Inverted exponentiated Weibull distribution with applications to lifetime data

Lee, Seunghyung,Noh, Yunhwan,Chung, Younshik The Korean Statistical Society 2017 Communications for statistical applications and me Vol.24 No.3

In this paper, we introduce the inverted exponentiated Weibull (IEW) distribution which contains exponentiated inverted Weibull distribution, inverse Weibull (IW) distribution, and inverted exponentiated distribution as submodels. The proposed distribution is obtained by the inverse form of the exponentiated Weibull distribution. In particular, we explain that the proposed distribution can be interpreted by Marshall and Olkin's book (Lifetime Distributions: Structure of Non-parametric, Semiparametric, and Parametric Families, 2007, Springer) idea. We derive the cumulative distribution function and hazard function and calculate expression for its moment. The hazard function of the IEW distribution can be decreasing, increasing or bathtub-shaped. The maximum likelihood estimation (MLE) is obtained. Then we show the existence and uniqueness of MLE. We can also obtain the Bayesian estimation by using the Gibbs sampler with the Metropolis-Hastings algorithm. We also give applications with a simulated data set and two real data set to show the flexibility of the IEW distribution. Finally, conclusions are mentioned.

A new flexible Weibull distribution

Park, Sangun,Park, Jihwan,Choi, Youngsik The Korean Statistical Society 2016 Communications for statistical applications and me Vol.23 No.5

Many of studies have suggested the modifications on Weibull distribution to model the non-monotone hazards. In this paper, we combine two cumulative hazard functions and propose a new modified Weibull distribution function. The newly suggested distribution will be named as a new flexible Weibull distribution. Corresponding hazard function of the proposed distribution shows flexible (monotone or non-monotone) shapes. We study the characteristics of the proposed distribution that includes ageing behavior, moment, and order statistic. We also discuss an estimation method for its parameters. The performance of the proposed distribution is compared with existing modified Weibull distributions using various types of hazard functions. We also use real data example to illustrate the efficiency of the proposed distribution.

Notes on a skew-symmetric inverse double Weibull distribution

Woo, Jung-Soo The Korean Data and Information Science Society 2009 한국데이터정보과학회지 Vol.20 No.2

For an inverse double Weibull distribution which is symmetric about zero, we obtain distribution and moment of ratio of independent inverse double Weibull variables, and also obtain the cumulative distribution function and moment of a skew-symmetric inverse double Weibull distribution. And we introduce a skew-symmetric inverse double Weibull generated by a double Weibull distribution.

Notes on a skew-symmetric inverse double Weibull distribution

우정수 한국데이터정보과학회 2009 한국데이터정보과학회지 Vol.20 No.2

For an inverse double Weibull distribution which is symmetric about zero, we obtain distribution and moment of ratio of independent inverse double Weibull variables, and also obtain the cumulative distribution function and moment of a skew-symmetric inverse double Weibull distribution. And we introduce a skew-symmetric inverse double Weibull generated by a double Weibull distribution.

Notes on a skew-symmetric inverse double Weibull distribution

Jung Soo Woo 한국데이터정보과학회 2009 한국데이터정보과학회지 Vol.20 No.2

For an inverse double Weibull distribution which is symmetric about zero, we obtain distribution and moment of ratio of independent inverse double Weibull variables, and also obtain the cumulative distribution function and moment of a skew-symmetric inverse double Weibull distribution. And we introduce a skew-symmetric inverse double Weibull generated by a double Weibull distribution.

The Exponentiated Weibull-Geometric Distribution: Properties and Estimations

Chung, Younshik,Kang, Yongbeen The Korean Statistical Society 2014 Communications for statistical applications and me Vol.21 No.2

In this paper, we introduce the exponentiated Weibull-geometric (EWG) distribution which generalizes two-parameter exponentiated Weibull (EW) distribution introduced by Mudholkar et al. (1995). This proposed distribution is obtained by compounding the exponentiated Weibull with geometric distribution. We derive its cumulative distribution function (CDF), hazard function and the density of the order statistics and calculate expressions for its moments and the moments of the order statistics. The hazard function of the EWG distribution can be decreasing, increasing or bathtub-shaped among others. Also, we give expressions for the Renyi and Shannon entropies. The maximum likelihood estimation is obtained by using EM-algorithm (Dempster et al., 1977; McLachlan and Krishnan, 1997). We can obtain the Bayesian estimation by using Gibbs sampler with Metropolis-Hastings algorithm. Also, we give application with real data set to show the flexibility of the EWG distribution. Finally, summary and discussion are mentioned.

Tree Size Distribution Modelling : Moving from Complexity to Finite Mixture

Friday Nwabueze Ogana,Onyekachi Chukwu,Samuel Ajayi 강원대학교 산림과학연구소 2020 Journal of Forest Science Vol. No.

