scholarly journals Estimation of Entropy for Inverse Lomax Distribution under Multiple Censored Data

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 601 ◽  
Author(s):  
Rashad A. R. Bantan ◽  
Mohammed Elgarhy ◽  
Christophe Chesneau ◽  
Farrukh Jamal

The inverse Lomax distribution has been widely used in many applied fields such as reliability, geophysics, economics and engineering sciences. In this paper, an unexplored practical problem involving the inverse Lomax distribution is investigated: the estimation of its entropy when multiple censored data are observed. To reach this goal, the entropy is defined through the Rényi and q-entropies, and we estimate them by combining the maximum likelihood and plugin methods. Then, numerical results are provided to show the behavior of the estimates at various sample sizes, with the determination of the mean squared errors, two-sided approximate confidence intervals and the corresponding average lengths. Our numerical investigations show that, when the sample size increases, the values of the mean squared errors and average lengths decrease. Also, when the censoring level decreases, the considered of Rényi and q-entropies estimates approach the true value. The obtained results validate the usefulness and efficiency of the method. An application to two real life data sets is given.

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Usman Shahzad ◽  
Nadia H. Al-Noor ◽  
Noureen Afshan ◽  
David Anekeya Alilah ◽  
Muhammad Hanif ◽  
...  

Robust regression tools are commonly used to develop regression-type ratio estimators with traditional measures of location whenever data are contaminated with outliers. Recently, the researchers extended this idea and developed regression-type ratio estimators through robust minimum covariance determinant (MCD) estimation. In this study, the quantile regression with MCD-based measures of location is utilized and a class of quantile regression-type mean estimators is proposed. The mean squared errors (MSEs) of the proposed estimators are also obtained. The proposed estimators are compared with the reviewed class of estimators through a simulation study. We also incorporated two real-life applications. To assess the presence of outliers in these real-life applications, the Dixon chi-squared test is used. It is found that the quantile regression estimators are performing better as compared to some existing estimators.


2021 ◽  
Vol 3 (2) ◽  
pp. 81-94
Author(s):  
Sule Ibrahim ◽  
Sani Ibrahim Doguwa ◽  
Audu Isah ◽  
Haruna, M. Jibril

Many Statisticians have developed and proposed new distributions by extending the existing distributions. The distributions are extended by adding one or more parameters to the baseline distributions to make it more flexible in fitting different kinds of data. In this study, a new four-parameter lifetime distribution called the Topp Leone Kumaraswamy Lomax distribution was introduced by using a family of distributions which has been proposed in the literature. Some mathematical properties of the distribution such as the moments, moment generating function, quantile function, survival, hazard, reversed hazard and odds functions were presented. The estimation of the parameters by maximum likelihood method was discussed. Three real life data sets representing the failure times of the air conditioning system of an air plane, the remission times (in months) of a random sample of one hundred and twenty-eight (128) bladder cancer patients and Alumina (Al2O3) data were used to show the fit and flexibility of the new distribution over some lifetime distributions in literature. The results showed that the new distribution fits better in the three datasets considered.


Author(s):  
Fathollah Salehirad ◽  
Sepehr Sadighi ◽  
Shahram Alijani

Abstract Light naphtha isomerization is a significant process in a crude oil refinery which is responsible for upgrading low-octane light naphtha to the high-octane and low-aromatic content gasoline. In this work, deactivation of an industrial chlorinated Pt/Al2O3 isomerization catalyst was studied in a laboratory scale plant. Experiments were carried out under temperatures in the range of 120–180 °C, liquid hourly space velocities (LHSV) of 0.7–2 h−1 and hydrogen to hydrocarbon molar ratios (H2/Oil) of 0.7–1.5. Moreover, the total water content of the combined feed, i. e. n-hexane and hydrogen, was 70 ppmwt. During 75 h time on stream (TOS), 42 data sets were collected and applied for training, testing and validating a hybrid-artificial neural network model (hybrid-ANN or HANN) to estimate the activity of the catalyst. Results showed that the activity decreased to 0.56 at the end of the operation mostly due to water poisoning. Furthermore, using the estimated activity, HANN could simulate the conversion and selectivity of the isomerization process with the absolute average deviations (AAD%) of 0.97 % and 0.0766 % and the mean squared errors (MSE) of 0.311 and 0.0156, respectively.


Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 309
Author(s):  
Sanaa Al-Marzouki ◽  
Farrukh Jamal ◽  
Christophe Chesneau ◽  
Mohammed Elgarhy

The last years have revealed the importance of the inverse Lomax distribution in the understanding of lifetime heavy-tailed phenomena. However, the inverse Lomax modeling capabilities have certain limits that researchers aim to overcome. These limits include a certain stiffness in the modulation of the peak and tail properties of the related probability density function. In this paper, a solution is given by using the functionalities of the half logistic family. We introduce a new three-parameter extended inverse Lomax distribution called the half logistic inverse Lomax distribution. We highlight its superiority over the inverse Lomax distribution through various theoretical and practical approaches. The derived properties include the stochastic orders, quantiles, moments, incomplete moments, entropy (Rényi and q) and order statistics. Then, an emphasis is put on the corresponding parametric model. The parameters estimation is performed by six well-established methods. Numerical results are presented to compare the performance of the obtained estimates. Also, a simulation study on the estimation of the Rényi entropy is proposed. Finally, we consider three practical data sets, one containing environmental data, another dealing with engineering data and the last containing insurance data, to show how the practitioner can take advantage of the new half logistic inverse Lomax model.


Author(s):  
Umar Kabir ◽  
Terna Godfrey IEREN

This article proposed a new distribution referred to as the transmuted Exponential Lomax distribution as an extension of the popular Lomax distribution in the form of Exponential Lomax by using the Quadratic rank transmutation map proposed and studied in earlier research. Using the transmutation map, we defined the probability density function (PDF) and cumulative distribution function (CDF) of the transmuted Exponential Lomax distribution. Some properties of the new distribution were extensively studied after derivation. The estimation of the distribution’s parameters was also done using the method of maximum likelihood estimation. The performance of the proposed probability distribution was checked in comparison with some other generalizations of Lomax distribution using three real-life data sets. The results obtained indicated that TELD performs better than the other distributions comprising power Lomax, Exponential-Lomax, and the Lomax distributions.


2020 ◽  
Vol 25 (4) ◽  
pp. 79 ◽  
Author(s):  
Jismi Mathew ◽  
Christophe Chesneau

The Lomax distribution is arguably one of the most useful lifetime distributions, explaining the developments of its extensions or generalizations through various schemes. The Marshall–Olkin length-biased Lomax distribution is one of these extensions. The associated model has been used in the frameworks of data fitting and reliability tests with success. However, the theory behind this distribution is non-existent and the results obtained on the fit of data were sufficiently encouraging to warrant further exploration, with broader comparisons with existing models. This study contributes in these directions. Our theoretical contributions on the the Marshall–Olkin length-biased Lomax distribution include an original compounding property, various stochastic ordering results, equivalences of the main functions at the boundaries, a new quantile analysis, the expressions of the incomplete moments under the form of a series expansion and the determination of the stress–strength parameter in a particular case. Subsequently, we contribute to the applicability of the Marshall–Olkin length-biased Lomax model. When combined with the maximum likelihood approach, the model is very effective. We confirm this claim through a complete simulation study. Then, four selected real life data sets were analyzed to illustrate the importance and flexibility of the model. Especially, based on well-established standard statistical criteria, we show that it outperforms six strong competitors, including some extended Lomax models, when applied to these data sets. To our knowledge, such comprehensive applied work has never been carried out for this model.


Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15 ◽  
Author(s):  
Ramadan A. ZeinEldin ◽  
Muhammad Ahsan ul Haq ◽  
Sharqa Hashmi ◽  
Mahmoud Elsehety

In this paper, a new three-parameter lifetime distribution, alpha power transformed inverse Lomax (APTIL) distribution, is proposed. The APTIL distribution is more flexible than inverse Lomax distribution. We derived some mathematical properties including moments, moment generating function, quantile function, mode, stress strength reliability, and order statistics. Characterization related to hazard rate function is also derived. The model parameters are estimated using eight estimation methods including maximum likelihood, least squares, weighted least squares, percentile, Cramer–von Mises, maximum product of spacing, Anderson–Darling, and right-tail Anderson–Darling. Numerical results are calculated to compare the performance of these estimation methods. Finally, we used three real-life datasets to show the flexibility of the APTIL distribution.


