A multi-objective programming approach to Weibull parameter estimation

Two-parameter Weibull distribution is the most widely adopted lifetime model for power transformers. An appropriate parameter estimation method is essential to guarantee the accuracy of a derived Weibull lifetime model. Six popular parameter estimation methods (i.e. the maximum likelihood estimation method, two median rank regression methods including the one regressing X on Y and the other one regressing Y on X, the Kaplan-Meier method, the method based on cumulative hazard plot, and the Li's method) are reviewed and compared in order to find the optimal one that suits transformer's Weibull lifetime modelling. The comparison took several different scenarios into consideration: 10000 sets of lifetime data, each of which had a sampling size of 40 1000 and a censoring rate of 90\%, were obtained by Monte-Carlo simulations for each scienario. Scale and shape parameters of Weibull distribution estimated by the six methods, as well as their mean value, median value and 90\% confidence band are obtained. The cross comparison of these results reveals that, among the six methods, the maximum likelihood method is the best one, since it could provide the most accurate Weibull parameters, i.e. parameters having the smallest bias in both mean and median values, as well as the shortest length of the 90\% confidence band. The maximum likelihood method is therefore recommended to be used over the other methods in transformer Weibull lifetime modelling.

This paper considers various estimation methods to estimate the unknown parameters of the DUS Inverse Weibull (DIW) distribution using the maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM) and the Anderson-Darling (AD) estimators. A Monte-Carlo simulation study is conducted to determine the most preferable estimators in terms of their efficiencies. Furthermore, the distribution of the error terms in the simple linear regression is assumed to be DIW to show the implementation of it to the linear models. We also carry out a simulation study for comparing the performances of the estimators of the unknown regression parameters.

In this article, an attempt is made to propose a novel method of lifetime distributions with maximum flexibility using a popular T–X approach together with an exponential distribution, which is known as the New Generalized Logarithmic-X Family (NGLog–X for short) of distributions. Additionally, the generalized form of the Weibull distribution was derived by using the NGLog–X family, known as the New Generalized Logarithmic Weibull (NGLog–Weib) distribution. For the proposed method, some statistical properties, including the moments, moment generating function (MGF), residual and reverse residual life, identifiability, order statistics, and quantile functions, were derived. The estimation of the model parameters was derived by using the well-known method of maximum likelihood estimation (MLE). A comprehensive Monte Carlo simulation study (MCSS) was carried out to evaluate the performance of these estimators by computing the biases and mean square errors. Finally, the NGLog–Weib distribution was implemented on four real biomedical datasets and compared with some other distributions, such as the Alpha Power Transformed Weibull distribution, Marshal Olkin Weibull distribution, New Exponent Power Weibull distribution, Flexible Reduced Logarithmic Weibull distribution, and Kumaraswamy Weibull distribution. The analysis results demonstrate that the new proposed model performs as a better fit than the other competitive distributions.

We revisit the problem of blind calibration of uniform linear sensors arrays for narrowband signals and set the premises for the derivation of the optimal blind calibration scheme. In particular, instead of taking the direct (rather involved) Maximum Likelihood (ML) approach for joint estimation of all the unknown model parameters, we follow Paulraj and Kailath’s classical approach in exploiting the special (Toeplitz) structure of the observed covariance. However, we offer a substantial improvement over Paulraj and Kailath’s Least Squares (LS) estimate by using asymptotic approximations in order to obtain simple, (quasi-)linear Weighted LS (WLS) estimates of the sensors’ gains and phases offsets with asymptotically optimal weighting. As we show in simulation experiments, our WLS estimates exhibit near-optimal performance, with a considerable improvement (reaching an order of magnitude and more) in the resulting mean squared errors, w.r.t. the corresponding ordinary LS estimates. We also briefly explain how the methodology derived in this work may be utilized in order to obtain (by certain modifications) the asymptotically optimal ML estimates w.r.t. the raw data via a (quasi)-linear WLS estimate.

One of the well-known methods for the determination of wind energy potential is the two-parameter Weibull distribution. It is clear that the success of the Weibull distribution for wind energy applications depends on the estimation of the parameters which can be determined by using various numerical methods. In the present study, Monte Carlo simulation method is performed by using six parameters estimation method that is used in the estimation of Weibull distribution parameters such as Maximum Likelihood Estimation (MLE), Least Squares Method (LSM), Method of Moments (MOM), Method of Logarithmic Moments (MLM), Percentile Method (PM), and L-Moment Method (LM), and is compared to Mean Squared Error (MSE) and Mean Absolute Percentage Error (MAPE). In this study, the wind energy potential of the Meşelik region in Eskişehir was modeled with two-parameter Weibull distribution. The average wind speed (m/s) data, which are gathered in 10-minute intervals from the measuring device installed 10 meters about the ground in Meşelik Campus of Eskişehir Osmangazi University, is used. As a result of the simulation study, it has been determined that MLE is the best parameter estimation method for two-parameter Weibull distribution in large sample sizes, and LM has the closest performance to MLE. The wind speed (m/h) data of the region has been successfully modeled with two-parameter Weibull distribution and the highest average wind power density has been obtained in July as 49.38295 (W/m2) while the lowest average wind power density has been obtained in October as 19.30044 (W/m2).

