A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter

Nine parametric estimators of the location and scale parameters of a two-parameter Weibull distribution are compared in terms of their bias and efficiency in a simulation study. The estimators considered are the maximum likelihood estimators (MLE), moment estimators (ME), generalized spacing estimators (GSE), modified maximum likelihood estimators I (MMLE-I), modified maximum likelihood estimators II (MMLE-II), Tiku's modified maximum likelihood estimators (TMMLE), least-squares estimators (LSE), weighted least-squares estimators (WLSE) and percentile estimators (PCE). The aim of the comparisons is to identify the most efficient estimators among these nine estimators for different shape parameters and sample sizes.     

Multistate least-squares parameter estimators

Multistage least-squares parameter estimation algorithms which are either recursive in the dimension of an assumed linear model, or both recursive in the model dimension and sequential in time are obtained. Different aspects of the following problems are considered. 1) Given a linear model,Z(k)= K(k)theta + V(k), withnunknown parameters,theta, and datum{Z(k), K(k)}, obtain least-square estimators (LSE's) which are recursive in the dimension oftheta. 2) Given the same conditions as in 1), a new measurement becomes available at timet_{k+1}, obtain a LSE forthetawhich is both recursive in the dimension ofthetaand sequential int.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

Using real-coded genetic algorithms for Weibull parameter estimation

Genetic algorithms (GAs) represent a class of adaptive search techniques based on a direct analogy to Darwinian natural selection and mutations in biological systems. “Standard” GAs have emphasized the utilization of binary codes. However, recent empirical results have indicated that a chromosome representation which utilizes real values have enhanced the performance of these GAs in certain engineering problems. A real-valued Genetic Algorithm method described in this paper estimates the parameter values from an unconstrained population of data points for a Weibull distribution function using a simultaneous random search function by integrating the principles of the Genetic Algorithm and the method of Maximum Likelihood Estimation. The results of the real-coded GA technique for parameter estimation are compared to the results of the Newton-Raphson Algorithm.     

Large-deviation inequalities for parameter estimators in stochastic systems

Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 95–112, March–April, 1994.     

Robust copyright marking using Weibull distribution

For multimedia document, watermarking provides many advantages; however, its vulnerability due to attacks using signal processing methods offers challenge in designing robust techniques. This paper presents a novel method for enhancing robustness in copyright marking into mid-frequency discrete cosine transform (DCT) coefficients of an image. We establish that Weibull distribution suitably models mid-frequency coefficients well and use it for hiding copyright data. Our method is robust against JPEG compression attacks at both high and low compressions. Further, it is statistically undetectable in comparison to existing techniques of data hiding based on DCT coefficients. This is because it insignificantly affects the DCT coefficients of the stego image without compromising with data hiding capacity of the host image. Simulation result on Lena, Gibbon, Boat, and Peppers images illustrates that our algorithm offers lower bit error rate (BER) and higher peak signal to noise ratio (PSNR) leading to improved imperceptibility. Thus, it is useful in robust copyright marking.     

On the design of LMS-based channel estimators using the Doppler spread parameter

An analysis of the optimization of the LMS (Least Mean Square) algorithm for the estimation of time-varying and frequency-selective communication channels is here presented. In contrast to previous works on this subject, in which the step-size optimization is rooted on the assumption of specific channel models, this analysis is much more generic in respect to the modelling of both the Doppler and delay spreadings. Besides, it addresses not only the steady-state performance of channel estimators, but also their transient behaviors. Several useful approximate expressions for designing LMS-based channel estimators are herein derived and validated by comparisons with simulation and with a previous work on this kind of analysis. These expressions are remarkably suitable for practical application, since they depend on a few parameters of the communication system. With regard to the channel variability, the knowledge of the Doppler spread parameter is shown to be enough to optimize the performance of a LMS channel estimator using the analysis here presented.     

Inference for the Weibull distribution with progressive hybrid censoring

Recently, progressive hybrid censoring schemes have become quite popular in life-testing and reliability studies. In this paper, we investigate the maximum likelihood estimation and Bayesian estimation for a two-parameter Weibull distribution based on adaptive Type-I progressively hybrid censored data. The Bayes estimates of the unknown parameters are obtained by using the approximation forms of Lindley (1980) and Tierney and Kadane (1986) as well as two Markov Chain Monte Carlo methods under the assumption of gamma priors. Computational formulae for the expected number of failures is provided and it can be used to determine the optimal adaptive Type-I progressive hybrid censoring schemes under a pre-determined budget of experiment.     

