Power Function Distribution Research Articles

Kolmogorov-Smirnov Goodness-of-Fit test for identifying distribution of the number of earthquakes and the losses due to earthquakes in Indonesia

Earthquakes of high magnitude often result in large casualties and huge economic losses in Indonesia. An earthquake risk assessment must be carried out. The risk assessment process requires a structured approach. This paper is containing of the first step of risk assessment using actuarial-statistical method. The data used as research variables in this paper are the number of earthquakes and the losses due to earthquakes. The earthquakes data used are earthquakes with magnitude > 5 Richter Scale. Data is collected within the last 40 years, from 1980 to 2019. Data is collected from Indonesian Disaster Data and Information, and NOAA National Centers for Environmental Information (NCEI). Descriptive statistics analysis obtains that the most frequently earthquakes with magnitude > 5 Richter Scale occurred in Indonesia in 2004, where as it is known that at the end of 2004 an earthquake with a magnitude of 9.1 Richter Scale occurred in Aceh which caused the highest loss, which was around 147485.500 billion rupiah. Normality test of the variables results that the number of earthquakes and the losses due to earthquakes are not normally distributed. Kolmogorov-Smirnov Goodness-of-Fit test results that the number of earthquakes follows Gamma Distribution with shape parameter α = 1.800 and scale parameter β = 2.236; while the losses due to earthquakes follows Power Function Distribution with shape parameter α = 0.072, and boundary parameter a = 0, b = 2.810 x 105.

Transmuted Topp-Leone Power Function Distribution: Theory and Application

Transmuted Topp-Leone Power Function Distribution: Theory and Application

Transmuted Weibull Power Function Distribution: its Properties and Applications

In this paper, we introduce a new four-parameter distribution called the transmuted Weibull power function (TWPF) distribution which e5xtends the transmuted family proposed by Shaw and Buckley [1]. The hazard rate function of the TWPF distribution can be constant, increasing, decreasing, unimodal, upside down bathtub shaped or bathtub shape. Some mathematical properties are derived including quantile functions, expansion of density function, moments, moment generating function, residual life function, reversed residual life function, mean deviation, inequality measures. The estimation of the model parameters is carried out using the maximum likelihood method. The importance and flexibility of the proposed model are proved empirically using real data sets.

Four Parameters Kumaraswamy Reciprocal Family Of Distributions

In this paper, kumaraswamy reciprocal family of distributions is introduced as a new continues model with some of approximation to other probabilistic models as reciprocal, beta, uniform, power function, exponential, negative exponential, weibull, rayleigh and pareto distribution. Some fundamental distributional properties, force of mortality, mills ratio, bowley skewness, moors kurtosis, reversed hazard function, integrated hazard function, mean residual life, probability weighted moments, bonferroni and lorenz curves, laplace-stieltjes transform of this new distribution with the maximum likelihood method of the parameter estimation are studied. Finally, four real data sets originally presented are used to illustrate the proposed estimators.

A Log-weighted Power Function Distribution and Its Statistical Properties

The Power function distribution is a flexible life time distribution that has applications in finance and economics. It is, also, used to model reliability growth of complex systems or the reliability of repairable systems. A new weighted Power function distribution is proposed using a logarithmic weight function. Statistical properties of the weighted power function distribution are obtained and studied. Location measures such as mode, median and mean, reliability measures such as reliability function, hazard and reversed hazard functions and the mean residual life are derived. Shape indices such as skewness and kurtosis coefficients and order statistics are obtained. Parametric estimation is performed to obtain estimators for the parameters of the distribution using three different estimation methods; namely: the maximum likelihood method, the L-moments method and the method of moments. Numerical simulation is carried out to validate the robustness of the proposed distribution.

Inference on Positive Exponential Family of Distributions (PEFD) through Transformation Method

The estimation of R(t) and R=Pr(Y>X) for the Positive Exponential Family of Distribution (PEFD) is considered. The UMVUES, MLES and Confidence Interval are derived. These estimators are derived through the method of Transformation. The α=Pr(X>γ), which is termed as probability of disaster is also derived when random stress X follows PEFD and finite strength follows Power function distribution.

