• Title/Summary/Keyword: weibull distribution

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RECURRENCE RELATIONS FOR QUOTIENT MOMENTS OF THE WEIBULL DISTRIBUTION BY RECORD VALUES

  • Chang, Se-Kyung
    • Journal of applied mathematics & informatics
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    • v.23 no.1_2
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    • pp.471-477
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    • 2007
  • In this paper we establish some recurrence relations satisfied by the quotient moments of the upper record values from the Weibull distribution. Suppose $X{\in}WEI({\lambda})\;then\;E(\frac {X^\tau_U(m)} {X^{s+1}_{U(n)}})=\frac{1}{(s-\lambda+1)}E(\frac {X^\tau_U(m)}{X^{s-\lambda+1}_{U(n-1)}})-\frac{1}{(s-\lambda+1)}+E(\frac{X^\tau_U(m)}{X^{s-\lambda+1}_{U(n)}})\;and\;E(\frac {X^{\tau+1}_{U(m)}}{X^s_{U(n)}})=\frac{1}{(r+\lambda+1)}E(\frac{X^{\tau+\lambda+1}_{U(m)}}{X^s_{U(n-1)}})-\frac{1}{(\tau+\lambda+1)}E(\frac{X^{\tau+\lambda+1}_{U(m-1)}}{X^s_{U(n-1)}})$.

Planning Practical Multiple-Stress Accelerated Life Tests (실용적 복합 가속수명시험 계획의 개발)

  • Bae, Bong-Soo;Seo, Sun-Keun
    • Journal of Applied Reliability
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    • v.17 no.2
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    • pp.112-121
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    • 2017
  • Purpose: The most previous works on designing accelerated life tests (ALTs) are focused on the application of a single stress. Because of the difficulty to obtain the sufficient information in a reasonable duration using single stress only, there is needed in practice to use multiple-stress ALTs frequently. This paper presents new practical plans with two stresses for Weibull distribution. Methods: The four-level practical plans based on rectangle test region are proposed and compared with the corresponding three-level statistically optimal plans. Sensitivity analyses for assumed design parameters and life-stress relationship are conducted. Results: A procedure to choose practical ALT plans is illustrated with a numerical example and guidelines for planning two-stress ALTs are provided. Conclusion: The proposed two-stress ALT plans on practical constraints to assess a quantile of Weibull lifetime distribution at the use condition are efficient and robust.

The Gringorten estimator revisited

  • Cook, Nicholas John;Harris, Raymond Ian
    • Wind and Structures
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    • v.16 no.4
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    • pp.355-372
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    • 2013
  • The Gringorten estimator has been extensively used in extreme value analysis of wind speed records to obtain unbiased estimates of design wind speeds. This paper reviews the derivation of the Gringorten estimator for the mean plotting position of extremes drawn from parents of the exponential type and demonstrates how it eliminates most of the bias caused by the classical Weibull estimator. It is shown that the coefficients in the Gringorten estimator are the asymptotic values for infinite sample sizes, whereas the estimator is most often used for small sample sizes. The principles used by Gringorten are used to derive a new Consistent Linear Unbiased Estimator (CLUE) for the mean plotting positions for the Fisher Tippett Type 1, Exponential and Weibull distributions and for the associated standard deviations. Analytical and Bootstrap methods are used to calibrate the bias error in each of the estimators and to show that the CLUE are accurate to better than 1%.

Bayesian and maximum likelihood estimation of entropy of the inverse Weibull distribution under generalized type I progressive hybrid censoring

  • Lee, Kyeongjun
    • Communications for Statistical Applications and Methods
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    • v.27 no.4
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    • pp.469-486
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    • 2020
  • Entropy is an important term in statistical mechanics that was originally defined in the second law of thermodynamics. In this paper, we consider the maximum likelihood estimation (MLE), maximum product spacings estimation (MPSE) and Bayesian estimation of the entropy of an inverse Weibull distribution (InW) under a generalized type I progressive hybrid censoring scheme (GePH). The MLE and MPSE of the entropy cannot be obtained in closed form; therefore, we propose using the Newton-Raphson algorithm to solve it. Further, the Bayesian estimators for the entropy of InW based on squared error loss function (SqL), precautionary loss function (PrL), general entropy loss function (GeL) and linex loss function (LiL) are derived. In addition, we derive the Lindley's approximate method (LiA) of the Bayesian estimates. Monte Carlo simulations are conducted to compare the results among MLE, MPSE, and Bayesian estimators. A real data set based on the GePH is also analyzed for illustrative purposes.

