Discrete half-logistic distributions with applications in reliability and risk analysis

In lifetime modeling, the observed measurements are usually discrete in nature, because the values are measured to only a finite number of decimal places and cannot really constitute all points in a continuum. For example, the survival time of a cancer patient can be measured as the number of months he/she survives. Then, even if the lifetime (of a patient, a device, etc.) is intrinsically continuous, it is reasonable to consider its observations as coming from a discretized distribution generated from an underlying continuous model. In this work, a discrete random distribution, supported on the non-negative integers, is obtained from the continuous half-logistic distribution by using a well-established discretization technique, which preserves the functional form of the survival function. Its main statistical properties are explored, with a special focus on the shape of the probability mass function and the determination of the first two moments; we discuss and compare, both theoretically and empirically, two different methods for estimating its unique parameter. This discrete random distribution can be used for modeling data exhibiting excess of zeros and over-dispersion, which are features often met in the insurance and ecology fields: an example of application is illustrated. An extension of this discrete distribution is finally suggested, by considering the generalized half-logistic distribution, which introduces a second shape parameter allowing for greater flexibility.

In many real-world applications, the phenomena of interest are continuous in nature and modeled through continuous probability distributions, but their observed values are actually discrete and hence it would be more reasonable and convenient to choose an appropriate (multivariate) discrete distribution generated from the underlying continuous model preserving one or more important features. In this paper, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely (1) the joint density function, or (2) the joint survival function. Examples of applications are presented, which involve two types of bivariate exponential distributions, in order to illustrate how the discretization procedures work and show whether and to which extent they alter the dependence structure of the original model. We also prove that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. Two real datasets, considering correlated discrete recurrence times (the former) and counts (the latter) are eventually fitted using two discrete analogues of a bivariate exponential distribution.

In this article we transmute the type I half logistic distribution using quadratic rank transmutation map to develop a transmuted type I half logistic distribution. The quadratic rank transmutation map enables the introduction of extra parameter into its baseline distribution to enhance more flexibility in the analysis of data in various disciplines such as reliability analysis in engineering, survival analysis, medicine, biological sciences, actuarial science, finance and insurance. The mathematical properties such as moments, quantile, mean, median, variance, skewness and kurtosis of this distribution are discussed. The reliability and hazard functions of the transmuted type I half logistic distribution are obtained. The probability density functions of the minimum and maximum order statistics of the transmuted type I half logistic distribution are established and the relationships between the probability density functions of the minimum and maximum order statistics of the parent model and the probability density function of the transmuted type I half logistic distribution are considered. The parameter estimation is done by the method of maximum likelihood estimation. The flexibility of the model in statistical data analysis and its applicability is demonstrated by using it to fit relevant data. The study is concluded by demonstrating that the transmuted type I half logistic distribution has a better goodness of fit than its parent model. We hope this model will serve as an alternative to the existing ones in the literature in fitting positive real data.

The uncertainty associated with many observed and modeled quantities of interest in Earth system prediction can be represented by mixed probability distributions that are neither discrete nor continuous. For instance, a forecast probability of precipitation can have a finite probability of zero precipitation, consistent with a discrete distribution. However, nonzero values are not discrete and are represented by a continuous distribution; the same is true for rainfall rate. Other examples include snow depth, sea ice concentration, the amount of a tracer, or the source rate of a tracer. Some Earth system model parameters may also have discrete or mixed distributions. Most ensemble data assimilation methods do not explicitly consider the possibility of mixed distributions. The quantile-conserving ensemble filter framework is extended to explicitly deal with discrete or mixed distributions. An example is given using bounded normal rank histogram probability distributions applied to observing system simulation experiments in a low-order tracer advection model. Analyses of tracer concentration and tracer source are shown to be improved when using the extended methods. A key feature of the resulting ensembles is that there can be ensemble members with duplicate values. An extension of the rank histogram diagnostic method to deal with potential duplicates shows that the ensemble distributions from the extended assimilation methods are more consistent with the truth. Significance Statement Data assimilation is a statistical method that is used to combine information from computer forecasts with measurements of the Earth system. The result is a better estimate of what is occurring in the physical system. As an example, data assimilation is used for making weather predictions. Some Earth system quantities, like precipitation, have special values that can occur very frequently. For instance, zero rainfall is quite common, while any other specific amount of rainfall, say, 0.42 in., is unusual. New data assimilation tools that work well for quantities like this are introduced and should lead to better estimates and predictions of the Earth system.

