Extended Half-Logistic Distribution with Theory and Lifetime Data Application

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.

This paper develops maximum likelihood estimators (MLEs) of unknown param-eters in an exponentiated half-logistic distribution based on a progressively type-IIcensored sample. We obtain approximate con dence intervals for the MLEs by usingasymptotic variance and covariance matrices. Using importance sampling, we obtainBayes estimators and corresponding credible intervals with the highest posterior densityand Bayes predictive intervals for unknown parameters based on progressively type-IIcensored data from an exponentiated half logistic distribution. For illustration pur-poses, we examine the validity of the proposed estimation method by using real andsimulated data.Keywords: Bayes predictive interval, exponentiated half-logistic distribution, HPD cred-ible interval, importance sampling, progressively type-II censored sample. 1. Introduction Many reliability and survival analysis studies have used half-logistic distributions, partic-ularly for censored data. Several studies have drawn inferences about half logistic distribu-tions. Balakrishnan and Puthenpura (1986) introduce the best linear unbiased estimator oflocation and scale parameters of half-logistic distributions by considering linear functionsof order statistics. Balakrishnan and Wong (1991) obtain approximate maximum likelihoodestimators (AMLEs) of location and scale parameters of half-logistic distributions by usinga type-II right-censored sample. Kang et al. (2008) derive AMLEs and MLEs of scale pa-rameters of half-logistic distributions by using progressively type-II censored samples. Kanget al. (2009) propose AMLEs of scale parameters of half-logistic distributions by consideringdouble-hybrid censored samples. Kang and Seo (2011) recently examine exponentiated half-logistic distributions and propose two types of AMLEs of scale parameters and reliabilityfunctions by using progressively type-II right-censored samples. Kim et al. (2011a) deriveBayesian estimators of shape parameters, reliability functions, and failure rate functions

For the half-logistic distribution, the maximum likelihood method does not provide an explicit estimator for the scale parameter based on a progressively Type-II censored sample. In this article, we first present a simple method of deriving an explicit estimator by approximating the likelihood function. We then examine through simulations the bias and variance of this estimator and show that this estimator is as efficient as the maximum likelihood estimator (MLE). Next, we show that the probability coverages of the pivotal quantities (for the scale parameter) based on asymptotic normality are unsatisfactory, especially when the effective sample size is small. Therefore, we suggest using unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate all the methods of inference discussed here.

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.

Considerations of sample size in medical research

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

A generalization of the half logistic distribution is developed through exponentiation of its cumulative distribution function and termed the Type I Generalized Half Logistic Distribution (GHLD). GHLD’s distributional characteristics and parameter estimation using maximum likelihood and modified maximum likelihood methods are presented with comparisons. Comparison of Type I GHLD and the exponential distribution is conducted via likelihood ratio criterion.

In the statistical literature, several discrete distributions have been developed so far for modeling non-negative integer-valued phenomena, yet there is still room for new counting models that adequately capture the diversity of real data sets. Here, we first discuss a count distribution derived as a discrete analogue of the continuous half-logistic distribution, which is obtained by preserving the expression of its survival function at each non-negative integer support point. The proposed discrete distribution has a mode at zero and allows for over-dispersion; these two features make it suitable for modeling purposes in many fields (e.g., insurance and ecology), when these conditions are satisfied by the data. In order to widen its spectrum of applications, a discrete analogue is also presented of the type I generalized half-logistic distribution (obtained by adding a shape parameter to the simple one-parameter half-logistic), which allows us to model count data whose mode is not necessarily zero. For these new count distributions, the main statistical properties are outlined, and parameter estimation along with related issues is discussed. Their feasibility is proved on two real data sets taken from the literature, which have already been fitted by other well-established count distributions. Finally, a possible application is illustrated in the insurance field, related to the exact/approximate determination of the distribution of the total claims amount through the well-known Panjer’s recursive formula, within the framework of collective risk models.

