Complete Sample Estimation Techniques for Reparameterized Weibull Distributions

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

The most applied principles for parameter estimation are maximum likelihood and least square error. This paper presents a new principle with regard to the parameter estimation of the three-parameter Weibull distribution. By transforming the cumulative distribution function, constructed is a mapping from the value of the random variable and its corresponding cumulative distribution probability to the scale parameter. The scale parameter estimated by such a mapping is the random variable value and the corresponding cumulative distribution probability dependent when the shape parameter and/ or location parameter applied in the mapping is subject to error. Given a set of random variable values or a set of sample values, a larger error in the shape/location parameter brings about larger differences between the scale parameter values obtained with the individual random variable values or sample values, respectively. Based on such a causal relationship between the discrepancy and the shape/ location parameter value applied in the mapping relation, a new parameter estimation method is proposed. For the Weibull distribution parameter estimation according to a set of sample values, the right shape parameter and location parameter are those minimizing the discrepancy between the scale parameter values obtained with the individual sample values, respectively. Case studies demonstrate that the proposed method outperforms the maximum likelihood method and the Weibull plot-based least squares method.

In fatigue testing of metals or composite materials under different levels of stress, the underlying distributions are often assumed to be Weibull with a common shape parameter. The problems of estimation and hypothesis tests of the common shape parameter and scale parameters in the Weibull distributions are considered in this paper. The results are: 1) the maximum likelihood equation for estimating the shape parameter, 2) a s-confidence interval and hypothesis test for the shape parameter, 3) s-bias factors and variances of the pooled estimator of shape parameter, and 4) s-confidence interval and hypothesis test for the scale parameters. Examples are given for composite material applications. Results are based on Monte Carlo simulation methods.

Introduction Assessing objective measures of sleep fragmentation could yield important features reflecting impaired sleep quality in people with insomnia. Survival analysis allows the specific examination of the stability of NREM sleep, REM sleep and wake. The objective of this study was to assess the differences between survival dynamics of NREM sleep, REM sleep and wake between people with insomnia and healthy controls. Methods We analyzed polysomnography recordings from 88 people with insomnia and 92 healthy controls. For each participant, survival dynamics of REM sleep, NREM sleep and wake were represented using Weibull distributions. We used lasso penalized linear regression to analyze the difference between participant groups with respect to the Weibull scale and shape parameters, while correcting for age, sex, total sleep time and relevant interaction effects. Because comparisons were done for scale and shape parameters of REM sleep, NREM sleep and wake, a Bonferroni correction was applied, resulting in an alpha value of 0.05/6 = 0.0083. Results Significant effects of group were found for the NREM scale parameter (unstandardized model coefficient B=-0.79, t=-3.0, p=0.0035), and for the scale and shape parameters of wake (scale parameter B=7.6, t=2.8, p=0.0065; shape parameter B=0.20, t=2.9, p=0.0048). Results indicated that people with insomnia had less stable NREM sleep and more stable wake after sleep onset compared to healthy controls. Additionally, the altered distribution of wake segment lengths indicated an increased difficulty to fall asleep after longer awakenings in the insomnia group. However, these differences were mainly observed in younger participants. Significant effects of group for the survival parameters of REM sleep were not found. Conclusion As illustrated by our results, survival analysis can be very useful for disentangling different types of sleep fragmentation in people with insomnia. For instance, the current findings suggest that people with insomnia have an increased fragmentation of NREM sleep, but not necessarily of REM sleep. Additional research into the underlying mechanisms of NREM sleep fragmentation could possibly lead to a better understanding of impaired sleep quality in people with insomnia, and consequently to improved treatment. Support (if any):

Article history: Received 1 January 2010 Received in revised form 8 June 2010 Accepted 9 June 2010 Available online 9 June 2010 This paper conducts a simulation study of the effects of violating the ANOVA normality assumption in the presence of Weibull data. Twelve specific Weibull distributions, characterizing the life data of a variety of real-world products and systems, are investigated. Confidence intervals on test significance and power are generated and compared against intervals from normally distributed data. The ANOVA procedure is found to be robust in the majority of cases. Furthermore, a designed experiment is conducted to isolate the effects of the Weibull shape and scale parameters within the preceding study. The shape parameter is found to have a significant effect on significance and power, whereas the scale parameter does not have a significant effect at the target α = 0.05 test significance level. © 2010 Growing Science Ltd. All rights reserved.

