scholarly journals Some New Facts about the Unit-Rayleigh Distribution with Applications

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1954
Author(s):  
Rashad A. R. Bantan ◽  
Christophe Chesneau ◽  
Farrukh Jamal ◽  
Mohammed Elgarhy ◽  
Muhammad H. Tahir ◽  
...  
Keyword(s):  
Closed Form ◽  
Stochastic Ordering ◽  
Rayleigh Distribution ◽  
Parameter Distribution ◽  
Data Sets ◽  
Simple Analytical Expression ◽  
Form Analysis ◽  
Beta Power ◽  
The One ◽  
Simple Ratio

The unit-Rayleigh distribution is a one-parameter distribution with support on the unit interval. It is defined as the so-called unit-Weibull distribution with a shape parameter equal to two. As a particular case among others, it seems that it has not been given special attention. This paper shows that the unit-Rayleigh distribution is much more interesting than it might at first glance, revealing closed-form expressions of important functions, and new desirable properties for application purposes. More precisely, on the theoretical level, we contribute to the following aspects: (i) we bring new characteristics on the form analysis of its main probabilistic and reliability functions, and show that the possible mode has a simple analytical expression, (ii) we prove new stochastic ordering results, (iii) we expose closed-form expressions of the incomplete and probability weighted moments at the basis of various probability functions and measures, (iv) we investigate distributional properties of the order statistics, (v) we show that the reliability coefficient can have a simple ratio expression, (vi) we provide a tractable expansion for the Tsallis entropy and (vii) we propose some bivariate unit-Rayleigh distributions. On a practical level, we show that the maximum likelihood estimate has a quite simple closed-form. Three data sets are analyzed and adjusted, revealing that the unit-Rayleigh distribution can be a better alternative to standard one-parameter unit distributions, such as the one-parameter Kumaraswamy, Topp–Leone, one-parameter beta, power and transmuted distributions.

scholarly journals Statistical Inference of the Half-Logistic Inverse Rayleigh Distribution

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 449 ◽  
Author(s):  
Abdullah M. Almarashi ◽  
Majdah M. Badr ◽  
Mohammed Elgarhy ◽  
Farrukh Jamal ◽  
Christophe Chesneau
Keyword(s):  
Statistical Inference ◽  
Goodness Of Fit ◽  
Stochastic Ordering ◽  
Linear Representation ◽  
Rayleigh Distribution ◽  
Model Parameters ◽  
Data Sets ◽  
New Model ◽  
Entropy Measures ◽  
Lifetime Studies

The inverse Rayleigh distribution finds applications in many lifetime studies, but has not enough overall flexibility to model lifetime phenomena where moderately right-skewed or near symmetrical data are observed. This paper proposes a solution by introducing a new two-parameter extension of this distribution through the use of the half-logistic transformation. The first contribution is theoretical: we provide a comprehensive account of its mathematical properties, specifically stochastic ordering results, a general linear representation for the exponentiated probability density function, raw/inverted moments, incomplete moments, skewness, kurtosis, and entropy measures. Evidences show that the related model can accommodate the treatment of lifetime data with different right-skewed features, so far beyond the possibility of the former inverse Rayleigh model. We illustrate this aspect by exploring the statistical inference of the new model. Five classical different methods for the estimation of the model parameters are employed, with a simulation study comparing the numerical behavior of the different estimates. The estimation of entropy measures is also discussed numerically. Finally, two practical data sets are used as application to attest of the usefulness of the new model, with favorable goodness-of-fit results in comparison to three recent extended inverse Rayleigh models.

scholarly journals An Inferential Aptness of a Weibull Generated Distribution and Application

2021 ◽  
Author(s):  
Brijesh P. Singh ◽  
Utpal Dhar Das
Keyword(s):  
Weibull Distribution ◽  
Continuous Distribution ◽  
Selection Criterion ◽  
Real Data ◽  
Rayleigh Distribution ◽  
Single Parameter ◽  
Parameter Distribution ◽  
Data Sets ◽  
Lifetime Distributions ◽  
Limiting Behavior

