HALF CAUCHY GENERALIZED RAYLEIGH DISTRIBUTION: THEORY AND APPLICATION

In this article, we have introduced a new probability distribution having three parameters using half Cauchy family of distribution named half Cauchy extended exponential distribution. The statistical properties and characteristics of the proposed distribution like the hazard rate function (HRF), cumulative hazard function, and the probability density function (PDF), and the cumulative distribution function (CDF), quantile function and the skewness, kurtosis are provided. The parameters of the proposed distribution are estimated using the Cramer-Von-Mises (CVM) least-square estimation (LSE), and maximum likelihood estimators (MLE) methods. A real data set is analyzed to test the goodness-of-fit of the proposed distribution. It is found that the half Cauchy extended exponential distribution performed well as compared to some competing distributions.

In this paper, we have derived and studied new probability distributions by extending the 2‐dimensional Rayleigh distribution (RD). First, we extend the RD to 3 dimensions and then generalize it to k dimensions for any positive integer k ≥ 3. The distributions are named the 3‐dimensional Rayleigh distribution (3‐DRD) and k‐dimensional Rayleigh distribution (k‐DRD), respectively. For both 3‐DRD and k‐DRD, detailed mathematical and statistical properties including derivations of the corresponding cumulative distribution, probability density, survival, and hazard functions, moments, moment generating functions, mode, skewness, kurtosis, and differential entropy are obtained in closed forms. Parameter estimation is done for both models using the maximum likelihood estimation method and some statistical properties of the estimator are discussed for each case. Interestingly, the commonly known Normal, Rayleigh, Maxwell–Boltzmann, chi‐square, gamma, and Erlang distributions are related to the newly developed extended RDs as special cases. For the 3‐DRD, plots of cumulative distribution, probability density, survival, and hazard functions are exhibited, a simulation study is carried out, and random samples are generated using the standard accept–reject (AR) algorithm to check the efficiency of the maximum likelihood estimates of the parameter. Moreover, the new 3‐DRD model is fitted to one simulated and three real datasets, revealing good performance compared to four existing Rayleigh‐based distributions. This study will contribute new knowledge to the field of applied statistics and probability, and the findings will be used as a basis for future research in the field.

Objectives: To investigate the Modified Inverse Weibull Distribution (MIWD) to determine its effectiveness in modelling real customer transaction data, which is often skewed and exhibits heavy-tailed characteristics. Methods: Parameters of the model were estimated using the Maximum Likelihood Estimation (MLE) technique. The model's performance was evaluated using goodness-of-fit measures, such as the log-likelihood (ℓ), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Kolmogorov–Smirnov (K–S) statistic. The probability density function (PDF), cumulative distribution function (CDF), hazard function, and survival function were derived analytically and visualized with Python to assess model behavior and data alignment. Findings: The MIWD demonstrated a better fit for the customer transaction data than its sub-models and other competing distributions. The estimated probability density function (PDF), cumulative distribution function (CDF), hazard, and survival functions highlighted the model's adaptability in representing both skewness and heavy-tailed characteristics in actual transaction data. Novelty: The main contribution of this study is the empirical validation of the four-parameter MIWD using actual customer transaction data, demonstrating its superiority over established sub-models based on various quantitative criteria. The findings provide compelling evidence that the MIWD is a robust and adaptable alternative for modelling skewed, heavy-tailed transaction and lifetime data, with practical applications in customer behavior analysis, reliability engineering, and risk modelling. Keywords: Modified Inverse Weibull Distribution, Maximum Likelihood Estimation, Customer transaction data, Parameters, Rayleigh distribution

This paper concerns with various techniques for estimations from the generalized linear exponential distribution (GLED) that can be used for modeling bathtub, increasing and decreasing hazard rate (HR) behavior and was first proposed by [3]. This distribution is important since it contains as special sub-models some widely well-known distributions such as the exponential distribution (ED), the Rayleigh distribution (RD), the linear exponential distribution (LED), and the Weibull distribution (WD). The various techniques for estimations can be considered as maximum likelihood estimation (MLE), least-square estimation (LSE), weighted least square estimation (WLSE), Cramer Von-Mises estimation (CVME), and Anderson Darling estimation (ADE). These methods of estimations are used to estimate the unknown parameters of the well-known GLED. Two applications are used to show that the GLED is a viable distribution in modeling lifetime data and to compare the varying methods of estimations based on the Kolmogorov-Simnorov test with the corresponding P-value to show the optimal method. Finally, a simulation study is presented to compare the varying methods of estimation based on the mean square error (MSE) and the average absolute bias (AAB).

