Distribution Of Order Statistics Research Articles

Stochastic Model for Novel Mixture Distribution with Properties and Its Application

Medical research is the one of the most important applications of statistical analysis. A great deal of occasions, specific statistical considerations are needed for cancer research in order to determine the model that best fits the survival data. In this paper, the proposed a new two parameter continuous distribution. It is called a new mixture distribution (NMD). Some statistical properties of the distribution are derived such as, the moments, moments generating function, reliability analysis. Also, the distribution of order statistics is presented and entropies are derived. The application of the maximum likelihood estimate technique to the performance of traditional parameter estimation. Two authentic data sets describing cancer patients' survival are used to empirically demonstrate the potential importance and usability of the proposed distribution. Comparing the new mixture distribution to some other competing distribution, the analysis's results demonstrated that it performed quite well.

A Generalized Class of Exponentiated Modified Weibull Distribution With Applications

Abstract: In this paper, a new class of five parameter gamma-exponentiated or generalized modified Weibull (GEMW) distribution which includes exponential, Rayleigh, Weibull, modified Weibull, exponentiated Weibull, exponentiated exponential, exponentiated modified Weibull, exponentiated modified exponential, gamma-exponentiated exponential, gamma exponentiated Rayleigh, gamma-modified Weibull, gamma-modified exponential, gamma-Weibull, gamma-Rayleigh and gamma-exponential distributions as special cases is proposed and studied. Mathematical properties of this new class of distributions including moments, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics and Renyi entropy are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to real data sets presented in order to illustrate the usefulness of this new class of distributions and its sub-models.

Multi-layer statistical analysis and surrogate model based fatigue crack growth reliability assessment of lifting lug structure under random load history

The reliability of fatigue crack growth (FCG) in lifting lug structures is a critical component of overall aircraft operational reliability. Assessing the FCG reliability of these structures poses a classic time-dependent challenge, necessitating the consideration of dynamic changes in load distribution, strength distribution, and failure correlations. To address these complexities, a novel FCG reliability assessment method for lifting lug structures has been developed, drawing upon multi-layer statistical analysis theory, artificial neural networks (ANN), and the concept of the minimum order statistic distribution of equivalent strength (Sequ). The introduced variable Sequ enables a direct comparison with applied loads to ascertain structural integrity. Findings suggest that the rate of FCG reliability change is governed by the load and Sequ distributions at any given moment. The predicted Sequ distribution from the new method closely aligns with experimental data, and it offers a more thorough characterization of random load histories. The reliability assessment outcomes of the new method are deemed more rational compared to those of traditional approaches.

Development of a new exponential generalized family of distribution with its properties and application

In the field of reliability analysis, the selection of an appropriate lifespan model is critical. With a multitude of lifetime distributions accessible, the hunt for a more suited distribution remains essential. In this paper, we offer a unique class of distributions derived from the notion of exponential generalization, improved with changes to boost flexibility. Our suggested distribution incorporates multiple hazard rate profiles, giving enhanced flexibility. Analytical characteristics including the rth moment, moment generating function, quantile function, distribution of order statistics, and Rényi entropy are obtained. Employing maximum likelihood estimation, we estimate the unknown parameters. Through simulation tests and analysis of real-world datasets, we exhibit the model's usefulness compared to five existing lifespan distributions, emphasising the better performance of the GAYUF distribution. This research underlines the GAYUF distribution as a better model in the field of lifespan analysis

Statistical analysis of the inverse power Zeghdoudi model: Estimation, simulation and modeling to engineering and environmental data

Abstract In this research, we investigate a brand-new two-parameter distribution as a modification of the power Zeghdoudi distribution (PZD). Using the inverse transformation technique on the PZD, the produced distribution is called the inverted PZD (IPZD). Its usefulness in producing symmetric and asymmetric probability density functions makes it the perfect choice for lifetime phenomenon modeling. It is also appropriate for a range of real data since the relevant hazard rate function has one of the following shapes: increasing, decreasing, reverse j-shape or upside-down shape. Mode, quantiles, moments, geometric mean, inverse moments, incomplete moments, distribution of order statistics, Lorenz, Bonferroni, and Zenga curves are a few of the significant characteristics and aspects explored in our study along with some graphical representations. Twelve effective estimating techniques are used to determine the distribution parameters of the IPZD. These include the Kolmogorov, least squares (LS), a maximum product of spacing, Anderson-Darling (AD), maximum likelihood, minimum absolute spacing distance, right-tail AD, minimum absolute spacing-log distance, weighted LS, left-tailed AD, Cramér-von Mises, AD left-tail second-order. A Monte Carlo simulation is used to examine the effectiveness of the obtained estimates. The visual representation and numerical results show that the maximum likelihood estimation strategy regularly beats the other methods in terms of accuracy when estimating the relevant parameters. The usefulness of the recommended distribution for modelling data is illustrated and displayed visually using two real data sets through comparisons with other distributions.