In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kthkt⁢h order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λλ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

Inverse Akash distribution and its applications

2021 ◽  
Vol 20 (2) ◽  
pp. 61-72
Author(s):  
E.W. Okereke ◽  
S.N. Gideon ◽  
J. Ohakwe
Keyword(s):  
Survival Function ◽  
Likelihood Estimation ◽  
Stochastic Ordering ◽  
Real Data ◽  
Rate Function ◽  
Entropy Measure ◽  
Parameter Distribution ◽  
Data Sets ◽  
Inverse Exponential Distribution ◽  
Inverse Lindley Distribution

A new one-parameter distribution named inverse Akash distribution, for modelling lifetime data, has been  introduced. Important statistical properties of the proposed distribution such as the density function, hazard rate function, survival function, stochastic ordering,  entropy   measure, stress-strength reliability and the maximum  likelihood estimation of the parameter of the distribution have been discussed. Two real data sets were employed in illustrating the usefulness of the new distribution. Comparatively, the inverse Akash distribution provided better fits to the data than each of the inverse exponential distribution and inverse Lindley distribution.

APPLICATIONS OF INVERSE WEIBULL RAYLEIGH DISTRIBUTION TO FAILURE RATES AND VINYL CHLORIDE DATA SETS

2021 ◽  
Vol 5 (2) ◽  
pp. 89-99
Author(s):  
Aminu Adamu ◽  
Abubakar Yahaya ◽  
Hussaini Garba Dikko
Keyword(s):  
Vinyl Chloride ◽  
Likelihood Estimation ◽  
Rayleigh Distribution ◽  
Parameter Distribution ◽  
Data Sets ◽  
Failure Rates ◽  
Data Set ◽  
Bathtub Shape ◽  
Asymptotically Efficient ◽  
New Distribution

In this work, a new three parameter distribution called the Inverse Weibull Rayleigh distribution is proposed. Some of its statistical properties were presented. The PDF plot of Inverse Weibull Rayleigh distribution showed that it is good for modeling positively skewed and symmetrical datasets. The plot of the hazard function showed that the proposed distribution can fit datasets with bathtub shape. Method of maximum likelihood estimation was employed to estimate the parameters of the distribution, the estimators of the parameters of Inverse Weibull Rayleigh distribution is asymptotically unbiased and asymptotically efficient from the result of the simulation carried out. Applying the new distribution to a positively skewed Vinyl Chloride data set shows that the distribution performs better than Rayleigh, Generalized Rayleigh, Weibull Rayleigh, Inverse Weibull, Inverse Weibull Weibull, Inverse Weibull Inverse Exponential and Inverse Weibull Pareto distribution in fitting the data as it has the smallest AIC value. Also, applying the new distribution to a negatively skewed bathtub shape failure rates data shows that the distribution is a competitive model after Weibull Rayleigh and Inverse Weibull Weibull distributions in fitting the data because it has the third least AIC value.

scholarly journals Classification of jujube defects in small data sets based on transfer learning

2021 ◽  
Author(s):  
Jianping Ju ◽  
Hong Zheng ◽  
Xiaohang Xu ◽  
Zhongyuan Guo ◽  
Zhaohui Zheng ◽  
...  
Keyword(s):  
Transfer Learning ◽  
Loss Function ◽  
Training Model ◽  
Parameter Distribution ◽  
Test Accuracy ◽  
Small Data ◽  
Data Sets ◽  
Data Set ◽  
Small Data Sets