Many Research and clinical studies have addressed the occurrence of failure (death). As a result of other (external) factors, which may introduce additional risks that compete with the event under study. The resulting data refers to the complete data on Competing risks, which are affected by time differences. The exact times of occurrence of these risks are known, meaning the times of failure are observed for all observations with certainty. The impact of these risks on the hazard function is estimated based on the Cox proportional hazards model, which is estimated using the partial maximum likelihood method and numerical algorithms for parameter estimation. This includes the effect of variables on the Cox hazard function and the nonparametric part, estimated by assessing the effect of time on the hazard function using the Kaplan-Meier formula and calculating competing risks through the cumulative hazard function. These methods were applied to experimental data through large-scale simulations of different sizes and parameters and for several arithmetic means and standard deviations models. Moreover, applied to real data from a sample of 80 individuals with breast cancer. Analyzing the simulation results and real data revealed that the Downhill algorithm outperforms the Newton-Raphson algorithm in terms of estimation accuracy and efficiency, based on the statistical criterion for comparison, the root mean square error. In addition, competitive risks explained the effect of common variables in increasing competitive risks beyond the hazard function of the Cox model of the Newton-Raphson estimator. While it is converging to the hazard function of the Cox model for the Downhill estimator . Paper type: Research paper

Strong‐motion attenuation relations are commonly derived from earthquake ground motion collected on triggered recorders. Parametric attenuation relations are estimated from these data using standard least squares methods. The variance of ground acceleration is relatively large and for any given earthquake, there is a distance range for which only stations with larger than average amplitudes will trigger recording. Consequently observed accelerations at these distances are higher than the mean ground acceleration and a bias may be introduced into an attenuation relation regression by non‐detection data censoring. Introduction of the log‐likelihood function as the objective function is a natural extension of current regression practices. Maximum likelihood estimation (MLE) of parametric forms fitted to data can be applied to peak acceleration data to lessen the bias introduced by the censoring of recordings by non‐triggering. In the absence of censoring information the MLE methods reduce to standard least squares estimation (LSE). An example is shown for peak acceleration data recorded from the Whittier Narrows earthquake of October 1, 1987. Roughly 33 percent of the CDMG stations located between 75 and 110 km from the epicenter did not trigger at the 1 percent g vertical acceleration level. No stations triggered beyond 115 km from the epicenter. A standard parametric form is fit to the vertical peak acceleration data using both LSE and MLE methods. It is found that inclusion of censoring data stabilizes the regression for attenuation parameters and that the MLE solution significantly differs from the LSE solution. For comparison, the LSE fit predicts a mean peak vertical acceleration of 1 percent g at 180 km while the MLE solution predicts a mean peak vertical acceleration of 1 percent g at only 115 km. By assuming a linear relationship between vertical and horizontal acceleration, a MLE fit for horizontal peak accelerations may also be determined.

This article sheds light on the mixed survival distribution, emphasizing its potential for describing the survival behavior of kidney failure patients under the stressors they encounter in their daily lives. The analysis focused on the mixed hazard function derived from the mixed survival distribution, with the parameters of the distribution and its associated functions estimated to better understand the behavior of the hazard function over time. A goodness-of-fit test was applied to evaluate how well the data fit the mixed distribution, while comparison criteria were employed to identify the most suitable distribution compared to classical models. The mixed hazard function was further compared with the Weibull hazard function. To illustrate its applicability, real data were used, representing the survival times to death for 153 patients at the Kidney Services Center in Al-Khums. The results revealed that the mixed distribution provided an excellent fit to the real data and outperformed the Weibull distribution in representing survival behavior. Moreover, the analysis showed that the hazard function increases over time, indicating a higher probability of death as the disease progresses. The hazard function curves for kidney patients at different stress levels (10%–99%) demonstrated that the probability of being at risk rises with time. At low stress levels (10%), the increase is gradual, whereas at high stress levels (95%), the risk grows substantially. This finding highlights stress as a significant factor that accelerates the rate of risk in kidney patients. Similarly, the cumulative hazard function across different survival periods showed that the accumulated probability of risk exposure increases with survival duration. The cumulative hazard function rises more rapidly at higher stress levels, while the increase is slower at lower stress levels. This indicates that the intensity of stress plays a clear role in accelerating the accumulation of risk among kidney patients.

