Estimation and prediction for Topp–Leone distribution using double Type-I hybrid censored data

<abstract> <p>In this paper, a new censoring test plan called progressively double Type-II hybrid censoring scheme is introduced for the first time. Based on this type of censored data, the maximum likelihood estimates of the unknown parameters and reliability for the two-parameter Pareto distribution are obtained. Using the Bayesian method, the Bayesian estimates of the unknown parameters and reliability are obtained under the symmetric and asymmetric loss functions. The failure times of all withdrawn units are predicted using the classical and Bayesian methods, including the predictive values and the prediction intervals. The mean values and mean square errors of the estimators are calculated by Monte-Carlo simulation, and the mean square errors between them are compared, and the results show that all Bayesian estimates are better than the corresponding maximum likelihood estimates. Using a real data set, we compute the Bayesian estimates of the unknown parameters and reliability, and predict the observations of the censored units.</p> </abstract>

In this paper, we propose an extension to the Topp-Leone distribution, as introduced by [20] using the Generalized-DUS transformation given by [8]. The Topp-Leone distribution is defined on interval (0,1) and has a characteristic J-shaped frequency curve. The newly extended version of Topp-Leone distribution accommodates a variety of shapes of hazard rate functions making it a versatile distribution. We have also derived explicit expressions for some properties like ordinary moments, conditional moments, distribution of order statistics, quantiles, mean deviation, and entropy. Further, we have also discussed results on identifiability, stress-strength reliability, and stochastic ordering that are concerned with two independent random variables. For inference regarding the unknown parameters of the distribution, we derive the equations which give their maximum likelihood estimators. We also present the asymptotic confidence intervals of the unknown parameters of the distribution, based on large sample property, using the Fisher information matrix. To facilitate further studies, a step-by-step algorithm is presented to produce a random sample from the distribution. Further, extensive simulation experiments are done to study the long-term behavior of the maximum likelihood estimators of the parameters through their mean squared error and mean absolute bias on the basis of large number of samples. The consistency of the MLEs is empirically proved. Lastly, the application of the proposed distribution is shown by fitting a real-life dataset over some existing distributions in the same range.

In this study, we propose a new family of distributions motivated by the Topp-Leone distribution. We present graphical illustrations of its probability density, reliability and hazard function, along with a discussion of their practical relevance in life testing applications. Explicit algebraic expressions are derived for the moments, exact moments based on order statistics, and the stress-strength reliability measure. Parameter estimation is carried out using multiple approaches, including the Method of maximum likelihood, Method of moments and Bayesian estimation under both Squared error and Stein loss functions. Finally, a simulation study is conducted to assess the performance of the proposed estimators and the applicability of the new model is demonstrated through the analysis of a real dataset.. KEYWORDS :Topp-Leone distribution, Stress-Strength reliability, Maximum likelihood estimation, Bayesian estimation.

ABSTRACTIn this paper, we have derived exact and explicit expressions for the ratio and inverse moments of dual generalized order statistics from Topp-Leone distribution. This result includes the single and product moments of order statistics and lower records . Further, based on n dual generalized order statistics, we have deduced the expression for Maximum likelihood estimator (MLE) and Uniformly minimum variance unbiased estimator (UMVUE) for the shape parameter of Topp-Leone distribution. Finally, based on order statistics and lower records, a simulation study is being carried out to check the efficiency of these estimators.

This paper aims to find a statistical model for the COVID-19 spread in the United Kingdom and Canada. We used an efficient and superior model for fitting the COVID 19 mortality rates in these countries by specifying an optimal statistical model. A new lifetime distribution with two-parameter is introduced by a combination of inverted Topp-Leone distribution and modified Kies family to produce the modified Kies inverted Topp-Leone (MKITL) distribution, which covers a lot of application that both the traditional inverted Topp-Leone and the modified Kies provide poor fitting for them. This new distribution has many valuable properties as simple linear representation, hazard rate function, and moment function. We made several methods of estimations as maximum likelihood estimation, least squares estimators, weighted least-squares estimators, maximum product spacing, Crame´r-von Mises estimators, and Anderson-Darling estimators methods are applied to estimate the unknown parameters of MKITL distribution. A numerical result of the Monte Carlo simulation is obtained to assess the use of estimation methods. also, we applied different data sets to the new distribution to assess its performance in modeling data.