Control Charts for the Shape Parameter of Power Function Distribution under Different Classical Estimators

In practice, the control charts for monitoring of process mean are based on the normality assumption. But the performance of the control charts is seriously affected if the process of quality characteristics departs from normality. For such situations, we have modified the already existing control charts such as Shewhart control chart, exponentially weighted moving average (EWMA) control chart and hybrid exponentially weighted moving average (HEWMA) control chart by assuming that the distribution of underlying process follows Power function distribution (PFD). By considering the situation that the parameters of PFD are unknown, we estimate them by using three classical estimation methods, i.e., percentile estimator (P.E), maximum likelihood estimator (MLE) and modified maximum likelihood estimator (MMLE). We construct Shewhart, EWMA and HEWMA control charts based on P.E, MLE and MMLE. We have compared all these control charts using Monte Carlo simulation studies and concluded that HEWMA control chart under MLE is more sensitive to detect an early shift in the shape parameter when the distribution of the underlying process follows power function distribution.

Estimation of Information Measures for Power-Function Distribution in Presence of Outliers and Their Applications

Entropy measurement plays an important role in the field of information theory. Furthermore, the estimation of entropy is an important issue in statistics and machine learning. This study estimated the Rényi and q-entropies of a power-function distribution in the presence of s outliers using classical and Bayesian procedures. In the classical method, the maximum likelihood estimators of the entropies were obtained and their performance was assessed through a numerical study. In the Bayesian method, the Bayesian estimators of the entropies under uniform and gamma priors were acquired based on different loss functions. The Bayesian estimators were computed empirically using a Monte Carlo simulation based on the Gibbs sampling algorithm. The simulated datasets were analyzed to investigate the accuracy of the estimates. The study results showed that the precision of the maximum likelihood and Bayesian estimates of both entropies improved with increasing the sample size and the number of outliers. The absolute biases and the mean squared errors of the estimates in the presence of outliers exceeded those of the corresponding estimates in the homogenous case (no-outliers). Furthermore, the Bayesian estimates of the Rényi and q-entropies under the squared error loss function were preferable to the other Bayesian estimates in a majority of the cases. Finally, analysis results of real data examples were consistent with those of the simulated data.

Control Charts for the Shape Parameter of Skewed Distribution

The weighted distributions are useful when the sampling is done using an unequal probability of the sampling units. The Weighted Power function distribution (WPFD) has applications in the fields of reliability engineering, management sciences and survival analysis. WPFD is more beneficial in Statistical process control (SPC). SPC is defined as the use of statistical techniques to control a process or production method. SPC tools and procedures can help to monitor process behaviour, discover problems in internal systems, and find solutions for production issues. To identify and remove the variation in different reliability processes and also to monitor the reliability of machines where the number of errors follows WPFD, we develop control charts to keep the process in control. A memory-based control chart like an exponentially weighted moving average (EWMA) control chart and an extended exponentially weighted moving average (EEWMA) control chart are discussed and compared each other. The proposal of these control charts is based on the modified maximum likelihood estimator (MMLE) under the shape parameter of WPFD. We have presented Monte Carlo simulation technique and a real-life application to compare the proposed control charts. This study shows that an EEWMA control chart based on MMLE performs better than EWMA control chart, when the underlying distribution of the errors in process monitoring follows WPFD. These findings can be useful for researchers and practitioners in dealing with production errors and optimizing the output.

Study on the Dynamic Response Characteristics of Cylindrical Coal-Rock Samples under Dynamic Loads

To research the dynamic response characteristics of cylindrical coal-rock samples under impact loads, the impact of rigid bars on cylindrical coal-rock samples is simulated under different speed conditions, based on LS-DYNA software, and the dynamic distribution characteristics of the stress, strain, and energy of cylindrical coal-rock samples are analyzed. The results demonstrated the following: (1) the cylindrical coal-rock sample failed at the center first, and the damage developed downward along the axial direction. (2) The critical effective stress and strain have an exponential function relationship with the velocity, and the critical time has a linear relationship with the velocity. (3) The energy change law of the cylindrical coal-rock sample is consistent with the destruction morphology. (4) The axial stress peaks in the severe damage part have a linear relationship with the speed, the axial stress attenuates rapidly after passing the stress yield point, and the axial strain does not increase continuously. (5) The peaks stress and strain on the central axis and the radial line obey the power function distribution, the axial stress produces tensile stress in the axial propagation direction, and the axial stress and strain peaks at the same position are larger than those of the radial stress and strain peaks. This research provides a reference for studying coal and rock dynamic disasters.