Comparative studies on the high-variability embedded robust parameter design from the perspective of estimators

Engineers and scientists often identify robust parameter design (RPD) as one of the most important process and quality improvement methods. Focused on determining the optimum operating conditions that facilitate target attainment with minimum variability, typical approaches to RPD use ordinary least squares methods to obtain response functions for the mean and variance by assuming that process data are normally distributed and exhibit reasonably low variability. Consequently, the sample mean and standard deviation are the most common estimators used in the initial tier of estimation, as they perform best when these assumptions hold. Realistically, however, industrial processes often exhibit high variability, particularly in mass production lines. If ignored, such conditions can cause the quality of the estimates obtained using the sample mean and standard deviation to deteriorate. This paper examines several alternatives to the sample mean and standard deviation, incorporating them into RPD modeling and optimization approaches to ascertain which tend to yield better solutions when highly variable conditions prevail. Monte Carlo simulation and numerical studies are used to compare the performances of the proposed methods with the traditional approach.     

Estimating turning points of the failure rate of the extended Weibull distribution

Marshall and Olkin (1997) proposed a way of introducing a parameter, called the tilt parameter, to expand a family of distributions. In this paper we compare the extended distribution and the original distribution with respect to some stochastic orderings. Also we investigate thoroughly the monotonicity of the failure rate of the resulting distribution when the baseline distribution is taken as Weibull. It turns out that the failure rate is increasing, decreasing, or non-monotonic with one or two turning points depending on the parameters. For non-monotonic types, the turning points of the failure rate are estimated and their confidence intervals are provided. Simulation studies are carried out to examine the performance of these intervals. An example is provided to illustrate the procedure.     

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.     

On the asymptotic distribution of the least-squares estimators in unidentifiable models

In order to analyze the stochastic property of multilayered perceptrons or other learning machines, we deal with simpler models and derive the asymptotic distribution of the least-squares estimators of their parameters. In the case where a model is unidentified, we show different results from traditional linear models: the well-known property of asymptotic normality never holds for the estimates of redundant parameters.     

Inference for Weibull distribution under generalized order statistics

Based on generalized order statistics from Weibull distribution the approach of Bayesian and non-Bayesian estimation are discussed. We present a simple and efficient simulational algorithm for generating a generalized order statistics sample from any continuous distribution. Specializations to Bayesian and non-Bayesian estimators, some lifetime parameters and confidence intervals of progressive II censoring and record values are obtained and compared with the existing results. Two examples are given to illustrate the proposed estimators and the simulation algorithm.     

Performance analysis of Nitzberg's clutter map for Weibull distribution

The clutter map analyzed by Nitzberg, which averages the detector outputs of the same resolution cell over several past scans to set the detection threshold, plays an important role in radar detection fields. In this work, the average detection threshold (ADT) of Nitzberg's clutter map is derived and its steady-state property is analyzed, moreover, the detection performance of Nitzberg's clutter map for Weibull background is analyzed by the importance sampling technique.     

Pathological behaviors of fisher confidence bounds for Weibull distribution

The Weibull distribution is widely used in reliability engineering. To estimate its parameters and associated reliability indices, the maximum likelihood (ML) approach is often employed, and the associated Fisher information matrix is used to obtain the confidence bounds on the reliability indices that are of interest. The estimates and the confidence bounds usually behave similarly in terms of monotonic and asymptotic properties. However, the confidence bounds may behave differently under certain circumstances. As a result, the Fisher matrix approach may not always be preferred in obtaining the desired confidence bounds. This paper provides some properties of Fisher confidence bounds for the Weibull distribution. These properties can be used as guidelines when implementing the ML approach and Fisher information matrix to analyze failure time data and plan life tests.     

Estimating the parameters of the Marshall-Olkin bivariate Weibull distribution by EM algorithm

In this paper we consider the Marshall-Olkin bivariate Weibull distribution. The Marshall-Olkin bivariate Weibull distribution is a singular distribution, whose both the marginals are univariate Weibull distributions. This is a generalization of the Marshall-Olkin bivariate exponential distribution. The cumulative joint distribution of the Marshall-Olkin bivariate Weibull distribution is a mixture of an absolute continuous distribution function and a singular distribution function. This distribution has four unknown parameters and it is observed that the maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms. In this paper we discuss about the computation of the maximum likelihood estimators of the unknown parameters using EM algorithm. We perform some simulations to see the performances of the EM algorithm and re-analyze one data set for illustrative purpose.     

指数威布尔分布参数的模糊最小二乘估计

讨论了当寿命数据含有异常数据时指数威布尔分布参数的估计问题。当寿命数据含有异常数据时,利用基于模糊聚类的最小二乘方法对参数进行估计,这种方法能削弱异常数据的影响。并获得了在完全样本情形下参数的最小二乘估计和加权最小二乘估计。最后给出数值模拟,结果显示模糊最小二乘估计是正确有效的。     

Fitting the generalized Pareto distribution to data using maximum goodness-of-fit estimators

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.