The events of interest in any survival analysis study are regularly subject to censoring. There are various censoring schemes, including right or left censoring, and interval censoring. The most frequent censoring scheme is the right censoring, where subjects might drop out of the study or simply because not all events of interest occur before the end of the study. Moreover, for each subject, additional information referred to as covariates is registered at the beginning or throughout the study, such as age, sex, undergoing treatment, etc. The classical model to study the distribution of the events of interest, while accounting for additional information, is the Cox model. The Cox model expresses the hazard function of a subject given a set of covariates in terms of a baseline hazard, for which all covariates are zero, and an exponential function of the covariates and corresponding regression parameters. The baseline hazard can be left completely unspecified while estimating the regression parameters. Nonetheless, in practice, there are numerous studies in which the baseline hazard appears to be monotone. Time to death or to the onset of a disease are observed to have a nondecreasing baseline hazard, while the survival or recovery time after a successful medical treatment usually exhibit a nonincreasing baseline hazard. The aim of this thesis is to study the behavior of nonparametric baseline hazard and baseline density estimators in the Cox model under monotonicity constraints. The event times are assumed to be right censored and the censoring mechanism is assumed to be independent of the event of interest and non-informative. The covariates are assumed to be time-independent, usually recorded at the beginning of the study. In addition to point estimates, interval estimates of a monotone baseline hazard will be provided, based on a likelihood ratio method, along with testing at a fixed point. Furthermore, kernel smoothed estimates of a monotone baseline hazard will be defined and their behavior will be investigated. In Chapter 2, we propose several nonparametric monotone estimators of a baseline hazard or a baseline density within the Cox model. We derive the nonparametric maximum likelihood estimator of a nondecreasing baseline hazard and we consider a Grenander-type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are then shown to be strongly consistent and asymptotically equivalent. Moreover, we derive their common limit distribution at a fixed point. The two equivalent estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired similarly. Furthermore, we introduce a Grenander-type estimator of a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function derived from the Breslow estimator. This estimator is proven to be strongly consistent and its asymptotic distribution at a fixed point is derived. Chapter 3 provides an asymptotic linear representation of the Breslow estimator of the baseline cumulative hazard function in the Cox model. This representation can be used to derive the asymptotic distribution of the Grenander type estimator of a monotone baseline hazard estimator. The representation consists of an average of independent random variables and a term involving the difference between the maximum partial likelihood estimator and the underlying regression parameter. The order of the remainder term is arbitrarily close to n^-1. Chapter 4 focuses on interval estimation and on testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point, via a likelihood ratio method. Nonparametric maximum likelihood estimators under the null hypothesis are defined for both nondecreasing and nonincreasing baseline hazard functions. These characterizations, along with those of the monotone nonparametric maximum likelihood estimators provide the asymptotic distribution of the likelihood ratio test. This asymptotic distribution enables, via inversion, the construction of pointwise confidence intervals. This method of constructing confidence intervals avoids the issue of estimating the nuisance parameters, as in the case of confidence intervals based on the asymptotic distribution of the estimators. Simulations indicate that the two methods yield confidence intervals with comparable coverage probabilities. Nonetheless, the confidence intervals based on the likelihood ratio are smaller, on average. Finally, in chapter 5 we consider smooth baseline hazard estimators. The estimators are obtained by kernel smoothing the maximum likelihood and Grenander-type estimators of a monotone baseline hazard function. Three different estimators are proposed for a nondecreasing baseline hazard, which are provided by the interchange of the smoothing and isotonization step. With this respect, we define a smoothed maximum likelihood estimator (SMLE), as well as a smoothed Grenander type (SG) estimator and a Grenander type smoothed (GS) estimator. All estimators are shown to be strongly pointwise or uniformly consistent.