AbstractAlthough convolutional neural networks have achieved success in the field of image classification, there are still challenges in the field of agricultural product quality sorting such as machine vision-based jujube defects detection. The performance of jujube defect detection mainly depends on the feature extraction and the classifier used. Due to the diversity of the jujube materials and the variability of the testing environment, the traditional method of manually extracting the features often fails to meet the requirements of practical application. In this paper, a jujube sorting model in small data sets based on convolutional neural network and transfer learning is proposed to meet the actual demand of jujube defects detection. Firstly, the original images collected from the actual jujube sorting production line were pre-processed, and the data were augmented to establish a data set of five categories of jujube defects. The original CNN model is then improved by embedding the SE module and using the triplet loss function and the center loss function to replace the softmax loss function. Finally, the depth pre-training model on the ImageNet image data set was used to conduct training on the jujube defects data set, so that the parameters of the pre-training model could fit the parameter distribution of the jujube defects image, and the parameter distribution was transferred to the jujube defects data set to complete the transfer of the model and realize the detection and classification of the jujube defects. The classification results are visualized by heatmap through the analysis of classification accuracy and confusion matrix compared with the comparison models. The experimental results show that the SE-ResNet50-CL model optimizes the fine-grained classification problem of jujube defect recognition, and the test accuracy reaches 94.15%. The model has good stability and high recognition accuracy in complex environments.

scholarly journals Kumaraswamy Generalized Power Lomax Distributionand Its Applications

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 28-45
Author(s):  
Vasili B.V. Nagarjuna ◽  
R. Vishnu Vardhan ◽  
Christophe Chesneau
Keyword(s):  
Hazard Rate ◽  
Real Data ◽  
Rate Function ◽  
Maximum Likelihood Estimates ◽  
Parameter Estimates ◽  
Parameter Distribution ◽  
Data Sets ◽  
Lomax Distribution ◽  
Entropy Measures ◽  
Modeling Behavior

In this paper, a new five-parameter distribution is proposed using the functionalities of the Kumaraswamy generalized family of distributions and the features of the power Lomax distribution. It is named as Kumaraswamy generalized power Lomax distribution. In a first approach, we derive its main probability and reliability functions, with a visualization of its modeling behavior by considering different parameter combinations. As prime quality, the corresponding hazard rate function is very flexible; it possesses decreasing, increasing and inverted (upside-down) bathtub shapes. Also, decreasing-increasing-decreasing shapes are nicely observed. Some important characteristics of the Kumaraswamy generalized power Lomax distribution are derived, including moments, entropy measures and order statistics. The second approach is statistical. The maximum likelihood estimates of the parameters are described and a brief simulation study shows their effectiveness. Two real data sets are taken to show how the proposed distribution can be applied concretely; parameter estimates are obtained and fitting comparisons are performed with other well-established Lomax based distributions. The Kumaraswamy generalized power Lomax distribution turns out to be best by capturing fine details in the structure of the data considered.

scholarly journals Analysis of Risk Factors in Dementia Through Machine Learning

2020 ◽  
pp. 1-17
Author(s):  
Francisco Javier Balea-Fernandez ◽  
Beatriz Martinez-Vega ◽  
Samuel Ortega ◽  
Himar Fabelo ◽  
Raquel Leon ◽  
...  
Keyword(s):  
Machine Learning ◽  
Optimization Algorithms ◽  
Progressive Increase ◽  
Control Group ◽  
Data Sets ◽  
Modifiable Factors ◽  
Validation Set ◽  
The One ◽  
And Control ◽  
Potential Tool

Background: Sociodemographic data indicate the progressive increase in life expectancy and the prevalence of Alzheimer’s disease (AD). AD is raised as one of the greatest public health problems. Its etiology is twofold: on the one hand, non-modifiable factors and on the other, modifiable. Objective: This study aims to develop a processing framework based on machine learning (ML) and optimization algorithms to study sociodemographic, clinical, and analytical variables, selecting the best combination among them for an accurate discrimination between controls and subjects with major neurocognitive disorder (MNCD). Methods: This research is based on an observational-analytical design. Two research groups were established: MNCD group (n = 46) and control group (n = 38). ML and optimization algorithms were employed to automatically diagnose MNCD. Results: Twelve out of 37 variables were identified in the validation set as the most relevant for MNCD diagnosis. Sensitivity of 100%and specificity of 71%were achieved using a Random Forest classifier. Conclusion: ML is a potential tool for automatic prediction of MNCD which can be applied to relatively small preclinical and clinical data sets. These results can be interpreted to support the influence of the environment on the development of AD.