Multilevel modelling of clustered grouped survival data using Cox regression model: an application to ART dental restorations

In this article, a new modified asymmetric Topp–Leone distribution is created and developed from a theoretical and inferential point of view. It has the feature of extending the remarkable flexibility of a special one-shape-parameter lifetime distribution, known as the inverse Topp–Leone distribution, to the bounded interval [0, 1]. The probability density function of the proposed truncated distribution has the potential to be unimodal and right-skewed, with different levels of asymmetry. On the other hand, its hazard rate function can be increasingly shaped. Some important statistical properties are examined, including several different measures. In practice, the estimation of the model parameters under progressive type-II censoring is considered. To achieve this aim, the maximum likelihood, maximum product of spacings, and Bayesian approaches are used. The Markov chain Monte Carlo approach is employed to produce the Bayesian estimates under the squared error and linear exponential loss functions. Some simulation studies to evaluate these approaches are discussed. Two applications based on real-world datasets—one on the times of infection, and the second dataset is on trading economics credit rating—are considered. Thanks to its flexible asymmetric features, the new model is preferable to some known comparable models.

The Topp-Leone distribution is widely used for modeling lifetime data across many disciplines. This paper presents the Exponentiated Cosine Topp-Leone Generalized family (ExCTLG), which is an extension of the Topp-Leone family. The research examines the mathematical properties of the ExCTLG, including the survival and hazard functions, moments, moment-generating functions, and Renyi’s entropy. Parameter estimation is carried out using the maximum likelihood estimation (MLE) techniques, and the estimated parameters consistency is validated using the Monte Carlo simulation method, thereby highlighting the superiority of MLE. The advantages and applicability of the proposed distribution are shown by analyzing two-lifetime datasets.

Study’s Excerpt/Novelty This study provides a comprehensive evaluation of parameter estimation methods for the Log-Topp-Leone (L-T-L), Topp-Leone-Exponential (T-L-E), and Topp-Leone-Weibull (T-L-W) distributions using maximum likelihood (ML) and maximum product spacing (MPS) techniques. Through simulations conducted in R-Studio, the research demonstrates the consistency and adequacy of these estimators, particularly highlighting the unbiased and efficient performance of the ML estimator. The findings underscore the recommendation of ML over MPS for parameter estimation in Topp-Leone distribution extensions, contributing valuable insights into statistical methodology for complex distribution models. Full Abstract The parameters for Log-Topp-leone (L-T-L) distribution, Topp-leone-exponential (T-L-E) distribution, and Top-plane-Weibull (T-L-W)distribution were estimated via the methods of ML (maximum likelihood) and MPS (maximum product spacing). Simulations were carried out using R-Studio to estimate the parameters and check the efficiency of the estimators. As a consequence, if the sample size (SS) increases, the outcome of the estimations tends to the true PV (parameter values), which proves the consistency and adequacy (C & A) of all the estimators. Similarly, the MSE and biases using the two estimators approach zero, even though some MPS bias values fluctuated. This shows the suitability of each estimator, and MLE is more unbiased and adequate. The estimates using MLE and MPS were efficient. Thus, we recommend that MLE be employed in assessing and computing the parameters of extensions of the Topp-Leone (TL) distribution.

SYNOPTIC ABSTRACTIn this study, we have obtained the uniformly minimum variance unbiased estimator (UMVUE) of reliability function and stress–strength reliability for one parameter proportional reversed hazard rate family of distribution based on lower record values. Further, the results are reduced for power function distribution, exponentiated Weibull distribution, generalized exponential distribution, generalized Rayleigh (also known as Burr type X) distribution and Topp–Leone distribution. Also, the UMVU estimator and maximum likelihood estimator are compared through simulation.

This paper aims at defining an optimal statistical model for the COVID-19 distribution in the United Kingdom, and Canada. A combining the inverted Topp–Leone distribution and the odd Weibull family introduces a new lifetime distribution with a three-parameter to formulate the odd Weibull inverted Topp–Leone (OWITL) distribution. As a simple linear representation, hazard rate function, and moment function, this new distribution has several nice properties. To estimate the unknown parameters of OWITL distribution, maximum likelihood, least-square, weighted least-squares, maximum product spacing, Cramér–von Mises estimators, and Anderson–Darling estimation methods are used. To evaluate the use of estimation techniques, a numerical outcome of the Monte Carlo simulation is obtained.