A STOCHASTIC MODEL FOR LENGTH BIASED LOAI DISTRIBUTION WITH PROPERTIES AND ITS APPLICATIONS

One of the most important applications of statistical analysis is in health research and application. Cancer studies are mostly required special statistical considerations in order to find the appropriate model for fitting the survival data. In this paper, we examine some general models leading to a weighted distribution. The developed distribution, also known as the length biased model, is referred to as the length biased Loai distribution. This distribution is a specific type of basic distribution, the Loai distribution. The length biased distribution has been compared with the original distribution. Some statistical properties of this distribution are derived, such as moments, the moment generating function, the reliability analysis, and the included functions. Also, the distribution of order statistics, quantile function, and likelihood ratio test are presented. The Bonferroni and Lorenz carvers, as well as the Rnyi and Tsallis entropies, are derived. The method of maximum likelihood estimation is used to estimate the distribution parameters. A simulation study is performed to investigate the performance of the estimation. The real data applications demonstrate that the proposed distribution can provide better results than several well-known distributions.

Topp-Leone Exponentiated Pareto Distribution: Properties and Application to Covid-19 Data

This paper proposes a new Topp-Leone Exponentiated Pareto (TLEtP) distribution. The new distribution family is derived by expanding the Topp Leone-G family of distributions with additional positive shape parameters. The corresponding density and distribution functions are derived and shown. Some of the derived mathematical properties of the distribution include quantile function, ordinary and incomplete moments generating function (mgf), hazard function, survival function, odd function, probability weighted moment, and distribution of order statistic. The parameters of the distribution are estimated using Maximum Likelihood method. The proposed distribution’s validity is demonstrated by fitting two sets of real data and comparing the results with two existing same-family distributions, the Topp-Leone Pareto type I(TLPI) and Pareto (P), with the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC), respectively. The comparison of the proposed Topp-Leone Exponentiated Pareto (TLEtP) to the Topp-Leone Pareto type I(TLPI) and Pareto (P) distribution demonstrate that the TLEtP distribution offers a better fit for the data sets than the other two distributions.

Estimation and comparison of Weibull-Normal distribution with some other probability models using Bayesian method of estimation

In statistical applications the Normal distribution is adjudged to be the best. Recent studies Terna (2017) using classical method indicated that Weibull-Normal distribution outperformed the Normal distribution. In this study we used the non-classical Bayesian method of estimation to estimate and compare the Weibull-Normal distribution with some other distributions including Normal and Gamma-Normal distributions. This study derived explicit expressions for basic statistical properties such as moments, moment generating function, the characteristic function, reliability analysis and the distribution of order statistics. It looks at estimation of confidence intervals for the parameters of the Weibull-Normal distribution and estimated the parameters of the new distribution using a non-classical approach for the purpose of theoretical comparisons. The two other distributions whose parameters were also estimated by using Bayesian estimation are the normal distribution and gamma distribution as well as the combination Gamma-Normal distribution. Based on the analyses and interpretations of the results obtained it was found that the parameters and other general properties of Normal distribution gives a better fit than other distributions. R-software was used; the models were written as an R code in R program using the rjags library, the distribution parameters were obtained from a Gibbs sampling of a Bayesian Fit for data set I and data set II. The criteria used in R for comparisons were the negative log-likelihood, AIC (Akaike information criterion), CAIC (Consistent Akaike Information Criterion) and BIC (Bayesian information Criterion).

A new bathtub and increasing failure rate model: An extension of the Mustapha type II distribution

This article introduces a new three-parameter lifetime model with an increasing and bathtub failure rate functions as an extension of the Mustapha type II distribution (MuII). The model can be very useful in statistical studies, reliability, computer sciences and engineering. Various mathematical and statistical properties of the distribution are discussed, such as moments, mean deviations, Bonferroni and Lorenz curves, entropy, order statistic, and extreme value distributions. Moreover, we consider the bivariate extension of the new model. Statistical inferences by the maximum likelihood method are discussed and assess by simulation studies. Applications of the proposed model to two right-skewed data are presented for illustration. The new model provides a better fit than some other existing distribution as measured by some model selection criteria and goodness of fits statistics.

A New Cubic Transmuted Inverse Weibull Distribution: Theory and Applications

This paper introduces a new cubic transmutation of the inverse Weibull distribution, known as a cubic transmuted inverse Weibull distribution. The model is thought to be useful for the analysis of complex life data, modeling failure times, accessing product reliability, and many other fields like economics, hydrology, biology, and engineering. Some statistical features of the proposed distribution are explored. These include moments, generating functions, quantile functions, reliability functions, and hazard rate functions. The distribution of order statistics for the proposed cubic transmuted inverse Weibull distribution is also studied. The maximum likelihood estimation approach is used to estimate the model parameters. The effectiveness of the estimation is investigated through extensive simulation study. The suitability of the proposed distribution has been studied by using five real-life datasets. It is found that the proposed distribution is the most suitable fit for the used data sets.

