DUS Inverse Weibull Distribution and Parameter Estimation in Regression Model

Weibull distribution is widely used in various areas such as life tables, failure rates, and definition of wind speed distribution. Therefore, parameter estimation for the Weibull distribution is an important problem in many real data applications. The least square (LS), the weighted least square (WLS) and the maximum likelihood (ML) are the most popular methods for the parameter estimation in the Weibull distribution. In this study, based on the LS, WLS and ML estimation methods, a multi-objective programming approach is proposed for the parameter estimation of two-parameter Weibull distribution. This new approach evaluates together LS, WLS and ML methods in the estimation process. NSGA-II method, which is a multi-objective heuristic optimization method, is used to solve the proposed multi-objective estimation model. To evaluate the applicability and performance of the proposed approach, a detailed Monte Carlo simulation study based on deficiency criteria and a real data application are designed. The results illustrated that the proposed multi-objective programming approach provides quite accurate parameter estimates for the two parameter Weibull distribution with respect to deficiency criteria.

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.

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.

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.

This study presents the Kumaraswamy Marshal–Olkin Flexible Weibull Extension (KMO-FWE) distribution, a newly developed probability model designed to extend existing lifetime distributions. This extension enhances the ability to model various hazard rate patterns with greater flexibility. The study explores the fundamental properties of this distribution, including the cumulative distribution function (CDF), probability density function (PDF), survival function, hazard function, moments, moment generating function, quantile function, and order statistics. To estimate the distribution’s parameters, several statistical methods are assessed, with a primary focus on Maximum Likelihood Estimation (MLE). Additionally, alternative estimation techniques such as Maximum Product Spacing (MPS), Least Squares (LS), Weighted Least Squares (WLS), and Percentile Estimation (PE) are considered. A Monte Carlo simulation study is conducted to evaluate the accuracy and efficiency of these estimation approaches under both complete data and Type-II censoring conditions. The results indicate that MLE is highly effective for large sample sizes but may introduce bias in smaller samples. In contrast, MPS provides stable parameter estimates under more complex conditions. While LS and WLS are computationally straightforward, their performance diminishes in the presence of censoring. Meanwhile, PE proves beneficial in reliability analysis based on percentile estimation. Overall, the findings highlight the KMO-FWE distribution as a versatile and reliable model, suitable for applications in reliability engineering, survival analysis, and biomedical research.

Normal distribution has a vital role for the most of statistical methods. Box-Cox power transformation is the most usually applied method when the distribution of data is not normal. In this study, a novel algorithm is proposed assembling different Box-Cox transformation estimates of the well performed six techniques through random effect model in meta analysis. These techniques include the use of goodness-of-fit tests for normality; Anderson–Darling, Lilliefors, Cramer-von Mises, Shapiro–Wilk, Jarque–Bera and Shapiro–Francia tests. For the estimation of Box-Cox parameter, we assemble all possible combinations (63 combinations) of estimates calculated by these six methods. A Monte-Carlo simulation study is implemented to investigate which combination performs better compared to the rest. The simulation study states that the combination of Shapiro–Wilk, Jarque–Bera and Ander-son–Darling tests performs well in most of the simulation scenarios constructed under different transformation parameters and sample sizes. In this study, this combination is proposed as ensemble based Box-Cox transformation via meta analysis. The proposed approach is implemented on white blood count data of leukaemia patients which are not normally distributed. Also, the proposed methodology is provided in AID R package with “box-coxmeta” function for public use.

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.

The standard Cramer-von Mises and Anderson-Darling goodness-of-fit tests require continuous underlying distributions with known parameters. In this paper, tables of critical values are generated for both tests for Weibull distributions with unknown location and scale parameters and known shape parameters. The powers of the Cramer-von Mises, Anderson-Darling, Kolmogorov-Smirnov, and Chi-Square tests for this situation are investigated. The Cramer-von Mises test has most power when the shape is 1.0 and the Anderson-Darling test has most power when the shape is 3.5. Finally, a relation between critical value and inverse shape parameter is presented.

Maximum-likelihood (ML), also given its connection to least-squares (LS), is widely adopted in parameter estimation of physiological system models, i.e., assigning numerical values to the unknown model parameters from the experimental data. A more sophisticated but less used approach is maximum a posteriori (MAP) estimation. Conceptually, while ML adopts a Fisherian approach, i.e., only experimental measurements are supplied to the estimator, MAP estimation is a Bayesian approach, i.e., a priori available statistical information on the unknown parameters is also exploited for their estimation. In this paper, after a brief review of the theory behind ML and MAP estimators, we compare their performance in the solution of a case study concerning the determination of the parameters of a sum of exponential model which describes the impulse response of C-peptide (CP), a key substance for reconstructing insulin secretion. The results show that MAP estimation always leads to parameter estimates with a precision (sometimes significantly) higher than that obtained through ML, at the cost of only a slightly worse fit. Thus, a three exponential model can be adopted to describe the CP impulse response model in place of the two exponential model usually identified in the literature by the ML/LS approach. Simulated case studies are also reported to evidence the importance of taking into account a priori information in a data poor situation, e.g., when a few or too noisy measurements are available. In conclusion, our results show that, when a priori information on the unknown model parameters is available, Bayes estimation can be of relevant interest, since it can significantly improve the precision of parameter estimates with respect to Fisher estimation. This may also allow the adoption of more complex models than those determinable by a Fisherian approach.