2020 ◽  
pp. 112067212092727
Author(s):  
Marko Lukic ◽  
Gwyn Williams ◽  
Zaid Shalchi ◽  
Praveen J Patel ◽  
Philip G Hykin ◽  
...  

Purpose To assess visual and optical coherence tomography–derived anatomical outcomes of treatment with intravitreal aflibercept (Eylea®) for diabetic macular oedema in patients switched from intravitreal ranibizumab (Lucentis®). Design Retrospective, cohort study. Participants Ninety eyes (of 67 patients) receiving intravitreal anti–vascular endothelial growth factor therapy were included. Methods This is a retrospective, real-life, cohort study. Each patient had visual acuity measurements and optical coherence tomography scans performed at baseline and 12 months after the first injection of aflibercept was given. Main Outcome Measures We measured visual acuities in Early Treatment Diabetic Retinopathy Study letters, central foveal thickness and macular volume at baseline and at 12 months after the first aflibercept injection was given. Results Ninety switched eyes were included in this study. The mean (standard deviation) visual acuity was 63 (15.78) Early Treatment Diabetic Retinopathy Study letters. At baseline, the mean (standard deviation) central foveal thickness was 417.7 (158.4) μm and the mean macular volume was 9.96 (2.44) mm3. Mean change in visual acuity was +4 Early Treatment Diabetic Retinopathy Study letters (p = 0.0053). The mean change in macular volume was −1.53 mm 3 in SW group (p = 0.21), while the change in central foveal thickness was −136.8 μm (p = 0.69). Conclusion There was a significant improvement in visual acuity and in anatomical outcomes in the switched group at 12 months after commencing treatment with aflibercept for diabetic macular oedema.


2021 ◽  
pp. 112067212110057
Author(s):  
Pierre Gascon ◽  
Prithvi Ramtohul ◽  
Charles Delaporte ◽  
Sébastien Kerever ◽  
Danièle Denis ◽  
...  

Purpose: To report the visual and anatomic outcomes in treatment-naïve neovascular age-related macular degeneration (nAMD) patients treated with aflibercept under a standardized Treat and Extend (T&E) protocol for up to 3 years of follow-up in “real-life” practice. Methods: This retrospective, observational, multicenter study included patients with treatment-naïve nAMD and at least 12 months of follow-up. T&E regimen adjustment was initiated after loading phase. At each visit best-corrected visual acuity (BCVA) and optical coherence tomography parameters were performed. Results: One hundred and thirty-six eyes of 115patients had at least 1 year of follow-up with 114 and 82 eyes completing at least 2 and 3 years of follow-up, respectively (mean follow-up duration: 2.7 ± 1.3 years). Mean age was 78.6 ± 8.6 years old and 52% were women. Mean BCVA increased from 60.6 ± 18.7 letters at diagnosis to 66.9 ± 16.2 letters at 1 year (+6.3 letters, p = 0.003) and remained stable throughout the follow-up period (63.1 ± 20.3 letters (+2.5, p = 0.1) and 64.0 ± 20.1 letters (+3.4, p = 0.27) at 2 and 3 years, respectively). The mean central retinal thickness decreased significantly from 358.2 ± 87.9 µm at baseline to 302 ± 71.7 µm at 12 months and maintained stable after 36 months of follow-up (297.1 ± 76 µm, p < 0.0001). Mean number of injections was 6.6 ± 2.2, 4.8 ± 1.9, and 5.6 ± 1.7 at 1, 2, and 3 years, respectively. Mean cumulative number of 16.4 ± 5.6 injections after 3 years. Mean treatment interval was 6.8 ± 2.5 weeks at 1 year. Eight-week and 12-week treatment interval were achieved in 59.5% and 19.1%, 65.8%, and 36.8% and 69.5% and 41.5% at 1, 2, and 3 years, respectively. Conclusions: Our study demonstrated that intravitreal injections of aflibercept initiated under a standardized T&E for patients with treatment-naïve nAMD allow for significant visual improvement at 12 months, which was maintained over a 3-year follow-up period.


