Different Estimation Methods for A New Extension Three Parameter Log-Logistic Distribution with Applications

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.

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.

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.

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.

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 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.

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.

The weighted least squares (WLS) algorithm is widely employed for designing quadrature mirror filters (QMF) in the frequency domain. The equiripple nature of the WLS design and its performance near the filter band edge have been reported to be better than that of the least squares (LS) design. It is known that these design techniques suffer from pass-band anomaly under different stop band frequencies. We have proposed an algorithm to eliminate pass-band anomaly of LS design. In this paper, we show that WLS can be modified in way similar to that proposed for the LS design. The characteristics of the resulting filters satisfy the stop band requirements and do not show anomaly in the pass band region.

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.

The odd exponentiated Rayleigh-G family: A flexible statistical framework for reliability and radiation modeling

ABSTRACTThe log-logistic distribution is commonly used to model lifetime data. We propose a wider distribution, named the exponentiated log-logistic geometric distribution, based on a double activation approach. We obtain the quantile function, ordinary moments, and generating function. The method of maximum likelihood is used to estimate the model parameters. We propose a new extended regression model based on the logarithm of the exponentiated log-logistic geometric distribution. This regression model can be very useful in the analysis of real data and could provide better fits than other special regression models. The potentiality of the new models is illustrated by means of two applications to real lifetime data sets.

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.

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.