rate functions
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2022 ◽  
Author(s):  
Sebastian Hoehna ◽  
Bjoern Tore Kopperud ◽  
Andrew F Magee

Diversification rates inferred from phylogenies are not identifiable. There are infinitely many combinations of speciation and extinction rate functions that have the exact same likelihood score for a given phylogeny, building a congruence class. The specific shape and characteristics of such congruence classes have not yet been studied. Whether speciation and extinction rate functions within a congruence class share common features is also not known. Instead of striving to make the diversification rates identifiable, we can embrace their inherent non-identifiable nature. We use two different approaches to explore a congruence class: (i) testing of specific alternative hypotheses, and (ii) randomly sampling alternative rate function within the congruence class. Our methods are implemented in the open-source R package ACDC (https://github.com/afmagee/ACDC). ACDC provides a flexible approach to explore the congruence class and provides summaries of rate functions within a congruence class. The summaries can highlight common trends, i.e. increasing, flat or decreasing rates. Although there are infinitely many equally likely diversification rate functions, these can share common features. ACDC can be used to assess if diversification rate patterns are robust despite non-identifiability. In our example, we clearly identify three phases of diversification rate changes that are common among all models in the congruence class. Thus, congruence classes are not necessarily a problem for studying historical patterns of biodiversity from phylogenies.


2022 ◽  
Author(s):  
Tamás Jónás ◽  
Christophe Chesneau ◽  
József Dombi ◽  
Hassan Salah Bakouch

This paper is devoted to a new flexible two-parameter lower-truncated distribution, which is based on the inversion of the so-called epsilon distribution. It is called the inverse epsilon distribution. In some senses, it can be viewed as an alternative to the inverse exponential distribution, which has many applications in reliability theory and biology. Diverse properties of the new lower-truncated distribution are derived including relations with existing distributions, hazard and reliability functions, survival and reverse hazard rate functions, stochastic ordering, quantile function with related skewness and kurtosis measures, and moments. A demonstrative survival times data example is used to show the applicability of the new model.


2021 ◽  
Author(s):  
Predrag Jelenković ◽  
Jané Kondev ◽  
Lishibanya Mohapatra ◽  
Petar Momčilović

Single-class closed queueing networks, consisting of infinite-server and single-server queues with exponential service times and probabilistic routing, admit product-from solutions. Such solutions, although seemingly tractable, are difficult to characterize explicitly for practically relevant problems due to the exponential combinatorial complexity of its normalization constant (partition function). In “A Probabilistic Approach to Growth Networks,” Jelenković, Kondev, Mohapatra, and Momčilović develop a novel methodology, based on a probabilistic representation of product-form solutions and large-deviations concentration inequalities, which identifies distinct operating regimes and yields explicit expressions for the marginal distributions of queue lengths. From a methodological perspective, a fundamental feature of the proposed approach is that it provides exact results for order-one probabilities, even though the analysis involves large-deviations rate functions, which characterize only vanishing probabilities on a logarithmic scale.


Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1662
Author(s):  
Ahmed Sayed M. Metwally ◽  
Amal S. Hassan ◽  
Ehab M. Almetwally ◽  
B M Golam Kibria ◽  
Hisham M. Almongy

The inverted Topp–Leone distribution is a new, appealing model for reliability analysis. In this paper, a new distribution, named new exponential inverted Topp–Leone (NEITL) is presented, which adds an extra shape parameter to the inverted Topp–Leone distribution. The graphical representations of its density, survival, and hazard rate functions are provided. The following properties are explored: quantile function, mixture representation, entropies, moments, and stress–strength reliability. We plotted the skewness and kurtosis measures of the proposed model based on the quantiles. Three different estimation procedures are suggested to estimate the distribution parameters, reliability, and hazard rate functions, along with their confidence intervals. Additionally, stress–strength reliability estimators for the NEITL model were obtained. To illustrate the findings of the paper, two real datasets on engineering and medical fields have been analyzed.


Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2836
Author(s):  
Rashad A. R. Bantan ◽  
Christophe Chesneau ◽  
Farrukh Jamal ◽  
Mohammed Elgarhy ◽  
Waleed Almutiry ◽  
...  

In this article, a structural modification of the Kumaraswamy distribution yields a new two-parameter distribution defined on (0,1), called the modified Kumaraswamy distribution. It has the advantages of being (i) original in its definition, mixing logarithmic, power and ratio functions, (ii) flexible from the modeling viewpoint, with rare functional capabilities for a bounded distribution—in particular, N-shapes are observed for both the probability density and hazard rate functions—and (iii) a solid alternative to its parental Kumaraswamy distribution in the first-order stochastic sense. Some statistical features, such as the moments and quantile function, are represented in closed form. The Lambert function and incomplete beta function are involved in this regard. The distributions of order statistics are also explored. Then, emphasis is put on the practice of the modified Kumaraswamy model in the context of data fitting. The well-known maximum likelihood approach is used to estimate the parameters, and a simulation study is conducted to examine the performance of this approach. In order to demonstrate the applicability of the suggested model, two real data sets are considered. As a notable result, for the considered data sets, statistical benchmarks indicate that the new modeling strategy outperforms the Kumaraswamy model. The transmuted Kumaraswamy, beta, unit Rayleigh, Topp–Leone and power models are also outperformed.


Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2112
Author(s):  
Ahmed Elshahhat ◽  
Berihan R. Elemary

Censoring mechanisms are widely used in various life tests, such as medicine, engineering, biology, etc., as they save (overall) test time and cost. In this context, we consider the problem of estimating the unknown xgamma parameter and some survival characteristics, such as reliability and failure rate functions in the presence of adaptive type-II progressive hybrid censored data. For this purpose, the maximum likelihood and Bayesian inferential approaches are used. Using the observed Fisher information under s-normal approximation, different asymptotic confidence intervals for any function of the unknown parameter were constructed. Using the gamma flexible prior, Bayes estimators against the squared-error loss were developed. Two procedures of Bayesian approximations—Lindley’s approximation and Metropolis–Hastings algorithm—were used to carry out the Bayes estimates and to construct the associated credible intervals. An extensive simulation study was implemented to compare the performance of the different methods. To validate the proposed methodologies of inference—two practical studies using datasets that form engineering and chemical fields are discussed.


2021 ◽  
Vol 34 (4) ◽  
pp. 93-107
Author(s):  
Lamyaa Khalid Hussein ◽  
Iden Hasan Hussein ◽  
Huda Abdulla Rasheed

In this paper, we used the maximum likelihood estimation method to find the estimation values ​​for survival and hazard rate functions of the Exponential Rayleigh distribution based on a sample of the real data for lung cancer and stomach cancer obtained from the Iraqi Ministry of Health and Environment, Department of Medical City, Tumor Teaching Hospital, depending on patients' diagnosis records and number of days the patient remains in the hospital until his death.


Author(s):  
Francesco Buono ◽  
Emilio De Santis ◽  
Maria Longobardi ◽  
Fabio Spizzichino

AbstractThe family of the multivariate conditional hazard rate functions often reveals to be a convenient tool to describe the joint probability distribution of a vector of non-negative random variables (lifetimes) in the absolutely continuous case. Such a tool can have in particular an important role in the study of the behavior of the minima among inter-dependent lifetimes. In this paper we introduce the concept of reversed multivariate conditional hazard rate functions, which extends the one-dimensional notion of reversed hazard rate of a single non-negative random variable. Several basic properties of this concept are proven. In particular, we point out a related role in the study of the behavior of the maximum value among inter-dependent lifetimes. In different applied fields, and in particular in the reliability literature, a remarkable class of dependence models for vectors of lifetimes is related with the load-sharing condition, which can be defined in terms of the multivariate conditional hazard rate functions. In the paper we define the class of reversedload-sharing models, which can be seen as natural extensions to the multivariate case of the univariate inverse exponential distributions. We analyze basic properties of such a class of dependence models. In particular we show a result related to the study of the inactivity time of a coherent system when the joint distribution of the components’ lifetimes is a reversed load-sharing model.


2021 ◽  
Author(s):  
Brandon Legried ◽  
Jonathan Terhorst

In a striking result, Louca and Pennell (2020) recently proved that a large class of birth-death models are statistically unidentifiable from lineage-through-time (LTT) data. Specifically, they showed that any pair of sufficiently smooth birth and death rate functions is "congruent" to an infinite collection of other rate functions, all of which have the same likelihood for any LTT vector of any dimension. This fact has distressing implications for the thousands of studies which have utilized birth-death models to study evolution. In this paper, we qualify their finding by proving that an alternative and widely used class of birth-death models is indeed identifiable. Specifically, we show that piecewise constant birth-death models can, in principle, be consistently estimated and distinguished from one another, given a sufficiently large extant time tree and some knowledge of the present day population. Subject to mild regularity conditions, we further show that any unidentifiable birth-death model class can be arbitrarily closely approximated by a class of identifiable models. The sampling requirements needed for our results to hold are explicit, and are expected to be satisfied in many contexts such as the phylodynamic analysis of a global pandemic.


2021 ◽  
Vol 19 (1) ◽  
Author(s):  
El-Sayed A. El-Sherpieny ◽  
Salwa Assar ◽  
Tamer Helal

A new method for generating family of distributions was proposed. Some fundamental properties of the new proposed family include the quantile, survival function, hazard rate function, reversed hazard and cumulative hazard rate functions are provided. This family contains several new models as sub models, such as the Weibull exponential model which was defined and discussed its properties. The maximum likelihood method of estimation is using to estimate the model parameters of the new proposed family. The flexibility and the importance of the Weibull-exponential model is assessed by applying it to a real data set and comparing it with other known models.


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