In this study, we construct a family of distributions that does not include additional parameter to any baseline distribution. This new family is a composition of two families of distributions, that is, the Kavya-Manoharan (KM) family and the Dinesh-Umesh-Sanjay (DUS) family and it is named the Kavya-Manoharan Dinesh-Umesh-Sanjay (KM-DUS) family. Further, an innovative two-parameter lifetime distribution being the KM-DUS Rayleigh-inverted Weibull (KM-DUS-RIW) distribution was proposed. In this new distribution, the Rayleigh-Inverted Weibull (RIW) was used as the baseline distribution. The new distribution is an improvement of the RIW in that it guarantees more flexibility and tractability with no increase in parameters hence offering a parsimonious but powerful tool for modeling reliability and survival data. The primary statistical characteristics of this KM-DUS-RIW distribution such as its probability density function, cumulative distribution function, quantile function, moments, and order statistic, have been derived and discussed in great detail. The methods named Maximum likelihood (ML), Least Squares (LS), Weighted Least Squares (WLS), Maximum Product Spacing (MPS), Cramér-von Mises (CVM), Anderson-Darling (AD), Right-Tailed Anderson-Darling (RTAD), Percentile (PERC) as well as Bayesian methods with different loss functions such as Squared Error, LINEX, and General Entropy are utilized for estimating the model parameters were explored. Of these, MPS consistently yields the most efficient estimates across various sample sizes. The practicality of the proposed distribution is exemplified by using real entomological data with adult progeny counts of Stegobium paniceum L. under choice and non-choice tests as application cases. Besides, the model is compared using the log-likelihood, AIC, BIC, HQIC, and several goodness-of-fit statistics, which show the KM-DUS-RIW model fits better than the competing available models. These findings attest to the KM-DUS-RIW distribution being a promising potential alternative to modeling complex lifetime data.

The probability distributions of wind speeds and the availability of wind turbines were investigated by considering the vertical wind shear. Based on the wind speed data at the standard height observed at a wind farm, the power-law process was used to simulate the wind speeds at a hub height of 60 m. The Weibull and Rayleigh distributions were chosen to express the wind speeds at two different heights. The parameters in the model were estimated via the least square (LS) method and the maximum likelihood estimation (MLE) method, respectively. An adjusted MLE approach was also presented for parameter estimation. The main indices of wind energy characteristics were calculated based on observational wind speed data. A case study based on the data of Hexi area, Gansu Province of China was given. The results show that MLE method generally outperforms LS method for parameter estimation, and Weibull distribution is more appropriate to describe the wind speed at the hub height.

In this paper, a modified internal rate of return (MIRR) has been presented on the assumption of Weibull distribution to investigate the investment's attractiveness in the Malaysian property development sector (MPDS). The research intends is to produce parameters estimates of the Weibull distribution for investment analysis for a long-time investment period. The MIRR data were obtained from the company financial report for 5 years investment period from 2010–2014. The Maximum likelihood estimation method has been incorporated with the Simulated annealing algorithm (SA) in estimating the parameters of Weibull distribution. The shape parameter of the Weibull distribution reflects the effectiveness in maximizing the investment return based on MIRR with lower returns and is represented as the slope of the fitted line on a Weibull probability plot. The Weibull parameter estimated using Simulated annealing (SAA) has been compared with the existing Weibull parameter estimation methods. The finding reveals that the proposed methods have good agreement with other methods used for Weibull parameter estimates based on MIRR data. The research is expected to provide an overview of the investment behaviour for the long term investment period. Therefore, the new approach based on SA in estimating the parameters Weibull function can serve as a good alternative approach for the assessment of the rate of return on investment potential.KeywordsModified internal rate of returnWeibull distributionMaximum likelihoodSimulated annealing algorithmGoodness-of-fit tests

Decoupled State-Estimation in Energy Control Centres

Flexural fatigue life analysis of unsaturated polyester-methyl methacrylate polymer concrete