This article deals with the problem of testing for the equality of the shape parameters in two inverse Weibull distributions. We propose Bayesian hypothesis testing procedures for the equality of the shape parameters under the noninformative prior. The noninformative prior is usually improper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. So we propose the default Bayesian hypothesis testing procedures based on the fractional Bayes factor and the intrinsic Bayes factors under the reference priors. Simulation study and an example are provided.

As an extension of the standard Weibull distribution, a new crucial distribution termed alpha power Weibull distribution has been presented. It can model decreasing, increasing, bathtub, and upside-down bathtub failure rates. This research investigates the estimation of model parameters and some of its reliability characteristics using progressively Type-II censored data. To get estimates of unknown parameters, reliability, and hazard rate functions, the maximum likelihood, and Bayesian estimation approaches are studied. To acquire estimated confidence intervals for unknown parameters and reliability characteristics, the maximum likelihood asymptotic properties are used. The Markov chain Monte Carlo approach is used in Bayesian estimation to provide Bayesian estimates under squared error and LINEX loss functions. Furthermore, the highest posterior density credible intervals of the parameters and reliability characteristics are determined. A Monte Carlo simulation study is used to investigate the accuracy of various point and interval estimators. In addition, various optimality criteria are used to choose the best progressive censoring schemes. Two real data from the engineering field are analyzed to demonstrate the applicability and significance of the proposed approaches. Based on numerical results, the Bayesian procedure for estimating the parameters and reliability characteristics of alpha power Weibull distribution is recommended. The analysis of two real data sets showed that the alpha power Weibull distribution is a good model to investigate engineering data in the presence of progressive Type-II censoring.

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).

Asymptotically efficient GNSS trilateration

The overlay modeling errors are commonly modeled as the sum of inter-field and intra-field errors in lithography process of wafer stepper. The inter-field errors characterize the global effect while the intra-field errors represent the local effect. To have a better resolution and alignment accuracy, it is important to model the overlay errors and compensate them into tolerances. This paper proposes a weighted least squares (WLS) estimator for two general overlay error models such that more accurate linear term parameters of the overlay error can be obtained. First, the least squares (LS) estimator is applied to obtain the parameters of linear and nonlinear terms. We intend to estimate the parameters of linear term while taking the nonlinear term as our modeling residual errors. Next, we use the WLS estimator to obtain more accurate parameters of linear term and thus reduced the modeling errors by choosing appropriate stepper control parameters. The WLS estimator is applied to a real data set collecting 537 wafers from a wafer fabrication facility. The test results demonstrate that the estimated linear term parameters of the WLS estimator are much more close to the assumed ones than the LS estimator.

This paper introduces a novel class of continuous probability distributions called the Log-Adjusted Polynomial (LAP) G family, with a focus on the LAP Weibull distribution as a key special case. The proposed family is designed to enhance the flexibility of classical distributions by incorporating additional parameters that control shape, skewness, and tail behavior. The LAP Weibull model is particularly useful for modeling lifetime data, extreme events, and insurance claims characterized by heavy tails and asymmetry. The paper presents the mathematical formulation of the new family, including its cumulative distribution function, probability density function, and hazard rate function. It also explores structural properties such as series expansions and tail behavior. Risk analysis is conducted using advanced risk measures, including Value-at-Risk (VaR), Tail VaR (TVaR), and tail mean-variance (TMVq), under various estimation techniques. Estimation methods considered include maximum likelihood (MLE), Cramer–von Mises (CVM), Anderson–Darling (ADE), ´ and their right-tail and left-tail variants. These methods are compared using both simulated and real-world insurance data to assess their sensitivity to tail events. The performance of each estimator is evaluated in terms of bias, accuracy, and robustness in capturing extreme risks. The LAP Weibull model demonstrates superior performance in fitting heavy-tailed data compared to traditional models. AD2LE emerges as the most risk-sensitive estimator, producing the highest values for all key risk indicators. ADE also performs well, offering a balance between sensitivity and stability. MLE and CVM tend to underestimate tail risks, which could lead to insufficient capital reserves in insurance applications. The study highlights the importance of selecting appropriate estimation techniques based on the specific goals of the risk analysis. With its enhanced flexibility and performance in modeling extreme risks, the LAP Weibull model offers a robust framework for modern risk assessment. The findings support the use of AD2LE or ADE in high-stakes risk management scenarios, especially when dealing with heavy-tailed insurance data. This work contributes to the growing literature on advanced statistical models for actuarial and financial risk analysis. The LAP Weibull model proves particularly useful in capturing the tail behavior of claim distributions, improving the accuracy of risk predictions. The paper provides a solid foundation for future applications of the LAP family in modeling complex real-world phenomena under uncertainty.