Multicomponent Stress-Strength Reliability Based on Progressive First Failure for Kumaraswamy Model: Bayesian and Non-Bayesian Estimation

It is highly common in many real-life settings for systems to fail to perform in their harsh operating environments. When systems reach their lower, upper, or both extreme operating conditions, they frequently fail to perform their intended duties, which receives little attention from researchers. The purpose of this article is to derive inference for multicomponent reliability where stress-strength variables follow unit Kumaraswamy distributions based on the progressive first failure. Therefore, this article deals with the problem of estimating the stress-strength function, R when X,Y, and Z come from three independent Kumaraswamy distributions. The classical methods namely maximum likelihood for point estimation and asymptotic, boot-p and boot-t methods are also discussed for interval estimation and Bayes methods are proposed based on progressive first-failure censored data. Lindly’s approximation form and MCMC technique are used to compute the Bayes estimate of R under symmetric and asymmetric loss functions. We derive standard Bayes estimators of reliability for multicomponent stress–strength Kumaraswamy distribution based on progressive first-failure censored samples by using balanced and unbalanced loss functions. Different confidence intervals are obtained. The performance of the different proposed estimators is evaluated and compared by Monte Carlo simulations and application examples of real data.

Study of a Modified Kumaraswamy Distribution

In this article, a structural modification of the Kumaraswamy distribution yields a new two-parameter distribution defined on (0,1), called the modified Kumaraswamy distribution. It has the advantages of being (i) original in its definition, mixing logarithmic, power and ratio functions, (ii) flexible from the modeling viewpoint, with rare functional capabilities for a bounded distribution—in particular, N-shapes are observed for both the probability density and hazard rate functions—and (iii) a solid alternative to its parental Kumaraswamy distribution in the first-order stochastic sense. Some statistical features, such as the moments and quantile function, are represented in closed form. The Lambert function and incomplete beta function are involved in this regard. The distributions of order statistics are also explored. Then, emphasis is put on the practice of the modified Kumaraswamy model in the context of data fitting. The well-known maximum likelihood approach is used to estimate the parameters, and a simulation study is conducted to examine the performance of this approach. In order to demonstrate the applicability of the suggested model, two real data sets are considered. As a notable result, for the considered data sets, statistical benchmarks indicate that the new modeling strategy outperforms the Kumaraswamy model. The transmuted Kumaraswamy, beta, unit Rayleigh, Topp–Leone and power models are also outperformed.

Classical and Bayesian Inference of Conditional Stress-Strength Model under Kumaraswamy Distribution

Stress-strength models have been frequently studied in recent years. An applicable extension of these models is conditional stress-strength models. The maximum likelihood estimator of conditional stress-strength models, asymptotic distribution of this estimator, and its confidence intervals are obtained for Kumaraswamy distribution. In addition, Bayesian estimation and bootstrap method are applied to the model.

The Probability Distribution of Maximum Temperature to Assess the Suitable Statistical Models: Take the North-East and Southern Regions of Pakistan

Precise maximum temperature probability distribution information is indeed of accurately significance for numerous temperature uses. The purpose of this research to assess the appropriateness of these functions likelihood for evaluating the temperature models at different sites in southern part of Pakistan. The Kumaraswamy distribution function is used initially to approximation the models of maximum temperature. Compare the presentation of the Kumaraswamy distribution with twelve commonly used the probability functions. The consequences obtained show that the more effective functions are not similar across all sites. The maximum temperature features, quality and quantity of the noted temperature observation can be regarded as a factors that affect the presentation of the function. Similarly, the skewness of the noted maximum temperature observations may affect the precision of Kumaraswamy distribution. For the Hyderabad, Lahore and Sialkot sites, the Kumaraswamy distribution obtainable the topmost presentation, however for the Karachi, Multan stations, the generalized extreme value (GEV) distributions provided the best fit, respectively. According to the calculations, the Kumaraswamy distribution usually be regarded as a valid distribution because it runs 3 best fit sites and ranks 2 to 3 among the remaining sites. Though, the tight presentation of the Kumaraswamy and GEV and the flexibility of the Weibull distribution which has been usually verified, more evaluations of the presentation of the Kumaraswamy distribution are needed.

Bayesian Inference for the Parameters of Kumaraswamy Distribution via Ranked Set Sampling

In this paper, we address the estimation of the parameters for a two-parameter Kumaraswamy distribution by using the maximum likelihood and Bayesian methods based on simple random sampling, ranked set sampling, and maximum ranked set sampling with unequal samples. The Bayes loss functions used are symmetric and asymmetric. The Metropolis-Hastings-within-Gibbs algorithm was employed to calculate the Bayes point estimates and credible intervals. We illustrate a simulation experiment to compare the implications of the proposed point estimators in sense of bias, estimated risk, and relative efficiency as well as evaluate the interval estimators in terms of average confidence interval length and coverage percentage. Finally, a real-life example and remarks are presented.