A New Neutrosophic Extended Rayliegh Distribution for Enhanced Productivity and Efficiency Across Industrial Sectors: A case study of Al-Kharj region

This paper introduces a new statistical distribution called the Neutrosophic Extended Rayleigh Distribution (NERD), which is specifically developed to handle uncertainty commonly found in industrial applications. We conduct a comprehensive examination of the statistical characteristics of NERD, including important measures such as the quantile function, moments, moment generating function, mean deviation, skewness, kurtosis, reliability measures, uncertainty measures, distributions of order statistics, and L-moments. Parameter estimation is conducted by maximum-likelihood estimation within a neutrosophic framework, guaranteeing resilient inference in practical situations. Through the application of NERD to actual industrial datasets, we evaluate its adaptability and efficiency in simulating industrial processes. A real case study of Al-Kharj region demonstrates the higher performance of NERD. This research highlights the capacity of NERD to greatly improve productivity and efficiency in several industrial sectors.

On Power Chris-Jerry Distribution: Properties and Parameter Estimation Methods

In this study, we introduce the "Power Chris-Jerry" distribution, conducting a comprehensive analysis of its fundamental mathematical characteristics and an extensive exploration of various crucial aspects. These encompass investigations into its mode, quantile function, moments, coefficient of skewness, kurtosis, moment generating function, stochastic ordering, distribution of order statistics, reliability analysis, and mean past lifetime. Furthermore, we provide an in-depth assessment of four distinct parameter estimation methodologies: maximum likelihood estimation (MLE), Least Squares (LS), maximum product spacing method (MPS), and the Method of Cram`er-von-Mises (CVM). Our investigation uncovers a consistent pattern wherein the MLE, LS, and CVM approaches consistently yield underestimated parameter values. Intriguingly, we observe a consistent trend of decreasing Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and BIAS across all estimation techniques as sample sizes increase. Remarkably, our simulation results consistently favor the Maximum Product Spacing (MPS) method, highlighting its superiority in generating estimates with smaller MSE values across a broad spectrum of parameter values and sample sizes. These findings emphasize the robustness and dependability of the MPS estimator, offering valuable insights and practical guidance for both practitioners and researchers engaged in probability distribution modeling.

Statistical Analysis of Covid-19 Data using the Odd Log Logistic Kumaraswamy Distribution

This paper presents a statistical analysis of Covid-19 data using the Odd log logistic kumaraswamy Kumaraswamy (OLLK) distribution. Some mathematical properties of the proposed OLLK distribution such as the survival and hazard functions, quantile function, ordinary and incomplete moments, moment generating function, probability weighted moment, distribution of order statistic and Renyi entropy were derived. Five estimators are examined for unknown model parameters. The performance of the estimators is compared using an extensive simulation study based on the bias and mean square error criteria. Two Covid-19 data sets representing the percentage of daily recoveries of Covid-19 patients are used to illustrate the applicability of the proposed OLLK distribution. Results revealed that the OLLK distribution is a better alternative to some existing models with bounded support.

Statistical analysis of Saudi Arabia and UK Covid-19 data using a new generalized distribution

In this article, we extended the Topp–Leone distribution to create the Odd-Topp–Leone–Gompertz-G distribution, a new family of distributions. We were motivated by the limitations of classical models in modeling real life datasets hence, the creation of a new flexible distribution. Expansion of density, distribution of order statistics, Rényi entropy, moments, quantile, generating functions, residual life function and reverse residual life function, and reliability model are some of its structural aspects that are obtained. We looked at some special cases of the new distribution and concluded that our new model can handle both highly skewed data and non-monotonic hazard rate functions. The maximum likelihood estimation technique was used to estimate the model parameters. A simulation exercise was carried out to evaluate the performance of the proposed family of distribution. The article used the Saudi Arabia and United Kingdom dataset to demonstrate the utility of the new model and it was found that the new model performs better in modeling these datasets as compared to non-nested models presented in this article.