A lifetime model with increasing failure rate

In recent decades we have seen an increased interest in the use of seemingly unrelated regressions models (SUR) in a spatial context, with compelling case studies in different fields. This upsurge has favoured the development of new and more efficient inference techniques. At present, the user has a basic toolkit to deal with this kind of model, that is, however, in need of improvement. This paper focuses on the question of estimating, quickly and accurately, spatial SUR models. The most popular procedure is maximum likelihood (ML) which guarantees precision at the cost of a high computational burden. This is especially true in cases of large sample size and strong spatial structure. We explore simpler estimation algorithms such as instrumental variables (IV), which expedites calculation at the cost of a certain loss in quality of the estimates. We focus on the importance of sample size and the trade-off between accuracy and speed. To that end, we perform a comprehensive simulation experiment in which we compare ML and IV algorithms, looking for their strengths and weaknesses. The paper includes two applications to the case of Airbnb in the urban area of Madrid. First, we estimate a spatial SUR hedonic model of accommodation prices, using micro-data for three different cross sections a la Anselin, that is, considering temporal correlation plus spatial structure. Then, the data are aggregated by neighbourhoods. We specify a spatial SUR model with two equations (apartments and rooms) also using three cross sections. In both cases, the models are estimated by ML and IV using the spsur R package (Angulo et al. in spSUR: spatial seemingly unrelated regression models. R Package version 1.0.0.3, 2019), with the aim of illustrating its capabilities.

본 연구는 비연속 분할함수 성장혼합모형의 접근법에 따라 계층 간 원격 결과변인 추정치에 어떠한 차이가 있는지를 확인하는 데 목적이 있다. 이를 위하여 본 연구는 3단계 ML접근법과 3단계 BCH 접근법을 적용하여 모의실험 연구를 수행하였다. 모의실험에 계층 분류정도(1.3, 2.6), 계층 혼합비율(90:10, 70:30, 50:50), 표본크기(200, 500, 1000, 2000)의 조건을 설정하였다. 분석결과, 3단계 ML접근법과 3단계 BCH접근법의 원격 결과변인에서 양호도의 차이는 미미한 것으로 나타났지만, 소표본일 때 3단계 BCH접근법이 3단계 ML접근법에 비해 상대적으로 효율성이 높은 것으로 나타났다. 본 연구는 원격 결과변인의 보조변수 접근법에 기존에 활용되지 않았던 비연속 분할함수 성장혼합모형을 적용하고, 다양한 모의실험 조건을 설정하여 모의실험 연구를 수행하였다는 점에서 의의가 있다. 본 연구가 연구자들에게 실질적인 모형 활용 방안을 제시할 수 있을 것으로 기대하며, 연구의 제한점을 바탕으로 후속연구를 제언하였다.The purpose of this study was to compare the estimates of distal outcome variable in discontinuous growth mixture model. For this purpose, this study carried out a simulation study using a three-step maximum likelihood (ML) approach and a three-step BCH approach. The manipulated conditions were the class separation (1.3, 2.6), the mixing proportions (90:10, 70:30, 50:50), and the sample size (200, 500, 1000, 2000). The differences between ML and a three-step BCH approaches were not substantial, however, the three-step BCH approach was relatively more efficient than the three-step ML approach in the conditions of relatively small sample sizes. In particular, the bias of standard error was smaller in the conditions of the relatively less mixing proportions when three-step BCH approach was used. Limitations of the current study and suggestions for future study were provided.

Chi-square tests based on maximum likelihood (ML) estimation of covariance structures often incorrectly over-reject the null hypothesis: Σ = Σ θ when the sample size is small. Reweighted least squares (RLS) avoids this problem. In some models, the vector of parameter must contain means, variances, and covariances, yet whether RLS also works in mean and covariance structures remains unexamined. This research extends RLS to mean and covariance structures, evaluating a generalized least squares function with ML parameter estimates. A Monte Carlo simulation study was carried out to examine the statistical performance of ML vs RLS with multivariate normal data. Based on empirical rejection frequencies and empirical averages of test statistics, this study shows that RLS performs much better than ML in mean and covariance structure models when sample sizes are small, whereas it does not perform better than ML to reject misspecified models.

Two regression methods for estimation of a two-parameter Weibull distribution for reliability of dental materials

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.