An assumption of ordinary least squares (OLS) is constant variance across the domain of a regressor. Often, however, the random error term variance is not constant when modeling biological data and therefore weighted least squares (WLS) is commonly used to estimate parameters. Less emphasis is placed on observations of the dependent variable with larger variances when using WLS, which is thought to produce more “efficient” parameter estimates. Statisticians often use WLS instead of OLS when the random error variance is not constant. However, as natural resource managers, we are often most interested in values of the dependent variable where the variance is greatest. What then are the biological, economic, and management implications of using WLS versus OLS in natural resource modeling? In this article, parameters of the combined-variable volume equation were estimated for individual trees in loblolly pine plantations using both OLS and WLS. A growth-and-yield model was used to predict stand development over time of two planting densities (303 and 1,090 seedlings/ac) and individual tree volumes were predicted using OLS and WLS parameter estimates. Economic analyses were then conducted. Our results show that the choice of OLS or WLS can have a substantial impact on predicted economic returns. Thus, when using WLS, landowners may decide to invest in additional management treatments that may, in fact, not be economically feasible. This study provides further evidence that landowners need to be aware of model structures when basing harvest-age decisions on growth and yield projections.

Sampling experiments were conducted to compare least squares (LS) and weighted least squares (WLS) estimators in heteroscedastic linear regression. Variance estimates obtained both from replicates, si2, and from a modification of Rao's [1970] MINQUE estimator, j2, were used as weighting factors in WLS estimation. Efficiency comparisons of the LS, WLS(s 2), and WLS(g 2) estimators were based on the generalized variance, the Euclidean norm, the variance of the slope estimator, and the variance of the estimator of E(y I x). It was found thatLS estimators are remarkably robust, and only for marked variance heterogeneity or many replicates at each point are weighted estimators uniformly better. In general, unless the number of replicates at each point is large (> 10), WLS (9 2) estimators are more efficient than WLS(s,2).

This article presents a new extension of the three-parameter log-logistic distribution, the so-called heavy-tailed log-logistic (HTLL) distribution. Some important mathematical properties of the new HTLL distribution are calculated. In addition, some numerical results of moments for the HTLL distribution are calculated. Extensive simulations were performed to investigate the estimation of the model parameters using many established approaches, including maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM), Anderson-Darling (AD), and right-tail Anderson-Darling (RTAD). The simulation results show that the AD approach has the highest efficiency among these approaches. The usefulness of the newly proposed model is demonstrated by analyzing two real data sets.

Modeling the relationship between vehicle speed and density on the road is a fundamental problem in traffic flow theory. Recent research found that using the least-squares (LS) method to calibrate single-regime speed-density models is biased because of the uneven distribution of samples. This paper explains the issue of the LS method from a statistical perspective: the biased calibration is caused by the correlations/dependencies in regression residuals. Based on this explanation, we propose a new calibration method for single-regime speed-density models by modeling the covariance of residuals via a zero-mean Gaussian Process (GP). Our approach can be viewed as a generalized least-squares (GLS) method with a specific covariance structure (i.e., kernel function) and is a generalization of the existing LS and the weighted least-squares (WLS) methods. Next, we use a sparse approximation to address the scalability issue of GPs and apply a Markov chain Monte Carlo (MCMC) sampling scheme to obtain the posterior distributions of the parameters for speed-density models and the hyperparameters (i.e., length scale and variance) of the GP kernel. Finally, we calibrate six well-known single-regime speed-density models with the proposed method. Results show that the proposed GP-based methods (1) significantly reduce the biases in the LS calibration, (2) achieve a similar effect as the WLS method, (3) can be used as a non-parametric speed-density model, and (4) provide a Bayesian solution to estimate posterior distributions of parameters and speed-density functions.

Increasing the amount of data with complex dynamics requires the constant updating of statistical distributions. This study aimed to introduce a new three-parameter distribution, named the new exponentiated Weibull (NEW) distribution, by applying the logarithmic transformation to the exponentiated Weibull distribution. The exponentiated Weibull distribution is a powerful generalization of the Weibull distribution that includes several classical distributions as special cases—Weibull, exponential, Rayleigh, and exponentiated exponential—which make it capable of capturing diverse forms of hazard functions. By combining the advantages of the logarithmic transformation and exponentiated Weibull, the new distribution offers great flexibility in modeling different forms of hazard functions, including increasing, J-shaped, reverse-J-shaped, and bathtub-shaped functions. Some mathematical properties of the NEW distribution were studied. Moreover, four different methods of estimation—the maximum likelihood (ML), least squares (LS), Cramer–Von Mises (CVM), and percentile (PE) methods—were employed to estimate the distribution parameters. To assess the performance of the estimates, three simulation studies were conducted, showing the benefit of the ML method, followed by the PE method, in estimating the model parameters. Additionally, five datasets were used to evaluate the effectiveness of the new distribution in fitting real data. Compared with some Weibull-type extensions, the results demonstrate the superiority of the new distribution in modeling various forms of real data and provide evidence for the applicability of the new distribution.