The Half-Logistic (HL) distribution is a renowned lifetime distribution widely used in various domains, including reliability engineering, biological sciences, and actuarial research. Despite its fundamental simplicity and easy interpretation, the HL distribution has limitations in terms of flexibility for modeling datasets with heavy tails or varying skewness. To address these constraints, this paper introduces a new extension of the HL distribution based on the Odd Beta Prime (OBP) family. The newly developed distribution has two shape parameters and is named the Odd Beta Prime Half-Logistic (OBPHL) distribution. Its several distributional properties, including the probability density function, cumulative distribution function, survival function, hazard function, and quantile function, alongside other various distributional characteristics, such as moments, measures of skewness and kurtosis, entropy measure ( Rényi and Tsallis), and order statistics, are explored. The maximum likelihood estimation technique is used to estimate the parameters of the new model. The performance of the estimators under different sample sizes has been evaluated using a Monte Carlo simulation study. In addition, the proposed OBPHL distribution is applied to two real-world datasets. The results are compared against several classical distributions using standard model selection criteria. The OBPHL distribution consistently outperforms the competing models based on evaluation metrics. This research contributes a new flexible model to the field of statistical distribution theory and demonstrates its efficacy in practical applications.

In this study, a new three-parameter mixed Poisson Cubic Rank Transmuted New Weighted Exponential Distribution is proposed. The new discrete distribution is obtained by mixing the Poisson distribution with a newly obtained Cubic Rank Transmuted New Weighted Exponential Distribution. Various shapes and mathematical properties of both mixing distribution and the new count distribution are examined. Special cases of the new proposition are also identified. The distribution along with its special cases and other count distributions are assumed for skewed and dispersed count observations. The maximum likelihood estimation is used to provide estimates for the parameters of all examined distributions. Results show that the new proposition along with some of its special cases provide good fit for all the examined data.

Lanthanide doped TiO2: Coexistence of discrete and continuous dopant distribution in anatase phase

We derive analytic expressions for the biases of the maximum likelihood estimators of the scale parameter in the half-logistic distribution with known location, and of the location parameter when the latter is unknown. Using these expressions to bias-correct the estimators is highly effective, without adverse consequences for estimation mean squared error. The overall performance of the first of these bias-corrected estimators is slightly better than that of a bootstrap bias-corrected estimator. The bias-corrected estimator of the location parameter significantly out-performs its bootstrapped-based counterpart. Taking computational costs into account, the analytic bias corrections clearly dominate the use of the bootstrap.

Preface. 1 Introduction. 1.1 The Importance of Risk and Uncertainty Assessments. 1.2 The Need to Develop a Proper Risk Analysis Framework. Bibliographic Notes. 2 Common Thinking about Risk and Risk Analysis. 2.1 Accident Risk. 2.1.1 Accident Statistics. 2.1.2 Risk Analysis. 2.1.3 Reliability Analysis. 2.2 Economic Risk. 2.2.1 General Definitions of Economic Risk in Business and Project Management. 2.2.2 A Cost Risk Analysis. 2.2.3 Finance and Portfolio Theory. 2.2.4 Treatment of Risk in Project Discounted Cash Flow Analysis. 2.3 Discussion and Conclusions. 2.3.1 The Classical Approach. 2.3.2 The Bayesian Paradigm. 2.3.3 Economic Risk and Rational Decision-Making. 2.3.4 Other Perspectives and Applications. 2.3.5 Conclusions. Bibliographic Notes. 3 How to Think about Risk and Risk Analysis. 3.1 Basic Ideas and Principles. 3.1.1 Background Information. 3.1.2 Models and Simplifications in Probability Considerations. 3.1.3 Observable Quantities. 3.2 Economic Risk. 3.2.1 A Simple Cost Risk Example. 3.2.2 Production Risk. 3.2.3 Business and Project Management. 3.2.4 Investing Money in a Stock Market. 3.2.5 Discounted Cash Flow Analysis. 3.3 Accident Risk. Bibliographic Notes. 4 How to Assess Uncertainties and Specify Probabilities. 4.1 What Is a Good Probability Assignment? 4.1.1 Criteria for Evaluating Probabilities. 4.1.2 Heuristics and Biases. 4.1.3 Evaluation of the Assessors. 4.1.4 Standardization and Consensus. 4.2 Modelling. 4.2.1 Examples of Models. 4.2.2 Discussion. 4.3 Assessing Uncertainty of Y. 4.3.1 Assignments Based on Classical Statistical Methods. 4.3.2 Analyst Judgements Using All Sources of Information. 4.3.3 Formal Expert Elicitation. 4.3.4 Bayesian Analysis. 4.4 Uncertainty Assessments of a Vector X. 4.4.1 Cost Risk. 4.4.2 Production Risk. 4.4.3 Reliability Analysis. 4.5 Discussion and Conclusions. Bibliographic Notes. 5 How to Use Risk Analysis to Support Decision-Making. 5.1 What Is a Good Decision? 5.1.1 Features of a Decision-Making Model. 5.1.2 Decision-Support Tools. 5.1.3 Discussion. 5.2 Some Examples. 5.2.1 Accident Risk. 5.2.2 Scrap in Place or Complete Removal of Plant. 5.2.3 Production System. 5.2.4 Reliability Target. 5.2.5 Health Risk. 5.2.6 Warranties. 5.2.7 Offshore Development Project. 5.2.8 Risk Assessment: National Sector. 5.2.9 Multi-Attribute Utility Example. 5.3 Risk Problem Classification Schemes. 5.3.1 A Scheme Based on Potential Consequences and Uncertainties. 5.3.2 A Scheme Based on Closeness to Hazard and Level of Authority. Bibliographic Notes. 6 Summary and Conclusions. Appendix A: Basic Theory of Probability and Statistics. A.1 Probability Theory. A.1.1 Types of Probabilities. A.1.2 Probability Rules. A.1.3 Random Quantities (Random Variables). A.1.4 Some Common Discrete Probability Distributions (Models). A.1.5 Some Common Continuous Distributions (Models). A.1.6 Some Remarks on Probability Models and Their Parameters. A.1.7 Random Processes. A.2 Classical Statistical Inference. A.2.1 Non-Parametric Estimation. A.2.2 Estimation of Distribution Parameters. A.2.3 Testing Hypotheses. A.2.4 Regression. A.3 Bayesian Inference. A.3.1 Statistical (Bayesian) Decision Analysis. Bibliographic Notes. Appendix B: Terminology. Bibliography. Index.

