On Square Logistic Distribution: Mathematical Properties, Estimation Methods and Applications in Biology

Statistical data analysis available in most scientific fields is often recorded with measurement error. The modeling of these statistical data by ignoring the measurement errors, leads to estimators of the parameters of the distributions, whose use does not achieve sufficient accuracy in the goodness of fit. In reliability criteria, one of the important issues is hazard rate function. It prompted us to investigate the hazard rate criterion in the presence of measurement error generated from the normal or logistic distribution. Now, while providing the estimator for the density function using local time polynomial estimator methods, the risk rate function is estimated according to the contamination degree of 15 or 30%. Finally, we present the numerical analysis.

One of the novel methods described by the researchers for comparing several independent populations is to compare their quantiles. Researchers have previously considered comparing quantiles for normal and exponential model setups. In some cases, the datasets could be modeled using a different distribution, such as a logistic distribution. In this study, we consider the problem of comparing the quantiles for several ( k ≥ 2 ) logistic populations. First, the conventional likelihood ratio test is applied, and based on its asymptotic property, the cutoff point is determined. We propose two modifications of the likelihood ratio test- a standardized likelihood ratio test and a parametric bootstrap likelihood ratio test. Further, we propose a computational approach test to test the equality of quantiles by leveraging technology. The sizes and powers of all proposed tests are compared using an extensive simulation study. Except for the asymptotic likelihood ratio test, all the tests achieve the nominal level which has been revealed from our simulation study. Finally, we illustrate the importance of our model problem using two real-life situations.

This paper presents a novel extension of the exponentiated inverse Rayleigh distribution called the half-logistic exponentiated inverse Rayleigh distribution. This extension improves the flexibility of the distribution for modeling lifetime data for both monotonic and non-monotonic hazard rates. The statistical properties of the half-logistic exponentiated inverse Rayleigh distribution, such as the quantiles, moments, reliability, and hazard function, are examined. In particular, we provide several techniques to estimate the half-logistic exponentiated inverse Rayleigh distribution parameters: weighted least squares, Cramér-Von Mises, maximum likelihood, maximum product spacings and ordinary least squares methods. Moreover, numerical simulations were performed to evaluate these estimation methods for both small and large samples through Monte Carlo simulations, and the finding reveals that the maximum likelihood estimation was the best among all estimation methods since it comprises small mean square error compared to other estimation methods. We employ real-world lifetime data to demonstrate the performance of the newly generated distribution compared to other distributions through practical application. The results show that the half-logistic exponentiated inverse Rayleigh distribution performs better than alternative versions of the Rayleigh distributions.

This study develops a novel lifespan known as the Type Two Half Logistic Topp-Leone exponentiated generalized-G distributions. The motivation behind generating the new probability distributions lies in the pursuit of overcoming limitations of other existing distributions. The proposed model is more flexible and better captures the complexities of real-world data. The recommended distribution’s moments, hazard functions, and several other statistical characteristics are identified. The maximum likelihood estimators of the model parameters are discussed. Three different special models are provided. We demonstrate the potentiality of the new distribution using simulation data and two actual sets of data.

In this chapter, we consider models for experiments in which the stress levels are altered at intermediate stages during the exposure. These experiments, referred to as step-stress tests, belong to the class of accelerated models that are extensively used in reliability and life-testing applications. Models for step-stress tests have largely relied on the cumulative exposure model (CEM) discussed by Nelson. Unfortunately, the assumptions of the model are fairly restrictive and quite unreasonable for applications in survival analysis. In particular, under the CEM the hazard function has discontinuities at the points at which the stress levels are changed. We introduce a new step-stress model where the hazard function is continuous. We consider a simple experiment with only two stress levels. The hazard function is assumed to be constant at the two stress levels and linear in the intermediate period. This model allows for a lag period before the effects of the change in stress are observed. Using this formulation in terms of the hazard function, we obtain the maximum likelihood estimators of the unknown parameters. A simple least squares-type procedure is also proposed that yields closed-form solutions for the underlying parameters. A Monte Carlo simulation study is performed to study the behavior of the estimators obtained by the two methods for different choices of sample sizes and parameter values. We analyze a real data set and show that the model provides an excellent fit.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

The impact of distribution on value-at-risk measures

Full and conditional likelihood approaches for hazard change-point estimation with truncated and censored data

In this paper, we introduce a new three-parameter distribution, the New Exponentiated Burr Type III distribution (NEBIII), which is a member of the Generalized (G)-family of continuous distributions. The mathematical features that are derived include the mode, actuarial measures, order statistics, the analytical forms of the density and hazard functions, and explicit formulations for the moment generating function (MGF). The Maximum Likelihood Estimation (MLE) approach is used to estimate the model parameters. A simulated research with different sample sizes is used to evaluate the estimation's efficacy. The versatility and adaptability of the proposed distribution family are demonstrated on four real-world data sets. Additionally, we examine a Mixture of the Exponential and Exponentiated Burr Type III Distributions, identifying various mathematical characteristics and demonstrating their application to actual data.