Estimation of the reliability parameter for three-parameter Weibull models

The two‐parameter Weibull distribution (2PWD), similar to an instantaneous unit hydrograph (IUH), is parameterized in terms of Horton catchment ratios on the basis of a geomorphologic model of catchment response. For this the shape and scale parameters of the Weibull distribution are expressed analytically in terms of Horton's catchment ratios. The two parameters of the IUH derived using Nash's model, which is a two‐parameter gamma distribution (2PGD), are also expressed analytically in terms of Horton ratios. The performance of the proposed methods is tested for describing a synthetic unit hydrograph (SUH) under limited data conditions. A comparison is made with the unit hydrographs derived from the event data of two real catchments, and with the existing geomorphological based 2PGD for developing SUH given by Rosso (1984). The sensitivity analysis of the 2PWD to the nondimensional parameter β of the UH (a product of peak discharge and time to peak) shows β to be more sensitive to the shape parameter a than the scale parameter b. Further examination to find any similarity between the behavior of 2PWD and 2PGD showed that a in 2PWD corresponds to the shape parameter n in the 2PGD, and b behaves similar to the scale parameter k in the 2PGD. Finally, practical applicability of the proposed approach to ungauged catchments is tested using field data.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper presents a novel approach based on Weibull distribution to determine the capacity of wind turbine generators (WTGs) using capacity factor (CF), normalized average power (<formula formulatype="inline"><tex> $P_{N}$</tex> </formula>), and the product of CF and <formula formulatype="inline"><tex>$P_N$</tex> </formula> under different values of tower height and rated wind speed. Five locations for installation of WTG in Taiwan are practically examined. The proposed Weibull distribution is employed to represent probability distribution of wind speed variation, while the important relationships among mean wind speed (MWS), standard deviation of wind speed, and both scale and shape parameters of the Weibull distribution are also derived. The cost of energy (COE) and capital costs of WTG under different tower heights and various locations are also determined. It can be concluded from the simulation results of five installation locations of WTG in Taiwan that suitable values for both shape parameter and scale parameters of Weibull distribution as well as wind turbine capacity are identically important for selecting locations of installing WTG. The scale parameter of Weibull distribution also significantly affects COE, and it is important to determine whether a wind farm is good or not. </para>

In this paper, we consider the Bayesian inference on the stress-strength parameter \(R = P(Y < X)\), when X and Y follow independent Weibull distributions. We have considered different cases. It is assumed that the random variables X and Y have different scale parameters and (a) a common shape parameter or (b) different shape parameters. Moreover, both stress and strength may depend on some known covariates also. When the two distributions have a common shape parameter, Bayesian inference on R is obtained based on the assumption that the shape parameter has a log-concave prior, and given the shape parameter, the scale parameters have Dirichlet-Gamma prior. The Bayes estimate cannot be obtained in closed form, and we propose to use Gibbs sampling method to compute the Bayes estimate and also to compute the associated highest posterior density (HPD) credible interval. The results have been extended when the covariates are also present. We further consider the case when the two shape parameters are different. Simulation experiments have been performed to see the effectiveness of the proposed methods. One data set has been analyzed for illustrative purposes and finally, we conclude the paper.

Weibull renewal function has attracted a lot of attention because the Weibull distribution describes in a relatively simple analytical manner a wide range of realistic behavior and its shape and scale parameters can be readily determined with graphical or statistical procedure. On the other hand, there are no closed form analytical solutions for the Weibull renewal function except for the special case of exponential distribution. Bounds, approximations, and tables are usually used. In this article, some approximations based on Normal approximation of Weibull distribution are studied. Such a procedure, which is different from that in the existing literature, is shown to be good for Weibull renewal function when the shape parameter is of moderate or large size. Series truncation expression and approximation bounds can be obtained as well. Numerical examples and comparisons are shown to illustrate the procedure.

Synthesizing dual-track road elevation data

This manuscript consists of two topics. Firstly, we explore the utility of internal pilot study (IPS) approach for reestimating sample size at an interim stage when a reliable estimate of the nuisance shape parameter of the Weibull distribution for modeling survival data is unavailable during the planning phase of a study. Although IPS approach can help rescue the study power, it is noted that the adjusted sample size can be as much as twice the initially planned sample size, which may put substantial practical constraints to continue the study. Secondly, we discuss Bayesian predictive probability for conducting interim analyses to obtain preliminary evidence of efficacy or futility of an experimental treatment warranting early termination of a clinical trial. In the context of single-arm clinical trials with time-to-event endpoints following Weibull distribution, we present the calculation of the Bayesian predictive probability when the shape parameter of the Weibull distribution is unknown. Based on the data accumulated at the interim, we propose two approaches which rely on the posterior mode or the entire posterior distribution of the shape parameter. To account for uncertainty in the shape parameter, it is recommended to incorporate its entire posterior distribution in our calculation.