Within the realm of average mean squared error (AMSE), generalized estimation (GE; Kupper and Meydrech1,2) is an attractive alternative to estimate the response surface. GE is based on the criterion that minimizes the variance of the fitted response$(\hat Y({\bf x}))$by weighting the parameter estimates in a restricted parameter space provided that an upper bound can be given. Sometimes, in practice, it is quite difficult to provide such an upper bound. In this paper, we relax the restriction on parameter space and then try to present some useful mathematical properties of GE for the multi‐factor design problem. To obtain a GE‐based optimal design, the average prediction variance (APV) and average squared bias (ASB) are computed to form the AMSE, acting as the performance metric. The comparison between the GE and the least squares estimation (LSE) methods for a two‐factor design problem is made to validate GE's merit under the AMSE criterion. In this particular case, the comparison results show that the GE method, on the whole, outperforms the LSE method. The improvement of the GE method over the LSE method becomes more prominent if the potential bias error is significantly present. Moreover, the comparison report provided in the paper can also serve as a practical guide for response surface methodology (RSM) practitioners to choose an appropriate design point arrangement for the GE and LSE methods that achieve the objective of minimum AMSE. Copyright © 2007 John Wiley & Sons, Ltd.

System identification of structures is one of the important aspects of structural health monitoring. The accuracy and efficiency of identification results is affected severely by measurement noises, especially when the structure system is large, such as bridge structures, and when online system identification is required. In this paper, the least square estimation (LSE) method is used combined with the substructure approach for identifying structural parameters of a cable-stay bridge with large degree of freedoms online. Numerical analysis is carried out by first dividing the bridge structure into smaller substructures and then estimates the parameters of each substructure online using LSE method. Simulation results demonstrate that the proposed approach is capable of identifying structural parameters, however, the accuracy and efficiency of identification results depend highly on the noise sensitivities of loading region, loading pattern as well as element size.

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.

This paper examines the role of reliability theory in system analysis, focusing on system representation, quantification, and uncertainty modeling. It explores various multiscenario reliability stress-strength models based on Topp-Leone and Generalized Rayleigh distributions, covering environments with multiple stresses, strength variability, and constrained stress conditions. The study addresses modeling complexity in stress-strength relationships for multi-component, n-standby, and cascade systems. Several parameter estimation methods are evaluated, including maximum likelihood estimation, the Jackknife, and Bayesian estimators, with a Monte Carlo simulation used to compare their performance. The results indicate that maximum likelihood estimation and Jackknife methods are superior due to their low bias and Mean Squared Error across various scenarios and sample sizes, demonstrating robustness and reliability. Bayesian methods offer flexibility but require careful management of priors and data volume, while non-parametric methods tend to have higher bias and MSE, particularly in complex scenarios or with smaller sample sizes.

Half-Cauchy generalized exponential (HCGE) distribution is a novel distribution that we have proposed on in this paper. The quantiles, the measures of skewness based on quartiles, and the measures of kurtosis based on octiles, survival function, the probability density function, hazard function, cumulative distribution function, cumulative hazard function, are just a few of the crucial statistical properties we have derived for the proposed distribution. To estimate the parameters of the half-Cauchy generalized exponential distribution, the maximum likelihood estimation method has been applied. For the evaluation of the new distribution’s potential, we have considered a real dataset and compared the goodness-of-fit attained by proposed distribution with some competing distribution. The suggested model fits the data much better and is more adaptable than some other models.

Research on the Method of Orbit Determination Based on the Self-adaptive Stable Least p-norm Estimate

In this paper, a composite distribution of Kumaraswamy and Lindley distributions namely, Kumaraswamy-Lindley Kum-L distribution is introduced and studied. The Kum-L distribution generalizes sub-models for some widely known distributions. Some mathematical properties of the Kum-L such as hazard function, quantile function, moments, moment generating function and order statistics are obtained. Estimation of parameters for the Kum-L using maximum likelihood estimation and least square estimation techniques are provided. To illustrate the usefulness of the proposed distribution, simulation study and real data example are used.