About the Authors

A new two-parameter family of discrete distributions.

Christoph and Schreiber (1998a) studied the discrete analogue of positive Linnik distribution and obtained its characterizations using survival function. In this paper, we introduce a bivariate form of the discrete Linnik distribution and study its distributional properties. Characterizations of the bivariate distribution are obtained using compounding schemes. Autoregressive processes are developed with marginals follow the bivariate discrete Linnik distribution.

We consider data modelling under one inflation for zero-truncated count data, as they typically arise in capture-recapture modelling. One-inflation in zero-truncated count data has recently found considerable attention. In this regard, zero-truncated New Discrete distribution and a distribution to a point mass at one are used to create a one-inflated model namely one-inflated zero-truncated New Discrete distribution. Its reliability characteristics, generating functions, and distributional properties are investigated in some detail. which includes survival function, hazard rate function, probability generating function, characteristic function, variance, skewness, and kurtosis. Monte Carlo Simulation have been undertaken to evaluate the effectiveness of the maximum likelihood estimators. To test the compatibility of our proposed model, the baseline model and the proposed model are distinguished by using the two different test procedures. The adaptability of the suggested model is demonstrated using two real-life datasets from separate domains by taking various performance measures into consideration.

This article introduces and discusses a new three-parameter lifespan distribution called Zero-Truncated Poisson Pareto distribution ZTPP. that is built on compounding Pareto distribution as a continuous distribution and Zero-Truncated Poisson distribution as a discrete distribution. Various statistical properties and reliability characteristics of the proposed distribution have been investigated including explicit expressions for the moments, moment generating function, quantile function, and median. With three parameters, the suggested distribution has an advantage over other distributions in that it makes estimating the model parameters simpler. To estimate the unknown parameters of the ZTPP distribution, the maximum likelihood method, and L. Moments method are employed. Moreover, a real data set is used to evaluate the significance and ensure the applicability of the proposed distribution as compared to other probability distributions. The derived model proved to be the best compared to other fitted models, where the criteria values of (AIC), (CAIC), and (BIC) are minimum values by using the ZTPP distribution. The proposed model is hoped to attract a wider application.

In decision and risk analysis, it is common to use discrete probability distributions to approximate uncertain events with continuous outcomes. This paper discusses how these approximations may be selected. A class of approximations based on a modification to Taguchi's work on tolerance analysis is shown to be optimal under assumptions of independent uncertainties with normally distributed outcomes. The approximation procedure is shown to be robust in many other situations and is extremely easy to use in practice. We also show how the approximation may be integrated into the process of subjective probability estimation by a ‘subject-matter expert’.