Generalized blending type Bernstein operators based on the shape parameter λ

The Weibull distribution, frequently employed to assign probabilities to the lifetimes of components and systems operating under stress, is habitually characterized by a pair of positive parameters, termed the scale and shape parameters. Two fundamental reparameterizations of the Weibull probability density function are proposed. The first reparameterization replaces the shape parameter by its inverse, the resulting positive parameter thereafter termed the shaping parameter. This permits a more facile exposition of the properties of parameter estimates, derived in the event that a complete random sample from the Weibull distribution is available. The characteristics of these parameter estimation techniques are then reviewed and compared, and their variances and distributional properties are delineated whenever possible. A second reparameterization extends the parameter space so as to include nonpositive values of the shape parameter. This extension augments the utility and applicability of the Weibull distribution without requiring radical alteration of the standard parameter estimation procedures applicable to the original parameter space.

In this paper, we have construct a new version of the Bézier‐type Baskakov–Schurer–Szász–Stancu operators. For this new class of operators, uniform convergence is shown in any compact subset or positive real line. We prove Korovkin‐type theorem, Voronovskaya‐type theorem, and Grüss–Voronovskaya‐type theorem. Moreover, at the end, we express the behavior of the operators in the Lipschitz‐type space using the modulus of continuity and smoothness.

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

Cubic extension Bezier curves (CE-Bezier curves) are constructed by a natural extension of the traditional cubic Bezier curves. While retaining noticeable properties of cubic Bezier curves, CE-Bezier curves have the advantages such as adjustable shape parameters and better approximation property. In this paper, terminal behaviors of CE-Bezier curves, which are closely related to shape parameters, are carefully examinated and geometric interpretations of shape parameters are explicitly presented. Subsequently, the necessary and sufficient condition ensuring G1 and G2 continuity requirements is respectively established for the first category CE-Bezier curves. Parallel to CE-Bezier curves, Geometric condition ensuring G1 continuity requirement between CE-Bezier surfaces are rigorously investigated, and relevant examples of geometric modeling are provided. Theoretical results and concrete examples have validated that, with appropriately selected control parameters, mosaic technique of CE-Bezier curves and surfaces can be widely applied to the representation and modeling of engineering curves and surfaces.

A new type Bézier bases with λ shape parameters have been defined <cite>ye</cite>. We slightly modify these bases to establish new Bézier bases with Schurer polynomials and λ shape parameters. We construct a new type Schurer operators via defined new Bézier-Schurer bases. Also, we study statistical convergence properties of these operators and obtain an estimate for the rate of weighted A-statistical convergence. Moreover, we prove two Voronovskaja-type theorems including a Voronovskaja-type approximation theorem using weighted A-statistical convergence.

A generalised approach is presented to derive coupled mixed mode cohesive laws described with physical parameters such as peak traction, critical opening, fracture energy and cohesive shape. The approach is based on deriving mix mode fracture resistance curves from an effective mix mode cohesive law at different mode mixities. From the fracture resistance curves, the normal and shear stresses of the cohesive laws can be obtained by differentiation. Since, the mixed mode cohesive laws are obtained from a fracture resistance curve (potential function), path independence is automatically satisfied. The effective mix mode cohesive law can have different shape and cohesive law parameters at different mode mixities so that the approach can be applied to various material failure models.

The reliability assessment for an automobile crankshaft provides an important understanding in dealing with the design life of the component in order to eliminate or reduce the likelihood of failure and safety risks. The failures of the crankshafts are considered as a catastrophic failure that leads towards a severe failure of the engine block and its other connecting subcomponents. The reliability of an automotive crankshaft under mixed mode loading using the Markov Chain Model is studied. The Markov Chain is modelled by using a two-state condition to represent the bending and torsion loads that would occur on the crankshaft. The automotive crankshaft represents a good case study of a component under mixed mode loading due to the rotating bending and torsion stresses. An estimation of the Weibull shape parameter is used to obtain the probability density function, cumulative distribution function, hazard and reliability rate functions, the bathtub curve and the mean time to failure. The various properties of the shape parameter is used to model the failure characteristic through the bathtub curve is shown. Likewise, an understanding of the patterns posed by the hazard rate onto the component can be used to improve the design and increase the life cycle based on the reliability and dependability of the component. The proposed reliability assessment provides an accurate, efficient, fast and cost effective reliability analysis in contrast to costly and lengthy experimental techniques.

Rate of approximation of blending type modified univariate and bivariate λ-Schurer-Kantorovich operators

We proposed a new family of distributions called the Topp Leone exponentiated-G family of distributions with two extra positive shape parameters, which generalizes and also extends the Topp Leone-G family of distributions. We derived some mathematical properties of the proposed family including explicit expressions for the quantile function, ordinary and incomplete moments, generating function and reliability. Some sub-models in the new family were discussed. The method of maximum likelihood was used to estimate the parameters of the sub-model. Further, the potentiality of the family was illustrated by fitting two real data sets to the mentioned sub-models.

This study proposes and investigates a New Modified Extended Generalised Inverse Exponential (NMEGIEx) distribution, a novel distribution. The basic one-parameter inverse exponential distribution is extended in the new model. Four additional positive shape parameters are added to an extended Topp-Leone exponentiated a generalised family of distributions to create the new model, which simultaneously controls the centre and tail weights. The model’s asymptotic behaviour, explicit formulations for ordinary moments, mean, quantile function, hazard function, survival function, median, Moment Generating Function (MGF), and Probability Density Function (PDF) of lowest and highest order statistics were among the many statistical properties derived. Monte Carlo simulation is used to test the estimators of the proposed distribution. As predicted, as the sample size increases, the Root Mean Square Error (RMSE) and biases approach zero, and the estimated parameter values approach the true values of the parameters. Maximum Likelihood Estimation (MLE) is used to determine the values of the unknown parameters that make the observed data most likely under the assumed model. The superiority of the proposed NMEGIEx distribution is demonstrated through application to two real-world quality control engineering datasets, and it is clear that the proposed model fits the datasets better than competing distributions.

This paper provides Monte Carlo results for the performance of the method of moments (MM), maximum likelihood (ML) and ordinary least squares (OLS) estimators of the credit loss distribution implied by the Merton (1974) and Vasicek (1987, 2002) framework when the common or idiosyncratic asset-return factor is non-Gaussian and, thus, the true credit loss distribution deviates from the theoretical one. We find that OLS and ML outperform MM in small samples when the true data-generating process comprises a non-Gaussian common factor. This result intensifies as the sample size increases and holds in all cases. We also find that all three estimators present a large bias and variance when the true data-generating process comprises a non-Gaussian idiosyncratic factor. This last result holds independently of the sample size, across different asset correlation levels, and it intensifies for positive shape parameter values.

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.

We study univariate integration with the Gaussian weight for a positive variance α. This is done for the reproducing kernel Hilbert space with the Gaussian kernel for a positive shape parameter γ. We study Gauss-Hermite quadratures, although this choice of quadratures may be questionable since polynomials do not belong to this space of functions. Nevertheless, we provide the explicit formula for the error of the Gauss-Hermite quadrature using n function values. In particular, for 2αγ2 1 we have an exponential divergence.

Bèzier curves based on Lupaş [formula omitted]-analogue of Bernstein functions in CAGD

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.&lt;/p&gt;&lt;/abstract&gt;