On Generalized Linear Exponential Distribution: Different Methods of Estimation

A new exponentiated generalized linear exponential distribution (NEGLED) is introduced, which poses increasing, decreasing, bathtub-shaped, and constant hazard rate. Its various mathematical properties such as moments, quantiles, order statistics, hazard rate function (HRF), stress–strength parameter, etc. are derived. Five distributions, exponential distribution (ED), generalized linear exponential distribution (GLED), Rayleigh distribution (RD), Weibull distribution (WD), and generalized linear failure rate distribution (GLFRD) were used for comparison with NEGLED model using a dataset of blood cancer patients. Estimation of the parameters using the maximum likelihood estimation (MLE) method was obtained and to evaluate their performance a simulation study has been carried out. Finally, a dataset of 40 Leukemia patients was analysed for illustrative purpose proving that the NEGLED outperforms compared distributions.

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.

In this study, we propose the Unit One Parameter Polynomial Exponential (unit-OPPE) distribution and look into some of its mathematical properties while taking into account the transformation X = Z 1 + Z , where the random variable Z follows OPPE distribution with parameter θ. Distributional properties like moments, mode, characterization using conditional moments, reliability measures, and stochastic ordering are addressed. Maximum likelihood (ML) and uniform minimum variance unbiased (UMVU) estimation methods are used to estimate the model parameter. The same are also used to estimate the probability density function, cumulative density function, and reliability function. Some other estimation methods like Least Square estimation (LSE), Weighted Least Square estimation (WLSE), Anderson–Darling estimation (ADE), Cramer-Von-Mises estimation (CME), Maximum Product of Spacings estimation (MPSE) of the parameter are also discussed. The bias and mean square error (MSE) of the parameter estimates are investigated using Monte Carlo simulation. Finally, four real-world applications demonstrate how our unit-OPPE model fits better than the beta, unit-Lindley, unit-Zeghdoudi, unit-Half Normal, Kumaraswamy, unit-Gompertz, and unit-Xgamma models.

This paper presents a novel extension of the exponentiated inverse Rayleigh distribution called the half-logistic exponentiated inverse Rayleigh distribution. This extension improves the flexibility of the distribution for modeling lifetime data for both monotonic and non-monotonic hazard rates. The statistical properties of the half-logistic exponentiated inverse Rayleigh distribution, such as the quantiles, moments, reliability, and hazard function, are examined. In particular, we provide several techniques to estimate the half-logistic exponentiated inverse Rayleigh distribution parameters: weighted least squares, Cramér-Von Mises, maximum likelihood, maximum product spacings and ordinary least squares methods. Moreover, numerical simulations were performed to evaluate these estimation methods for both small and large samples through Monte Carlo simulations, and the finding reveals that the maximum likelihood estimation was the best among all estimation methods since it comprises small mean square error compared to other estimation methods. We employ real-world lifetime data to demonstrate the performance of the newly generated distribution compared to other distributions through practical application. The results show that the half-logistic exponentiated inverse Rayleigh distribution performs better than alternative versions of the Rayleigh distributions.

Exponentiated generalized linear exponential distribution

: The objective of this paper is to develop acceptance procedures based on a sequential probability ratio test when the system failures follow Weibull probability distribution. The acceptance procedures in both Military Handbook 108 and Mil-Std-781C are based on exponential distribution of failures. Exponential distribution implies a constant hazard rate. The Weibull distribution is much more comprehensive and, with suitable choice of parameters, can describe constant, increasing or decreasing hazard rates.

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.

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 new information age, high data rate and strong reliability features our wireless communication systems and is becoming the dominant factor for a successful deployment of commercial networks. MIMO-OFDM (multiple input multiple output-orthogonal frequency division multiplexing), a new wireless broadband technology, has gained great popularity for its capability of high rate transmission and its robustness against multi-path fading and other channel impairments. A major challenge to MIMO-OFDM systems is how to obtain the channel state information accurately and promptly for coherent detection of information symbols and channel synchronization. In the first part, this dissertation formulates the channel estimation problem for MIMO-OFDM systems and proposes a pilot-tone based estimation algorithm. A complex equivalent base-band MIMO-OFDM signal model is presented by matrix representation. By choosing equally-spaced and equally-powered pilot tones from sub-carriers in one OFDM symbol, a down-sampled version of the original signal model is obtained. Furthermore, this signal model is transformed into a linear form solvable for the LS (least-square) estimation algorithm. Based on the resultant model, a simple pilot-tone design is proposed in the form of a unitary matrix, whose rows stand for different pilot-tone sets in the frequency domain and whose columns represent distinct transmit antennas in the spatial domain. From the analysis and synthesis of the pilot-tone design in this dissertation, our estimation algorithm can reduce the computational complexity inherited in MIMO systems by the fact that the pilot-tone matrix is essentially a unitary matrix, and is proven an optimal channel estimator in the sense of achieving the minimum MSE (mean squared error) of channel estimation for a fixed power of pilot tones. In the second part, this dissertation addresses the wireless location problem in WiMax (worldwide interoperability for microwave access) networks, which is mainly based on the MIMO-OFDM technology. From the measurement data of TDOA (time difference of arrival), AOA (angle of arrival) or a combination of those two, a quasi-linear form is formulated for an LS-type solution. It is assumed that the observation data is corrupted by a zero-mean AWGN (additive white Gaussian noise) with a very small variance. Under this assumption, the noise term in the quasi-liner form is proved to hold a normal distribution approximately. Hence the ML (maximum-likelihood) estimation and the LS-type solution are equivalent. But the ML estimation technique is not feasible here due to its computational complexity and the possible nonexistence of the optimal solution. Our proposed method is capable