Introduction and objectives The objective measurement of cough frequency has changed the standards by which we evaluate anti-tussive treatments and study the mechanisms driving cough. The distribution of individual cough events over time can vary substantially between patients with identical cough frequencies but this has received little attention. The aim of this study was to explore the differences in the temporal distribution of cough in 24 hour acoustic recordings across a range of respiratory conditions using the Weibull Distribution. Methods We studied 24 hour acoustic recordings made using the VitaloJAK™ cough monitoring system from 62 participants [healthy volunteers (n=11), chronic cough (n=15), asthma (n=10) chronic obstructive pulmonary disease (COPD) (n=16), Idiopathic Pulmonary Fibrosis (IPF) n=10)]. Using the exact location of the explosive phase of each cough sound, the lengths of inter-cough intervals were calculated in each recording and the fit to Weibull and exponential distributions assessed; the shape parameter which describes the ‘clustering’ of events was examined. Results The number of coughs per 24 recording ranged from 0 to 3133, median 158.0 coughs and was increased compared with healthy controls in all disease groups [healthy volunteers median 13.0(IQR 1.0–22.0) coughs/24 hour versus chronic cough 509.0 (296.0–916.0), asthma 35.0 (13.5–144.0) COPD 158.0 (41.0–791.5), Idiopathic Pulmonary Fibrosis 193.5 (101.3–542.8), all p Conclusions The intervals between cough events are not random and can be described by a Weibull distribution. Significant differences in the Weibull shape parameter suggest differences in the temporal clustering of coughs between different respiratory conditions and therefore may be an important novel parameter from objective cough monitoring.

Today, most electrical energy is generated by burning huge fossil fuels and special weather conditions such as acid rain and snow, climate change, urban smog, regional haze, several tornados, etc., have happened around the whole world. It is now clear that the installation of a number of wind turbine generators can effectively reduce environmental pollution, fossil fuel consumption, and the costs of overall electricity generation. Although wind is only an intermittent source of energy, it represents a reliable energy resource from a long-term energy policy viewpoint. Among various renewable energy resources, wind power energy is one of the most popular and promising energy resources in the whole world today. At a specific wind farm, the available electricity generated by a wind power generation system depends on mean wind speed (MWS), standard deviation of wind speed, and the location of installation. Since year-to-year variation on annual MWS is hard to predict, wind speed variations during a year can be well characterized in terms of a probability distribution function (pdf). This paper also addresses the relations among MWS, its standard deviation, and two important parameters of Weibull distribution. The wind resource varies with of the day and the season of the year and even some extent from year to year. Wind energy has inherent variances and hence it has been expressed by distribution functions. In this paper, we present some methods for estimating Weibull parameters, namely, shape parameter ( k ) and scale parameter ( c ). The Weibul distribution is an important distribution especially for reliability and maintainability analysis. The suitable values for both shape parameter and scale parameters of Weibull distribution are important for selecting locations of installing wind turbine generators. The scale parameter of Weibull distribution also important to determine whether a wind farm is good or not. The presented method is the analytical methods and computational experiments on the presented methods are reported.

The wind resource varies with of the day and the season of the year and even some extent from year to year. Wind energy has inherent variances and hence it has been expressed by distribution functions. In this paper, we present some methods for estimating Weibull parameters in case of a low wind speed characterization, namely, shape parameter (k), scale parameter (c) and characterize the discrete wind data sample by the discrete Hilbert transform. We know that the Weibull distribution is an important distribution especially for reliability and maintainability analysis. The suitable values for both shape parameter and scale parameters of Weibull distribution are important for selecting locations of installing wind turbine generators. The scale parameter of Weibull distribution also important to determine whether a wind farm is good or not. Thereafter the use of discrete Hilbert transform (DHT) for wind speed characterization provides a new era of using DHT besides its application in digital signal processing. Basically in this paper, discrete Hilbert transform has been applied to characterize the wind sample data measured on College of Engineering and Management, Kolaghat, East Midnapore, India in January 2011.