Tail Conditional Expectations Based on Kumaraswamy Dispersion Models

Recently, there seems to be an increasing amount of interest in the use of the tail conditional expectation (TCE) as a useful measure of risk associated with a production process, for example, in the measurement of risk associated with stock returns corresponding to the manufacturing industry, such as the production of electric bulbs, investment in housing development, and financial institutions offering loans to small-scale industries. Companies typically face three types of risk (and associated losses from each of these sources): strategic (S); operational (O); and financial (F) (insurance companies additionally face insurance risks) and they come from multiple sources. For asymmetric and bounded losses (properly adjusted as necessary) that are continuous in nature, we conjecture that risk assessment measures via univariate/bivariate Kumaraswamy distribution will be efficient in the sense that the resulting TCE based on bivariate Kumaraswamy type copulas do not depend on the marginals. In fact, almost all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copula’s domain of definition. In this article, we examined the above risk measure in the case of a univariate and bivariate Kumaraswamy (KW) portfolio risk, and computed TCE based on bivariate KW type copulas. For illustrative purposes, a well-known Stock indices data set was re-analyzed by computing TCE for the bivariate KW type copulas to determine which pairs produce minimum risk in a two-component risk scenario.

Multivariate probabilistic rock physics models using Kumaraswamy distributions

Rock physics models are physical equations that map petrophysical properties into geophysical variables, such as elastic properties and density. These equations are generally used in quantitative log and seismic interpretation to estimate the properties of interest from measured well logs and seismic data. Such models are generally calibrated using core samples and well log data and result in accurate predictions of the unknown properties. Because the input data are often affected by measurement errors, the model predictions are often uncertain. Instead of applying rock physics models to deterministic measurements, I propose to apply the models to the probability density function of the measurements. This approach has been previously adopted in literature using Gaussian distributions, but for petrophysical properties of porous rocks, such as volumetric fractions of solid and fluid components, the standard probabilistic formulation based on Gaussian assumptions is not applicable due to the bounded nature of the properties, the multimodality, and the non-symmetric behavior. The proposed approach is based on the Kumaraswamy probability density function for continuous random variables, which allows modeling double bounded non-symmetric distributions and is analytically tractable, unlike the Beta or Dirichtlet distributions. I present a probabilistic rock physics model applied to double bounded continuous random variables distributed according to a Kumaraswamy distribution and derive the analytical solution of the posterior distribution of the rock physics model predictions. The method is illustrated for three rock physics models: Raymer’s equation, Dvorkin’s stiff sand model, and Kuster-Toksoz inclusion model.

Greenness as a Differentiating Strategy

In a vertical differentiation model, we study a market where consumers, depending on their level of environmental consciousness, value the greenness of the product they consume and are distributed according to a Kumaraswamy distribution. Three scenarios are studied: only one firm takes some green measures and firms compete upon prices; only one firm takes some green measures, and this firm acts as the leader of the price competition; and finally, both firms choose their level of greenness and compete upon their location and price. The results suggest that as consumers become more environmentally conscious, the marginal consumer and the greener firm’s location move to the right. In contrast, the less green firm’s response is non-monotonic. In fact, when the two firms choose their location along with their prices, the latter firm chooses to produce a less green product in response to more environmentally conscious consumers. In the extreme case where all consumers are fully environmentally conscious, the latter firm produces a brown product and sells it at a price equal to its marginal cost. In this case, the greener firm’s price and location choices make the consumers indifferent between the two products. These results could explain why despite all the improvements in the consumers’ environmental consciousness, brown (in its general term) products are still widely produced and consumed, even by environmentally conscious consumers.

Bayesian Estimation and Prediction Based on Constant Stress-Partially Accelerated Life Testing for Topp Leone-Inverted Kumaraswamy Distribution

Accelerated life testing or partially accelerated life tests is very important in life testing experiments because it saves time and cost. Partially accelerated life tests are used when the data obtained from accelerated life tests cannot be extrapolated to usual conditions. This paper proposes, constant–stress partially accelerated life test using Type II censored samples, assuming that the lifetime of items under usual condition have the Topp Leone-inverted Kumaraswamy distribution. The Bayes estimators for the parameters, acceleration factor, reliability and hazard rate function are obtained. Bayes estimators based on informative priors is derived under the balanced square error loss function as a symmetric loss function and balanced linear exponential loss function as an asymmetric loss function. Also, Bayesian prediction (point and bounds) is considered for a future observation based on Type-II censored under two samples prediction. Numerical studies are given and some interesting comparisons are presented to illustrate the theoretical results. Moreover, the results are applied to real data sets.