Tree size distribution modelling is an integral part of forest management. Most distribution yield systems rely on some flexible probability models. In this study, a simple finite mixture of two components two-parameter Weibull distribution was compared with complex four-parameter distributions in terms of their fitness to predict tree size distribution of teak (Tectona grandis Linn f) plantations. Also, a system of equation was developed using Seemingly Unrelated Regression wherein the size distributions of the stand were predicted. Generalized beta, Johnson’s SB, Logit-Logistic and generalized Weibull distributions were the four-parameter distributions considered. The Kolmogorov-Smirnov test and negative log-likelihood value were used to assess the distributions. The results show that the simple finite mixture outperformed the four-parameter distributions especially in stands that are bimodal and heavily skewed. Twelve models were developed in the system of equation-one for predicting mean diameter, seven for predicting percentiles and four for predicting the parameters of the finite mixture distribution. Predictions from the system of equation are reasonable and compare well with observed distributions of the stand. This simplified mixture would allow for wider application in distribution modelling and can also be integrated as component model in stand density management diagram.

혼합 와이블 분포를 이용한 고장률 산출 기법에 관한 연구

채희석(Hui-seok Chai),신중우(Joong-woo Shin),임태진(Tae-jin Lim),김재철(Jae-chul Kim) 대한전기학회 2017 전기학회논문지 Vol.66 No.3

In 2014, ISO 55000s has been enacted and the power plant asset management is becoming a hot issue for all over the world. The asset management system is being developed as a combination of CBM(Condition Based Maintenance) and RCM(Reliability Centered Maintenance). Therefore, the research on the calculation of the failure rate which is the most basic index of RCM is actively carried out. The failure rate calculation has been going on for a long time, and the most widely used probability distribution is the Weibull distribution. In the Weibull distribution, the failure rate function is determined in three types according to the value of the shape parameter. However, the Weibull distribution has a limitation that it is difficult to apply it when the trend of failure rate changes-such as bathtub curves. In this paper, the failure rate is calculated using the mixture Weibull distribution which can appropriately express the change of the shape of the failure rate. Based on these results, we propose the necessity and validity of applying mixture Weibull distribution.

세라믹스의 파괴강도에 관한 확률론적 해석

김선진(Seon-Jin Kim) 한국해양공학회 1996 韓國海洋工學會誌 Vol.10 No.2

Weibull distribution function is applied very successfully to the strength of brittle materials such as ceramics and the weakest link model is applied to explain the events. This paper deals with the effect of specimen size on the strength of ceramics. The values of tensile strength were calculated by the Monte-Calro simulation. The tensile strength obtained was plotted on Weibull probability papers and represented by the 3-parameter Weibull distribution. The strength distribution function was compared with the theoretical Weibull distribution. As a result, it was found that the Weibull shape parameter was changed due to the size and there was a possibility of a false indication as if the weakest link model holds good. We should be very careful when we apply the Weibull statistics to estimate the strength of products.

Weibull Model for Wind Speed Data Analysis of Different Locations in India

Arnab Sarkar,Gaurav Gugliani,Sneh Deep 대한토목학회 2017 KSCE JOURNAL OF CIVIL ENGINEERING Vol.21 No.7

Wind speed data should be fitted by a suitable statistical model like Weibull to determine expected number of hours per year in the critical wind speed range for a slender structure, which is required to determine the expected number of stress cycles in the projected working life of the structure. Apart from this, for the assessment of wind energy potential wind speed data should be fitted by an appropriate probability distribution. In the present scope of study, wind data of various locations of India have been fitted by Weibull model. Wind speed data are initially sampled in knot by Indian Meteorological Department and later converted into integer km/h before supplying them to the end user. Due to this conversion, wind speed data cannot be properly fitted by Weibull distribution and in this regard, the choice of appropriate class width becomes very much important. Without the choice of appropriate class width, estimated Weibull parameters become biased which would yield incorrect estimation of expected number of hours in critical wind speed ranges as well as wind energy potential. After taking appropriate class width of 4 km/h, it has been found that Weibull model is an adequate model to describe wind speed distributions of India. Weibull model has also been compared with other models such as Gamma and inverse Weibull distributions to establish its suitability than the others. In this study, the values of Weibull shape parameters vary from 1.3 to 2.3, whereas the values of scale parameters vary from 1.4 m/s to 6.5 m/s. The validity of Weibull model is also verified with a target confidence interval of 90%. The uncertainties involved in the estimation of available wind energy potential as well as the expected number of hours per year in critical wind speed ranges have also been considered due to random variation of wind climate in each year.