scholarly journals On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 953
Author(s):  
Rashad A. R. Bantan ◽  
Christophe Chesneau ◽  
Farrukh Jamal ◽  
Mohammed Elgarhy
Keyword(s):  
Statistical Models ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Exponential Distributions ◽  
New Family ◽  
Proposed Model ◽  
Complete Study ◽  
Probability And Statistics ◽  
Definition Of ◽  
Analyze Data

This paper develops the exponentiated Mfamily of continuous distributions, aiming to provide new statistical models for data fitting purposes. It stands out from the other families, as it depends on two baseline distributions, with the use of ratio and power transforms in the definition of the main cumulative distribution function. Thanks to the joint action of the possibly different baseline distributions, flexible statistical models can be created, motivating a complete study in this regard. Thus, we discuss the theoretical properties of the new family, with emphasis on those of potential interest to the overall probability and statistics. Then, a new three-parameter lifetime distribution is derived, with the choices of the inverse exponential and exponential distributions as baselines. After pointing out the great flexibility of the related model, we apply it to analyze an actual dataset of current interest: the daily COVID-19 cases observed in Pakistan from 21 March to 29 May 2020 (inclusive). As notable results, we demonstrate that the proposed model is the best among the 15 top ranked models in the literature, including the inverse exponential and exponential models, several modern extensions of them depending on more parameters, and the “unexponentiated” version of the proposed model as well. As future perspectives, the proposed model can be of interest to analyze data on COVID-19 cases in other countries, for possible comparison studies.

scholarly journals A Load-Share Reliability Model under the Changeable Piecewise Smooth Load

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Sergey V. Gurov ◽  
Lev V. Utkin
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Continuous Functions ◽  
Piecewise Smooth ◽  
Cumulative Distribution ◽  
Residual Lifetime ◽  
Time To Failure ◽  
Reliability Model ◽  
Piecewise Constant ◽  
Proposed Model

A new load-share reliability model of systems under the changeable load is proposed in the paper. It is assumed that the load is a piecewise smooth function which can be regarded as an extension of the piecewise constant and continuous functions. The condition of the residual lifetime conservation, which means continuity of a cumulative distribution function of time to failure, is accepted in the proposed model. A general algorithm for computing reliability measures is provided. Simple expressions for determining the survivor functions under assumption of the Weibull probability distribution of time to failure are given. Various numerical examples illustrate the proposed model by different forms of the system load and different probability distributions of time to failure.

scholarly journals Certain parameterized inequalities arising from fractional integral operators with exponential kernels

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1707-1724
Author(s):  
Zhengrong Yuan ◽  
Taichun Zhou ◽  
Qiang Zhang ◽  
Tingsong Du
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Cumulative Distribution ◽  
Digamma Function ◽  
Fractional Integral Operators ◽  
Definition Of ◽  
Fractional Type

We utilize the definition of a fractional integral operators, which was presented by Ahmad et al., to investigate a general fractional-type identity with a parameter. We establish certain parameterized fractional integral inequalities based on this identity, and provide two examples to illustrate the obtained results. Also, these results derived in this paper are applied to the estimations of q-digamma function, divergence measures and cumulative distribution function, respectively.

scholarly journals A Comparative Study on Computation of Cumulative Distribution Function in Predicting Time of Failure of Engineering Systems

2019 ◽  
Vol 11 (1) ◽  
Author(s):  
Gina Katherine Sierra Paez ◽  
Matthew Daigle ◽  
Kai Goebel
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Degradation Process ◽  
Cumulative Distribution ◽  
System State ◽  
Underlying Assumption ◽  
Hazard Zone ◽  
The Difference ◽  
Definition Of ◽  
Time Of Failure

Estimating accurate Time-of-Failure (ToF) of a system is key in making the decisions that impact operational safety and optimize cost. In this context, it is interesting to note that different approaches have been explored to tackle the problem of estimating ToF. The difference is in part characterized by different definitions of the hazard zones. The conventional definition for the cumulative distribution function (CDF) calculation is assumed to have well-defined hazard zones, that is, hazard zones defined as a function of the system state trajectory. An alternate method suggests the use of hazard zones defined as a function of the system state at time , instead of hazard zones defined as a function of system state up to and including time k (Acuña and Orchard 2018, 2017). This paper explores these differences and their impact on ToF estimation. Results for the conventional CDF definition indicated that, (i) the cumulative distribution function is always an increasing function of time, even when realizations of the degradation process are not monotonic, (ii) the sum of all probabilities is always 1 and does not need to be normalized, and (iii) all probabilities are positive and less than or equal to 1. Similar results are not observed for CDF calculation with hazard zones defined as a function only of the system state at time k. Results for ToF estimation using Acuña's definition differ, suggesting that there is an underlying assumption of independence in the hazard zone definition.  Therefore, we present an alternate definition of hazard zone which guarantees the properties of a well-defined CDF with a more straightforward ToF definition.

