the use and advantages of the proposed maintenance strategy.