Design of Median Control Chart for Unsymmetrical Weibull Distribution (비대칭(非對稱)와이블분포공정(分布工程)에서 메디안특수관리도(特殊管理圖)의 설계(設計))

  • Sin, Yong-Baek;Hwang, Ui-Mi
    • Journal of Korean Society for Quality Management
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    • v.14 no.2
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    • pp.2-8
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    • 1986
  • This thesis is concerned with the design of control chart based on the sample median which is easy to use in practical situations and to analyze the properties for non-normally distributed Weibull process. In this cases are use to the quality characteristics of the process are not normally distributed but skewed due to the intermitted production, small lot size and sample size is small one n=3 or n=5, etc. And when it relates unsymmetrically distributed process, model designed median control chart is more effective than Shewhart $\bar{x}$-chart which assumed on normal distribution, when we exactly should be known Weibull distribution or estimated. The median control chart in this thesis is more robustness compared with other conventionally developed control chart.

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A Piecewise Weibull Distribution in Reliability and its Estimation (신뢰성이론에서의 피스와이즈 와이블분포와 그 추정)

  • Jeong, Hai-Sung
    • Journal of Korean Society for Quality Management
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    • v.24 no.2
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    • pp.65-76
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    • 1996
  • In general, most industrial products exhibit bath-tub shaped curve for their failure rate functions. This distributional life model can be obtained by the Piecewise Weibull distribution. The least squares, maximum likelihood, and mixed methods of estimating the parameters of the Piecewise Weibull distribution are compared. The comparison is made by using the empirical mean squared errors of (a) the parameter estimates and (b) the estimated change-points, to summarize the results of 1000 simulated samples of three sizes - each 100, 150 and 200. The results are that the mixed method estimation comes to be the best as the sample sizes increase.

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CHARACTERIZATIONS OF THE WEIBULL DISTRIBUTION BY THE INDEPENDENCE OF THE UPPER RECORD VALUES

  • Chang, Se-Kyung;Lee, Min-Young
    • The Pure and Applied Mathematics
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    • v.15 no.2
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    • pp.163-167
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    • 2008
  • This paper presents characterizations of the Weibull distribution by the independence of record values. We prove that $X\;{\in}\;W\;EI ({\alpha})$, if and only if $\frac {X_{U(n+l)}} {X_{U(n+1)}\;+\;X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent or $\frac {X_{U(n)}} {X_{U(n+1)}\;+\;X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent. And also we establish that $X\;{\in}\;W\;EI({\alpha})$, if and only if $\frac {X_{U(n+1)}\;-\;X_{U(n)}} {X_{U(n+1)}\;+\;X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent.

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Comparison of Parameter Estimation for Weibull Distribution

  • Wang, Fu-Kwun;J. Bert Keats;B. Y. Leu
    • International Journal of Reliability and Applications
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    • v.4 no.1
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    • pp.41-50
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    • 2003
  • This paper represents the first comprehensive comparison of the Newton-Raphson's method and Simple Iterative Procedure (SIP) in the maximum likelihood estimation of the two-parameter Weibull distribution. Computer simulation is employed to compare these two methods for multiply censored, singly censored data (Type I or Type Ⅱ censoring) and complete data. Results indicate the Newton-Raphson's with the Menon's estimated value, as an initial point remains the effective iterative procedure for estimating the parameters.

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Wave Analysis Method for Offshore Wind Power Design Suitable for Suitable for Ulsan Area

  • Woobeom Han;Kanghee Lee;Seungjae Lee
    • New & Renewable Energy
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    • v.20 no.2
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    • pp.2-16
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    • 2024
  • Various loads induced by marine environmental conditions, such as waves, currents, and wind, are crucial for the operation and viability of offshore wind power (OWP) systems. In particular, waves have a significant impact on the stress and fatigue load of offshore structures, and highly reliable design parameters should be derived through extreme value analysis (EVA) techniques. In this study, extreme wave analyses were conducted with various Weibull distribution models to determine the reliable design parameters of an OWP system suitable for the Ulsan area. Forty-three years of long-term hindcast data generated by a numerical wave model were adopted as the analyses data, and the least-squares method was used to estimate the parameters of the distribution function for EVA. The inverse first-order reliability method was employed as the EVA technique. The obtained results were compared among themselves under the assumption that the marginal probability distributions were 2p, 3p, and exponentiated Weibull distributions.

A Study on Statistical Distribution of Muzzle Velocity of 155mm Propelling Charge (155mm 추진장약 포구속도의 확률분포 특성 연구)

  • Park, Sung-Ho;Park, No-Seok;Choi, Beong-Doo;Kim, Jae-Hoon
    • Proceedings of the Korean Society of Propulsion Engineers Conference
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    • 2009.11a
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    • pp.339-343
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    • 2009
  • The aim of the study was to investigate the statistical distribution of muzzle velocity of 155mm propelling charge K676 which is for the use of K9, a korean 155mm self-propelled artillery. A plenty of muzzle velocity data were collected from lot assessment test of propelling charge. The muzzle velocity of each test round is compensated by reference round. In the present work, the detailed statistical analysis of the muzzle velocity data is carried out using probability models including normal, Weibull 2-parameter and Weibull 3-parameter distributions. The results of goodness of fit test showed that the normal distribution described more appropriately the experimentally measured muzzle velocity data and the Weibull distribution is also applicable. The coefficient of variation showed that the mass production capability of each propelling charge lot has been maintained.

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