scholarly journals The tenets of indirect inference in Bayesian models

2021 ◽  
Author(s):  
Dmytro Perepolkin ◽  
Benjamin Goodrich ◽  
Ullrika Sahlin
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Bayesian Models ◽  
Quantile Function ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Indirect Inference ◽  
Root Finding ◽  
Sampling Distributions ◽  
Definition Of

This paper extends the application of Bayesian inference to probability distributions defined in terms of its quantile function. We describe the method of *indirect likelihood* to be used in the Bayesian models with sampling distributions which lack an explicit cumulative distribution function. We provide examples and demonstrate the equivalence of the "quantile-based" (indirect) likelihood to the conventional "density-defined" (direct) likelihood. We consider practical aspects of the numerical inversion of quantile function by root-finding required by the indirect likelihood method. In particular, we consider a problem of ensuring the validity of an arbitrary quantile function with the help of Chebyshev polynomials and provide useful tips and implementation of these algorithms in Stan and R. We also extend the same method to propose the definition of an *indirect prior* and discuss the situations where it can be useful

A new gamma degradation process with random effect and state-dependent measurement error

2022 ◽  
pp. 1748006X2110672
Author(s):  
Nicola Esposito ◽  
Agostino Mele ◽  
Bruno Castanier ◽  
Massimiliano Giorgio
Keyword(s):  
Distribution Function ◽  
Measurement Error ◽  
Particle Filter ◽  
Cumulative Distribution Function ◽  
Random Effect ◽  
Degradation Process ◽  
Remaining Useful Life ◽  
Cumulative Distribution ◽  
Proposed Model ◽  
Useful Life

In this paper, a new gamma-based degradation process with random effect is proposed that allows to account for the presence of measurement error that depends in stochastic sense on the measured degradation level. This new model extends a perturbed gamma model recently suggested in the literature, by allowing for the presence of a unit to unit variability. As the original one, the extended model is not mathematically tractable. The main features of the proposed model are illustrated. Maximum likelihood estimation of its parameters from perturbed degradation measurements is addressed. The likelihood function is formulated. Hence, a new maximization procedure that combines a particle filter and an expectation-maximization algorithm is suggested that allows to overcome the numerical issues posed by its direct maximization. Moreover, a simple algorithm based on the same particle filter method is also described that allows to compute the cumulative distribution function of the remaining useful life and the conditional probability density function of the hidden degradation level, given the past noisy measurements. Finally, two numerical applications are developed where the model parameters are estimated from two sets of perturbed degradation measurements of carbon-film resistors and fuel cell membranes. In the first example the presence of random effect is statistically significant while in the second example it is not significant. In the applications, the presence of random effect is checked via appropriate statistical procedures. In both the examples, the influence of accounting for the presence of random effect on the estimates of the cumulative distribution function of the remaining useful life of the considered units is also discussed. Obtained results demonstrate the affordability of the proposed approach and the usefulness of the proposed model.

scholarly journals Hybrid Clayton-Frank Convolution-Based Bivariate Archimedean Copula

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Maxwell Akwasi Boateng ◽  
Akoto Yaw Omari-Sasu ◽  
Richard Kodzo Avuglah ◽  
Nana Kena Frempong
Keyword(s):  
Convolution Operator ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Archimedean Copula ◽  
The Other ◽  
Cumulative Distribution ◽  
Entire Real Line ◽  
Proposed Model ◽  
To Come ◽  
The Right

This study exploits the closure property of the converse convolution operator to come up with a hybrid Clayton-Frank Archimedean copula for two random variables. Pairs of random variables were generated and the upper tail observation of the cumulative distribution function (CDF) was used to assess the right skew behavior of the proposed model. Various values of the converse convolution operator were used to see their effect on the proposed model. The simulation covered lengths n=10i,  i=2,3,4,5, and 6. The proposed model was compared with about 40 other bivariate copulas (both Archimedean and elliptical). The proposed model had parameters that spanned the entire real line, thus removing restrictions on the parameters. The parameters theta and omega were varied for a selected interval and the hybrid Clayton-Frank model was, in most cases, found to outperform the other copulas under consideration.

scholarly journals The Transmuted Odd Fréchet-G Family of Distributions: Theory and Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 958 ◽  
Author(s):  
Majdah M. Badr ◽  
Ibrahim Elbatal ◽  
Farrukh Jamal ◽  
Christophe Chesneau ◽  
Mohammed Elgarhy
Keyword(s):  
Cumulative Distribution Function ◽  
Parametric Estimation ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Additional Parameter ◽  
New Family ◽  
Rate Functions ◽  
Special Distribution ◽  
Standard Exponential Distribution ◽  
Modeling Strategy