Recent developments in applied statistics have given rise to the continuous Bernoulli distribution, a one-parameter distribution with support of [0, 1]. In this paper, we use it for a more general purpose: the creation of a family of distributions. We thus exploit the flexible functionalities of the continuous Bernoulli distribution to enhance the modeling properties of well-referenced distributions. We first focus on the theory of this new family, including the quantiles, expansion of important functions, and moments. Then we exemplify it by considering a special baseline: the Topp–Leone distribution. Thanks to the functional structure of the continuous Bernoulli distribution, we create a new two-parameter distribution with support for [0, 1] that possesses versatile shape capacities. In particular, the corresponding probability density function has left-skewed, N-type and decreasing shapes, and the corresponding hazard rate function has increasing and bathtub shapes, beyond the possibilities of the corresponding functions of the Topp–Leone distribution. Its quantile and moment properties are also examined. We then use our modified Topp–Leone distribution from a statistical perspective. The two parameters are supposed to be unknown and then estimated from proportional-type data with the maximum likelihood method. Two different data sets are considered, and reveal that the modified Topp–Leone distribution can fit them better than popular rival distributions, including the unit-Weibull, unit-Gompertz, and log-weighted exponential distributions. It also outperforms the Topp–Leone and continuous Bernoulli distributions.

The importance of providing structural validity evidence for test score(s) derived from psychometric test instruments is highlighted by several institutions; for example, the American Psychological Association (2014) demands that evidence for the validity of an instruments' internal structure and its underlying measurement model must be provided before it is applied in psychological assessment. The knowledge about the latent structure of data obtained with tests addressing the major question "What is/are the construct[s] being measured" by psychological tests under investigation (Ziegler, 2014 (Ziegler, , 2020)) . The study of structural validity is typically addressed with factor analyses when the test scores reflect continuous latent traits. As most submissions to Psychological Test Adaptation and Development (PTAD) deal with the adaptation and further development of existing measures, authors typically test a measurement model that is based on theoretical considerations and prior findings on original versions (or adaptations) of the test under investigation. Our literature review of PTAD's publications showed that more than 90% of the articles contain at least one confirmatory factor analysis (CFA). As editor and reviewers of PTAD, we appreciate that authors are rigorous in providing evidence on the structural validity of their tests' data. However, since PTAD's inception in 2019, we experience that one comment is frequently communicated to authors during the review process, namely, the request to adjust the analytic approach in CFA from maximum likelihood (ML) estimation toward using the mean-and variance-adjusted weighted least squares (WLSMV; Muthén et al., 1997) estimator to account for the ordinal nature of the data that psychological instruments typically generate on the item level. In this editorial, we discuss the rationale behind choosing the WLSMV estimator when analyzing test adaptations and developments that are based on ordinal categorical data and concisely illustrate the problems associated with using the ML estimator (potentially in combination with robust tests of model fit) for such data.

The existence of nonmonotonic hazard rates was recognized from the study of human mortality three centuries ago. Among such hazard rates, ones with bathtub or upside-down bathtub shape have received considerable attention during the last five decades. Several models have been suggested to represent lifetimes possessing bathtub-shaped hazard rates. In this chapter, we review the existing results and also discuss some new models based on quantile functions. We discuss separately bathtub-shaped distributions with two parameters, three parameters, and then more flexible families. Among the two-parameter models, the Topp-Leone distribution, exponential power, lognormal, inverse Gaussian, Birnbaum and Saunders distributions, Dhillon’s model, beta, Haupt-Schabe models, loglogistic, Avinadev and Raz model, inverse Weibull, Chen’s model and a flexible Weibull extension are presented along with their quantile functions. The quadratic failure rate distribution, truncated normal, cubic exponential family, Hjorth model, generalized Weibull model of Mudholkar and Kollia, exponentiated Weibull, Marshall-Olkin family, generalized exponential, modified Weibull extension, modified Weibull, generalized power Weibull, logistic exponential, generalized linear failure rate distribution, generalized exponential power, upper truncated Weibull, geometric-exponential, Weibull-Poisson and transformed model are some of the distributions considered under three-parameter versions. Distributions with more than three parameters introduced by Murthy et al., Jiang et al., Xie and Lai, Phani, Agarwal and Kalla, Kalla, Gupta and Lvin, and Carrasco et al. are presented as more flexible families. We also introduce general methods that enable the construction of distributions with nonmonotone hazard functions. In the case of many of the models so far specified, the hazard quantile functions and their analysis are also presented to facilitate a quantile-based study. Finally, the properties of total time on test transforms and Parzen’s score function are utilized to develop some new methods of deriving quantile functions that have bathtub hazard quantile functions.

Topp–Leone distribution is a continuous unimodal distribution with bounded support (recently rediscovered) which is useful for modelling life-time phenomena. In this paper we study some reliability measures of this distribution such as the hazard rate, mean residual life, reversed hazard rate, expected inactivity time, and their stochastic orderings.