2021 ◽  
Vol 80 (Suppl 1) ◽  
pp. 617.1-617
Author(s):  
H. Wohland ◽  
N. Leuchten ◽  
M. Aringer

Background:Fatigue is among the top complaints of patients with systemic lupus erythematosus (SLE), but only in part associated with SLE disease activity. Physical activity can help to reduce fatigue and should therefore be recommended to SLE patients. Vice versa, fatigue may arguably lead to reduced physical activity.Objectives:To investigate the extent of physical activity and the perception of fatigue and sleep quality in patients with SLE.Methods:Starting in February 2019, SLE patients were invited to participate in a cross-sectional survey study of fatigue and physical exercise during their routine outpatient clinic visits. Participants filled out a ten-page paper questionnaire focused on physical activity. To evaluate fatigue, we primarily used a 10 cm visual analogue scale (0-100 mm, with 100 meaning most fatigued), but also the FACIT fatigue score (range 0-52). Sleep quality was estimated using grades from 1 (excellent) to 6 (extremely poor).Results:93 SLE patients took part in the study. All patients fulfilled the European League Against Rheumatism/ American College of Rheumatology (EULAR/ACR) 2019 classification criteria for SLE. 91% of the patients were female. Their mean (SD) age was 45.5 (14.3) years and their mean disease duration 12.1 (9.4) years. The mean BMI was 25.2 (5.6). Of all patients, 7.5% had a diagnosis of (secondary) fibromyalgia. The mean fatigue VAS was 32 (27) mm and the mean FACIT fatigue score 35.7 (10.3). As expected, fatigue by VAS and FACIT was correlated (Spearman r=-0.61, p<0.0001). The mean SLEDAI was 1 (1) with a range of 0 to 6. Median glucocorticoid doses were 2 mg prednisolone equivalent, with a range from 0 to 10 mg.Out of 66 patients in payed jobs, 64 (97%) reported details on their working space. One person (2%) worked in a predominanty standing position, 37 (58%) worked in essentially sedentary jobs and 26 (40%) were in positions where they were mildly physically active in part. The mean fatigue VAS was 31 (24) mm for patients with partly active jobs and 27 (30) mm for those in sedentary jobs. Sleep was graded 2.9 (0.9) by those with active and 3.1 (1.3) by those with sedentary jobs.Half of the patients (51%) reported more than one physical recreational activity. 44 (47%) were walking and for five persons (5%) this was the only form of activity. Cycling was reported by 19 patients (20%), 18 of whom also practiced other activities. For transport, 52 (56%) in part chose active modes, such as walking and cycling. Patients who reported any of the above activities showed a mean fatigue VAS of 28 (25) mm, compared to 36 (28) mm in the patient group without a reported activity. Sleep quality was very similar: 3.1 (1.2) and 3.2 (1.1) for more active and more passive patients, respectively.65 (70%) patients regularly practiced sports. Of these, 39 (60%) practiced one kind of sport, 15 (23%) two, 7 (11%) three, and 2 (3%) each four and five kinds of sports. Fatigue VAS of patients practicing sports was 27 (25) mm versus 43 (28) in those who did not (p=0.0075). Sleep quality was 2.9 (1.1) in the sports cohort and 3.5 (1.1) in the no-sports cohort (p=0.0244).Conclusion:A majority of SLE patients in remission or low to moderate disease activity regularly practiced sports, and those doing so reported lesser fatigue and better sleep quality. The absolute values on the fatigue VAS were in a moderate range that made fatigue as the main cause of not performing sports rather unlikely for most patients.Disclosure of Interests:Helena Wohland: None declared, Nicolai Leuchten Speakers bureau: AbbVie, Janssen, Novartis, Roche, UCB, Consultant of: AbbVie, Janssen, Novartis, Roche, Martin Aringer Speakers bureau: AbbVie, Astra Zeneca, BMS, Boehringer Ingelheim, Chugai, Gilead, GSK, HEXAL, Lilly, MSD, Novartis, Pfizer, Roche, Sanofi, UCB, Consultant of: AbbVie, Astra Zeneca, BMS, Boehringer Ingelheim, GSK, Lilly, MSD, Roche, Sanofi, UCB


Sign in / Sign up

Export Citation Format

Share Document