The last years, the odd Fréchet-G family has been considered with success in various statistical applications. This notoriety can be explained by its simple and flexible exponential-odd structure quite different to the other existing families, with the use of only one additional parameter. In counter part, some of its statistical properties suffer of a lack of adaptivity in the sense that they really depend on the choice of the baseline distribution. Hence, efforts have been made to relax this subjectivity by investigating extensions or generalizations of the odd transformation at the heart of the construction of this family, with the aim to reach new perspectives of applications as well. This study explores another possibility, based on the transformation of the whole cumulative distribution function of this family (while keeping the odd transformation intact), through the use of the quadratic rank transmutation that has proven itself in other contexts. We thus introduce and study a new family of flexible distributions called the transmuted odd Fréchet-G family. We show how the former odd Fréchet-G family is enriched by the proposed transformation through theoretical and practical results. We emphasize the special distribution based on the standard exponential distribution because of its desirable features for the statistical modeling. In particular, different kinds of monotonic and nonmonotonic shapes for the probability density and hazard rate functions are observed. Then, we show how the new family can be used in practice. We discuss in detail the parametric estimation of a special model, along with a simulation study. Practical data sets are handle with quite favorable results for the new modeling strategy.

scholarly journals On stability Conditions of Burr X Autoregressive model

2019 ◽  
Vol 24 (5) ◽  
pp. 91
Author(s):  
Zena. S. Khalaf ◽  
, Azher. A. Mohammad
Keyword(s):  
Limit Cycle ◽  
Autoregressive Model ◽  
Cumulative Distribution Function ◽  
Linearization Method ◽  
Cumulative Distribution ◽  
Stability Conditions ◽  
Proposed Model ◽  
Nonlinear Autoregressive Model ◽  
Nonlinear Autoregressive ◽  
Local Linearization Method

This article deals with proposed nonlinear autoregressive model based on Burr X cumulative distribution function known as Burr X AR (p), we demonstrate stability conditions of the proposed model in terms of its parameters by using dynamical approach known as local linearization method to find stability conditions of a nonzero fixed point of the proposed model, in addition the study demonstrate stability condition of a limit cycle if Burr X AR (1) model have a limit cycle of period greater than one.   http://dx.doi.org/10.25130/tjps.24.2019.096

scholarly journals Constructing a New Exponentiated Family Distribution with Reliability Estimation

2018 ◽  
Vol 29 (2) ◽  
pp. 155
Author(s):  
Shurooq Ahmed Al-Sultany
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Reliability Estimation ◽  
Least Square ◽  
Cumulative Distribution ◽  
New Family ◽  
Regression Approach ◽  
Two Parameters ◽  
The Given

This paper deals with using one method for transforming two parameters given distribution to another form with three parameters distribution, through using idea of reparameterization with powering the given cumulative distribution function by new parameter, where this work gives a new family through using new parameter which is necessary for generating values of the r.v from given CDF through smoothing the values of the given random variable to obtain new values of r.v using three set of parameters rather than two. The new model Frechet p.d.f is obtained and also its Cumulative distribution function is found then we apply three methods of estimation (which are Maximum likelihood , moments estimator , and the third method is depend on using least square regression approach). Different set of initial values of parameters ( , , ) and different samples size (n=25,50,75,100). The simulation procedure is done using matlab-R2014b, and results are compared using integrated Mean square error.

The Raybit Model and the Assessment of its Quality in Comparison with the Logit and Probit Models

2017 ◽  
Vol 64 (3) ◽  
pp. 305-322
Author(s):  
Jan Purczyński ◽  
Kamila Bednarz-Okrzyńska
Keyword(s):  
Cumulative Distribution Function ◽  
Random Number ◽  
Random Number Generator ◽  
The Other ◽  
Cumulative Distribution ◽  
Number Generator ◽  
Probit Models ◽  
Proposed Model ◽  
Logit And Probit Models

A new model for a dependent variable taking the value 0 or 1 (binary, dichotomous) was proposed. The name of the proposed model – the raybit model – stems from the fact that the probability corresponds to the Rayleigh cumulative distribution function. The assessment of the quality of selected models was conducted with the use of four definitions of error: MSE, MAE, WMSE, WMAE. Two computational examples were considered, which proved that the raybit model yields smaller values of error than the logit and probit models. Computer simulations were conducted using a random number generator with a binomial distribution. They proved that for the values of the theoretical probabilityfor the interval Pi ∈ [0; 0.8] the raybit model outperforms the other two models yielding a smaller value of error.

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