We develop a new three parameter distribution using the half-Cauchy family of distributions, called the half-Cauchy generalized Rayleigh distribution. Some statistical properties and characteristics of the proposed distribution are provided and obtained, including the explicit expressions for the survival function, median, hazard function, mode, moments, mean deviation, order statistics, cumulative hazard function, quantiles, and the measures of dispersion based on quartiles and octiles. For the parameter estimation of the proposed model, three widely used estimation techniques namely, maximum likelihood estimators (MLE), Cramer-Von-Mises (CVM), and least-square estimation (LSE) methods are applied. Two real datasets are used for the illustration, and the goodness-of-fit test is run. It is discovered that the proposed model fits the real data very well and is more flexible than some well-known models under consideration.

In this paper, we offer generalized exponentiated gamma distribution. We consider the maximum likelihood and Bayesian estimators of unknown parameters and propose Markov chain Monte Carlo (MCMC) method to generate samples from the posterior distribution. The mean square error of estimations is computed and comparisons are made using Monte Carlo simulation. Key words: Maximum likelihood estimator, Bayesian estimator, squared error loss function, generalized exponentiated gamma distribution.

In this paper, it was obtained estimators of the scale parameter of Extended Poisson Exponential distribution by four methods using complete data, these methods are the Maximum Likelihood Estimator (MLE), the Least Squares Estimator (LSE), Plotting Position Estimator (PPE) and White Estimator (WE), these four methods have been compared by the criteria of Mean Square Error (MSE) using Monte Carlo simulation, and from the results on the samples in the simulation, the comparison in this study showed that Maximum Likelihood Estimator method is best from other methods followed by the White Estimator method then the Least Squares Estimator method.

This paper introduces a novel approach by implementing Type II censoring on both stages of the stage life testing (SLT) experiment. First, we have assumed that the lifetime of the experimental units follows a combination of Weibull distribution on initial stage and an exponential distribution on the accelerated stage. Furthermore, we have also considered a case where the lifetime of the experimental units on both stages follows Weibull distribution with same shape parameter and different scale parameters. The distributions under the two stress levels are connected through the cumulative exposure model. We derive maximum likelihood estimation (MLE) in the SLT model under different lifetime distributions. As an alternative to MLE, we also derive the maximum product of spacings (MPS) estimation method and compare the performance of the MPS and MLE methods through a simulation study. Additionally, we derive asymptotic confidence intervals (CIs) based on the asymptotic normality of MLE, and we also employ bootstrap methods to obtain the boot‐ and boot‐ CIs. Further, we proposed an optimal design for the SLT experiment based on two optimality criteria. Finally, an extensive simulation study and a real data analysis have been carried out to demonstrate the effectiveness of our approach.

The continuous development of flexible probability distributions has become essential for accurately describing complex stochastic phenomena. In this study, a new theoretical based on the New Sine Type II Generalized Topp-Leone distribution is investigated. The model is constructed by integrating the sine trigonometric transformation with the Type II Topp-Leone generator applied to an arbitrary baseline distribution. This combination introduces additional shape parameters that enhance the adaptability of the resulting family of distributions. Several mathematical properties of the proposed model are derived. These include explicit expressions for the cumulative distribution function (cdf), probability density function (pdf), survival function, and hazard rate function. In addition, analytical forms of the quantile function, moments, moment generating function, and order statistics are established. The author’s adopted two estimation techniques in this work: the maximum likelihood estimation (MLE) and the maximum product of spacing (MPLS) estimation methods. The structural characteristics of the distribution reveal that it is capable of producing a wide range of density and hazard rate shapes depending on the selected baseline model and parameter values. The theoretical results presented in this work provide a general framework for constructing new members of the family and offer a foundation for further statistical developments involving estimation techniques and practical implementations. Plots of the pdf, cdf as well as the validity check on the pdf of the new model indicate that the model is adequate capable of handling all the intricate associated with real-life datasets.

Laser pointing systems for small targets are mainly confronted with two pointing errors, jitter and boresight, arising due to atmospheric turbulence, mechanical vibrations and errors in optical alignment. Knowledge of these parameters provides information about the quality of the pointing and tracking system. In the past, several techniques have been investigated to estimate these parameters from returned laser signals. These include the key ratio technique, the chi-squared method and the maximum likelihood (ML) estimation technique. These techniques have been studied in the literature for Gaussian irradiance profiles. In particular, the ML estimation technique has been found to provide excellent results. In this paper, we extend the ML estimation technique from Gaussian profiles to near-Gaussian irradiance profiles. Our results show that the modified estimator performs much better than a direct application of the original ML estimator.