Statistical inference for the generalized Rayleigh distribution based on upper record values

Surles and Padgett recently considered two-parameter Burr Type X distribution by introducing a scale parameter and called it the generalized Rayleigh distribution. It is observed that the generalized Rayleigh and log-normal distributions have many common properties and both distributions can be used quite effectively to analyze skewed data set. In this paper, we mainly compare the Fisher information matrices of the two distributions for complete and censored observations. Although, both distributions may provide similar data fit and are quite similar in nature in many aspects, the corresponding Fisher information matrices can be quite different. We compute the total information measures of the two distributions for different parameter ranges and also compare the loss of information due to censoring. Real data analysis has been performed for illustrative purposes.

ABSTRACTThe generalized Rayleigh distribution was introduced and studied quite effectively in the literature. The closeness and separation between the distributions are extremely important for analyzing any lifetime data. In this spirit, both the generalized Rayleigh and Weibull distributions can be used for analyzing skewed datasets. In this article, we compare these two distributions based on the Fisher information measures and use it for discrimination purposes. It is evident that the Fisher information measures play an important role in separating between the distributions. The total information measures and the variances of the different percentile estimators are computed and presented. A real life dataset is analyzed for illustration purposes and a numerical comparison study is performed to assess our procedures in separating between these two distributions.

During the manufacturing process of a product, the manufacturer is very careful from the initial raw material to the final stage of the product. He always tries to enhance the quality level of the product. But if even the product is manufactured using the same machines, in the same laboratory, using the same manpower and environment and at the same temperature level, there may be still variation in the lifetime/failure time of the product. These failure times or lifetimes of the product are always fitted to some statistical distributions. Failure and repair times of electronic components are random and unpredictable in nature. But, the failure time of an electronic component can be modeled using statistical distributions and they are used in acceptance sampling. Distribution theory plays an important role in life testing of products. Many lifetime distributions have been widely used in the areas of quality control, reliability analysis and acceptance sampling plans which include such distributions as Birnbaum–Saunders distribution, Burr type XII distribution, generalized Birnbaum–Saunders distribution, generalized exponential distribution, generalized Rayleigh distribution, inverse Rayleigh distribution, inverse Gaussian distribution, log-logistic distribution and lognormal distribution. In this chapter, we will focus on these distributions to develop the acceptance sampling plans based on truncated life tests.

The generalized Rayleigh (GR) distribution [V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, I, Appl. Math. 21 (1976), pp. 395–412; V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, II, Appl. Math. 21 (1976), pp. 413–419] has been applied in several areas such as health, agriculture, biology and other sciences. For the first time, we propose the Kumaraswamy GR (KwGR) distribution for analysing lifetime data. The new density function can be expressed as a mixture of GR density functions. Explicit formulae are derived for some of its statistical quantities. The density function of the order statistics can be expressed as a mixture of GR density functions. We also propose a linear log-KwGR regression model for analysing data with real support to extend some known regression models. The estimation of parameters is approached by maximum likelihood. The importance of the new models is illustrated in two real data sets.

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.

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.

This paper interest to estimation the unknown parameters for generalized Rayleigh distribution model based on censored samples of singly type one . In this paper the probability density function for generalized Rayleigh is defined with its properties . The maximum likelihood estimator method is used to derive the point estimation for all unknown parameters based on iterative method , as Newton – Raphson method , then derive confidence interval estimation which based on Fisher information matrix . Finally , testing whether the current model ( GRD ) fits to a set of real data , then compute the survival function and hazard function for this real data.

This paper explores the likelihood and Bayesian estimation of the stress–strength reliability parameter (R) based on a lower record ranked set sampling scheme from the generalized Rayleigh distribution. Maximum likelihood and Bayesian estimators as well as confidence intervals of R are derived and their properties are studied. Furthermore, two parametric bootstrap confidence intervals are introduced in the paper. A comparative simulation study is conducted to assess the effectiveness of these four confidence interval methodologies in estimating R. The application of the methods is demonstrated using real data on fiber strength to showcase their practicability and relevance in the industry.

ABSTRACTMotivated by a central banker with an inflation target, we show that the optimal forecast bias under non‐quadratic loss functions and non‐normal forecast errors can decrease or initially increase and then decrease with the forecast horizon. We initially proof that, if the variable to forecast can be described by a generalized Rayleigh distribution, its conditional mean does in general not constitute the optimal prediction under a symmetric target zone loss function. Subsequently, we approximate the target zone loss function to show the potential for variation in optimal bias over the forecast horizon.

In this chapter, we establish limit theorems for some statistics based on order statistics and record values. The finite-sample as well as asymptotic properties of statistics based on invariant confidence intervals are investigated and their use in statistical inference is outlined.

In this paper, problems of estimation and prediction for record values are discussed for a two-parameter unit generalized Rayleigh distribution. Maximum likelihood estimators of model parameters along with their existence and uniqueness are established, and corresponding approximate confidence intervals are constructed based on asymptotic theory. In addition, alternative generalized inferential approach is provided for comparison based on the proposed pivotal quantities. Further, prediction intervals of remaining useful life are also presented under classical and generalized inferential methods when the unit generalized Rayleigh record values are available. Extensive numerical studies and two real-life examples are conducted to show the performance of the proposed methods. The results indicate that both classical and generalized estimation methods work satisfactorily and the proposed generalized inferential approach performs superior to traditional estimates in general.

In the literature, there are well-developed estimation techniques that can be used to measure the reliability in multicomponent stress-strength models when the information about all experimental units are available. However, in real applications, only observations that exceed (or fall below) the current value may be recorded. In this article, assuming that the components of the system follow generalized Rayleigh distribution, we investigate classical and Bayesian estimation of the reliability of a multicomponent stress-strength model when the available data are reported in terms of record values. The maximum likelihood estimate of the reliability parameter and its asymptotic confidence interval are obtained. Moreover, two confidence intervals based on the parametric bootstrap methods are presented. The Bayes estimate of the reliability parameter is derived using different loss functions. Because there is no closed form for the Bayes estimate, we use the Markov chain Monte Carlo method to obtain an approximate Bayes estimate and the highest posterior density credible interval of the reliability. To evaluate the performances of different procedures, simulation studies are conducted, and an example of real data sets is provided.

This paper introduces a new non-homogeneous stochastic diffusion process related the generalized Rayleigh distribution called the stochastic generalized Rayleigh diffusion process, thanks to the idea of extending the Rayleigh distribution families by introducing a positive shape parameter. The explicit forms of some important probabilistic characteristics of the process such as the analytical form of the process, the probability density function, and the conditional and non-conditional mean functions are obtained analytically. Then, we provide a computational statistical approach to estimate the process parameters via the maximum likelihood method through the discrete sampling. In order to obtain the estimators of the parameters, we find a nonlinear equation. To achieve this, we propose the simulated annealing algorithm to maximize the log-likelihood function once the parameter space has been bounded by a proposed stepwise procedure. We will end up by exploring the possibility of using this new GSRDP to fit and predict simulated data.

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.

An observation that shows greater than all the preceding observations, is called record. The characterization results via conditional expectation based on record values are expressed for family of Rayleigh distribution. Moreover, entropies based on distribution function are discussed and enumerated.