An Improved XShanker Distribution with Applications to Rainfall and Vinyl Chloride Data

This article is an improvement on the XShanker distribution having a single parameter which is the scale parameter and also in the class of Lindley distributions. It is named Double XShanker following the approach that generated it. The distributional properties which include the non-central moment with the associated statistics, the moment generating function, characteristic function, mean residual life function, stress-strength reliability function, Bonferroni and Lorenz curve functions, odd function, stochastic ordering, distribution of order statistics, and Reny entropy. The parameter was estimated using the method of maximum likelihood. Some visualizations based on theoretical values were presented. Some statistics were computed from theoretical values with convergence behavior recorded. A simulation study was conducted with 1000 samples of different sizes and the behavior observed is that as sample size increases the estimates decrease indicating precision. Data on rainfall and Vinyl chloride were used to validate the usefulness of the suggested distribution.

A New Logistic Generalization Arising from Distributions of Order Statistics: Properties and Applications

Distributions with the variable support x∈R that exhibit strict symmetricity are versed in literature; and serve as model-fit for various forms of bell shaped outcomes; where normal and logistic distributions are renowned examples. This strictness, however, limits the application of these probability models to a particular kind of data; hence, its minimal utility. In this paper, therefore, a new generalization for the logistic distribution termed the Jones generalized logistic distribution is proposed. This new distribution is conditionally symmetric; which entails that the distribution attains symmetricity, only at equal parameter combinations. By implication, the proposed distribution serves the dual purpose of modeling both symmetric and asymmetric outcomes. Some properties of the proposed model have been derived. Finally, JGLD were fit to two different lifetime data as a demonstration to its relevance.

A New Modification of Shanker Distribution with Applications to Increasing Failure Rate Data

In this paper, a new distribution is proposed by mixing the exponential distribution and the Shanker distribution with a mixing proportion being the same as those that yielded the Shanker distribution. The proposed distribution is referred to as the XShanker distribution. The distributional properties of the XShanker distribution namely, quantile function, moments and their associated measures,the mode, moment generating function, characteristic function, distribution of order statistics, and entropy are derived and studied. The reliability analysis shows that the failure rate is a strictly increasing function. The parameter of the model was estimated using the maximum likelihood function. We illustrated the usefulness of XShanker distribution using data on waiting times of 100 bank customers and vinyl chloride from clean ungradient ground-water monitoring wells in (g/L).

Distribution of number of observations required to obtain a cover for the support of a uniform distribution

For a given positive number ‘δ′, we consider a sequence of δ − neighborhoods of the independent and identically distributed (i.i.d.) random variables, from a U ( 0 , 1 ) distribution, and “stop as soon as their union contains the interval ( 0 , 1 ) . ” We call such a union “a cover.” To find the distributions of N ( δ ) , the stopping time random variable, we need the joint distribution of order statistics from a U ( 0 , 1 ) distribution. For each δ > 0 and n = 1 , 2 , … , we obtain a general expression for P ( N ( δ ) ≤ n ) , and for a fixed value of δ , it is the distribution function of N ( δ ) . For a given n, let Δ ( n ) be the minimum value of δ , so that the union of the n δ − neighborhoods of the first n observations contains the interval ( 0 , 1 ) . Because N ( δ ) ≤ n if and only if Δ ( n ) ≤ δ , the distributions of Δ ( n ) can be obtained by fixing n in the general expression for P ( N ( δ ) ≤ n ) . To describe the impact of δ on the distribution of N ( δ ) and that of n on Δ ( n ) , we sketch the graphs of distribution functions and the empirical distribution functions.

The Inverse Burr-Generalized Family of Distributions: Theory and Applications

This paper presents a new class of generalized distributions based on the inverse Burr (Burr III) distribution. Statistical properties of the proposed family of distributions such as the density and cumulative distribution functions, survival and hazard rate functions, quantile, moments, moment generating function, probability weighted moments, Renyi entropy and distribution of order statistics are derived. The maximum likelihood estimation method is employed to obtain the parameter estimates of the family of distributions. A Monte Carlo simulation study is conducted in order to investigate the asymptotic behaviour of the parameter estimates of a sub-model from the proposed family of distributions. Finally, the utility of proposed family of distributions in lifetime data fittings is illustrated using two real data sets and the results obtained were compared to some existing non-nested models. Based on some model selection criteria and goodness of fit test statistics, it was evident that the sub-model from the proposed family of distributions performed reasonably better than the competitor distributions in fitting the two data sets.

The Type II Exponentiated Half Logistic-Marshall-Olkin-G Family of Distributions with Applications

A new generalized family of distributions called the type II exponentiated half logistic-Marshall-Olkin-G distribution is developed. Some special cases of the new model are presented. We explore some statistical properties of the new family of distributions. The statistical properties studied include expansion of the density function, hazard rate and quantile functions, moments, moment generating functions, probability weighted moments, stochastic ordering, distribution of order statistics and Rényi entropy. The maximum likelihood, ordinary and weighted least-squares techniques for the estimation of model parameters are presented, and Monte Carlo simulations for the new family of distributions are conducted. The importance of the new family of distributions is examined by means of applications to two real data sets.