scholarly journals AN INDUSTRY QUESTION: THE ULTIMATE AND ONE-YEAR RESERVING UNCERTAINTY FOR DIFFERENT NON-LIFE RESERVING METHODOLOGIES

2014 ◽  
Vol 44 (3) ◽  
pp. 495-499 ◽  
Author(s):  
Eric Dal Moro ◽  
Joseph Lo
Keyword(s):  
Closed Form ◽  
Internal Models ◽  
Cape Cod ◽  
Best Estimate ◽  
To Come ◽  
Chain Ladder ◽  
One Year ◽  
The One ◽  
Best Estimates

AbstractIn the industry, generally, reserving actuaries use a mix of reserving methods to derive their best estimates. On the basis of the best estimate, Solvency 2 requires the use of a one-year volatility of the reserves. When internal models are used, such one-year volatility has to be provided by the reserving actuaries. Due to the lack of closed-form formulas for the one-year volatility of Bornhuetter-Ferguson, Cape-Cod and Benktander-Hovinen, reserving actuaries have limited possibilities to estimate such volatility apart from scaling from tractable models, which are based on other reserving methods. However, such scaling is technically difficult to justify cleanly and awkward to interact with. The challenge described in this editorial is therefore to come up with similar models like those of Mack or Merz-Wüthrich for the chain ladder, but applicable to Bornhuetter-Ferguson, mix Chain-Ladder and Bornhuetter-Ferguson, potentially Cape-Cod and Benktander-Hovinen — and their mixtures.

scholarly journals Theory and Applications of the Unit Gamma/Gompertz Distribution

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1850
Author(s):  
Rashad A. R. Bantan ◽  
Farrukh Jamal ◽  
Christophe Chesneau ◽  
Mohammed Elgarhy
Keyword(s):  
Stochastic Ordering ◽  
Real Data ◽  
Rate Function ◽  
The Other ◽  
Likelihood Method ◽  
Model Parameters ◽  
Data Sets ◽  
Gompertz Distribution ◽  
Probability And Statistics ◽  
Analytical Behavior

Unit distributions are commonly used in probability and statistics to describe useful quantities with values between 0 and 1, such as proportions, probabilities, and percentages. Some unit distributions are defined in a natural analytical manner, and the others are derived through the transformation of an existing distribution defined in a greater domain. In this article, we introduce the unit gamma/Gompertz distribution, founded on the inverse-exponential scheme and the gamma/Gompertz distribution. The gamma/Gompertz distribution is known to be a very flexible three-parameter lifetime distribution, and we aim to transpose this flexibility to the unit interval. First, we check this aspect with the analytical behavior of the primary functions. It is shown that the probability density function can be increasing, decreasing, “increasing-decreasing” and “decreasing-increasing”, with pliant asymmetric properties. On the other hand, the hazard rate function has monotonically increasing, decreasing, or constant shapes. We complete the theoretical part with some propositions on stochastic ordering, moments, quantiles, and the reliability coefficient. Practically, to estimate the model parameters from unit data, the maximum likelihood method is used. We present some simulation results to evaluate this method. Two applications using real data sets, one on trade shares and the other on flood levels, demonstrate the importance of the new model when compared to other unit models.

Closed-form Analysis of Age of Information in Energy Harvesting Network

2020 ◽  
Author(s):  
Siyu Wang ◽  
Xin Wang ◽  
Tianyi Peng ◽  
Jiaxi Zhou ◽  
Qi Qin ◽  
...  
Keyword(s):  
Energy Harvesting ◽  
Closed Form ◽  
Form Analysis ◽  
Age Of Information
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