Weibull distribution is one of the most classical distributions in the field of reliability applications. In order to improve the capability to fit various failure data, a number of extended forms of Weibull distribution have been proposed. In this paper, we will focused on the four-parameter Weibull distribution. It is difficult to estimate the parameters of fourparameter Weibull model accurately under limited failure data, but we can treat the parameter as a random variable and use the gamma distribution as a prior distribution to get the imprecise ones. From the view of impreciseness and Bayesian theory, we proposed a novel imprecise reliability analytical method named the Four-parameter Weibull-Gamma model which used for incomplete life data. The upper and lower bounds of the cumulative density function, the reliability function and the hazard function can also be obtained. Aiming at two cases that the hazard function is monotonically increasing or a bathtub shape, the experiments demonstrate the fitting effect of the proposed model, and the modeling capability of components or systems for different hazard function. Furthermore, it also verifies the multi-parameter of the proposed model can increase the adjustable flexibility of the distribution.

Often preventive maintenance (PM) performed to a pavement does not restore a condition to as good as new. This study presents a practical pavement maintenance policy based on imperfect PM and mechanistic–empirical analysis. The international roughness index is used as the condition indicator. The policy is based on an increased hazard rate model from one cycle to the next and on a minimum reliability threshold. Monte Carlo simulations are used to develop the reliability function of flexible pavements, from which the hazard rate function is determined. The change of hazard rate from one cycle to the next is modeled with the age reduction factor and the hazard-rate increase factor. The expected cost per unit time of the pavement is minimized such that the minimum reliability is always maintained. The minimization procedure produces the number of PM cycles required. Case studies are presented with analyses of results showing the application of the procedure. The study indicates that hazard-rate function increases over time, and more-frequent PM actions are needed when the selected threshold reliability level is higher.

<p>Ordinal response variables abound in scientific and quantitative analyses, whose outcomes comprise a few categorical values that admit a natural ordering, so that their values are often represented by non-negative integers, for instance, pain score (0-10) or disease severity (0-4) in medical research. Ordinal variables differ from rational variables in that its values delineate qualitative rather than quantitative differences. In this thesis, we develop new statistical methods for variable selection in a high-dimensional cumulative link regression model with an ordinal response. Our study is partly motivated by the needs for exploring the association structure between disease phenotype and high-dimensional medical covariates.</p><p>The cumulative link regression model specifies that the ordinal response of interest results from an order-preserving quantization of some latent continuous variable that bears a linear regression relationship with a set of covariates. Commonly used error distributions in the latent regression include the normal distribution, the logistic distribution, the Cauchy distribution and the standard Gumbel distribution (minimum). The cumulative link model with normal (logit, Gumbel) errors is also known as the ordered probit (logit, complementary log-log) model. While the likelihood function has a closed-form solution for the aforementioned error distributions, its strong nonlinearity renders direct optimization of the likelihood to sometimes fail. To mitigate this problem and to facilitate extension to penalized likelihood estimation, we proposed specific minorization-maximization (MM) algorithms for maximum likelihood estimation of a cumulative link model for each of the preceding 4 error distributions.</p><p>Penalized ordinal regression models play a role when variable selection needs to be performed. In some applications, covariates may often be grouped according to some meaningful way but some groups may be mixed in that they contain both relevant and irrelevant variables, i.e., whose coefficients are non-zero and zero, respectively. Thus, it is pertinent to develop a consistent method for simultaneously selecting relevant groups and the relevant variables within each selected group, which constitutes the so-called bi-level selection problem. We have proposed to use a penalized maximum likelihood approach with a composite bridge penalty to solve the bi-level selection problem in a cumulative link model. An MM algorithm was developed for implementing the proposed method, which is specific to each of the 4 error distributions. The proposed approach is shown to enjoy a number of desirable theoretical properties including bi-level selection consistency and oracle properties, under suitable regularity conditions. Simulations demonstrate that the proposed method enjoys good empirical performance. We illustrated the proposed methods with several real medical applications.</p>

A Flexible Probability Model for Reliability Data Analysis: The Extended Half-Normal Distribution with Further Results

In this paper, we propose a new method for generating distributions based on the idea of alpha power transformation introduced by Mahdavi and Kundu (Commun Stat Theory Methods 46(13):6543–6557, 2017). The new method can be applied to any distribution by inverting its quantile function as a function of alpha power transformation. We apply the proposed method to the Weibull distribution to obtain a three-parameter alpha power within Weibull quantile function. The new distribution possesses a very flexible density and hazard rate function shapes which are very useful in cancer research. The hazard rate function can be increasing, decreasing, bathtub or upside down bathtub shapes. We derive some general properties of the proposed distribution including moments, moment generating function, quantile and Shannon entropy. The maximum likelihood estimation method is used to estimate the parameters. We illustrate the applicability of the proposed distribution to complete and censored cancer data sets.

The hazard function describes the instantaneous rate of failure at a time t, given that the individual survives up to the instant t. The effect of the covariates produces a variation in the hazard function, hence a change point might occur. When dealing with survival analysis, it is of interest to identify where a change point has occurred. This paper proposes a new method for estimating the change point in the Cox proportional hazard model, which is based on maximum likelihood estimation combined with moments estimation (ME) and a numerical mehtod to minimize an objective function given by ME. The mean square error of the estimator is obtained by Monte Carlo simulation, considering different scenarios. For the purpose of studying the behavior of the proposed estimator in terms of its mean square error, a comparative study against a known method to estimate the change point is included. A real data application is also included.