Dynamic fragmentation of a Zr-based metallic glass under various impact velocities

The dynamic fragmentation of ZrCuNiAl bulk metallic glass (BMG) subjected to various impact velocities was investigated in argon atmosphere. The results demonstrated that the ZrCuNiAl metallic glass generated debris after impact. The cumulative mass distribution curves of ZrCuNiAl BMG samples after impact under various velocities were obtained; the results showed that the cumulative mass of BMG samples conforms to the Schuhmann power function distribution. The characteristic fragment size λ governing the fragment distributions was determined by fracture toughness and expansion strain rate. The results demonstrated that characteristic fragment size decreases with an increase in impact velocity, which was well described by the model. A relationship between the cumulative mass distribution and the impact velocity of Zr55Cu30Ni5Al10 alloy was obtained based on energy principles. This formula is feasible to predict the cumulative mass distribution of Zr55Cu30Ni5Al10 alloy after impact under a wide velocity range.

Numerical simulation of the dynamic distribution characteristics of the stress, strain and energy of coal mass under impact loads

To research the dynamic response characteristics of coal mass under impact loads, based on LS-DYNA software, rigid body bars are simulated to impact coal mass under different speed conditions, and the dynamic distribution characteristics of the stress, strain and energy of coal mass are analyzed. The results demonstrate that (1) the peaks of the axial and radial stresses and strain on the central axis and the radial line obey the power function distribution; at the same position, the axial and the radial stress peaks are close, and the axial strain peak is from much larger than the radial strain peak to close to. (2) The axial and radial stresses generate tensile stresses in the axial and radial propagation directions, respectively, and the coal mass is prone to damage under tensile stress. (3) When the speed is large, the axial stress–strain curve is similar to that of the dynamic load experiment. The axial stress peak, axial strain peak, critical effective stress, critical time and secant modulus have a linear relationship with the velocity. (4) When the dynamic load is large, most of the energy is in the form of kinetic energy, and the total energy loss also increases.

A view on characterizations of the J shaped statistical distribution

Objectives: In recent years, characterization of any distribution has become important in the field of probability distribution.The objective of the study is to characterize the power function distribution to see its usefulness under different real life situations such as Engineering and medical sciences. Methods: The study proposed the characterization of Power function distribution based on mean inactivity times (MIT), mean residual function (MRF), conditional moments, conditional variance (CV), doubly truncated mean (DTM), incomplete moments and reverse hazard function. Findings:We have characterized the power function distribution using different method, and conclude that thesufficient and necessary conditions of different methods mentioned above meet the results of Power function distribution. Application: Power function distribution has wide applicability in the field of Engineering. The findings of the paper may help the Engineers to know more about the Power function distribution. Keywords: Characterization; mean inactivity time; mean residual function; power function distribution

A Spline Finite Point Method for Nonlinear Bending Analysis of FG Plates in Thermal Environments Based on a Locking-free Thin/Thick Plate Theory

A spline finite point method (SFPM) based on a locking-free thin/thick plate theory, which is suitable for analysis of both thick and thin plates, is developed to study nonlinear bending behavior of functionally graded material (FGM) plates with different thickness in thermal environments. In the proposed method, one direction of the plate is discretized with a set of uniformly distributed spline nodes instead of meshes and the other direction is expressed with orthogonal functions determined by the boundary conditions. The displacements of the plate are constructed by the linear combination of orthogonal functions and cubic B-spline interpolation functions with high efficiency for modeling. The locking-free thin/thick plate theory used by the proposed method is based on the first-order shear deformation theory but takes the shear strains and displacements as basic unknowns. The material properties of the FG plate are assumed to vary along the thickness direction following the power function distribution. By comparing with several published research studies based on the finite element method (FEM), the correctness, efficiency, and generality of the new model are validated for rectangular plates. Moreover, uniform and nonlinear temperature rise conditions are discussed, respectively. The effect of the temperature distribution, in-plane temperature force, and elastic foundation on nonlinear bending under different parameters are discussed in detail.

Moments of Generalized Upper Record Values from Weibull- Power Function Distribution and Characterization

Moments of Generalized Upper Record Values from Weibull- Power Function Distribution and Characterization

Step Stress Partially Accelerated Life Tests and Estimating Costs of Maintenance Service Policy for the Power Function Distribution under Progressive Type-II Censoring

Step Stress Partially Accelerated Life Tests and Estimating Costs of Maintenance Service Policy for the Power Function Distribution under Progressive Type-II Censoring

Beta Lehmann-2 power function distribution with application to bladder cancer susceptibility and failure times of air-conditioned system

Objectives: Probability distributions have great use in reliability engineering where the researchers try to find the distribution of the different processes. To meet the needs of the reliability engineers, we have proposed a simple probability distribution named as Beta Lehman-2 which may be proved more useful as compared to already existing models of the probability distributions. The aim of the study is to show the performance of the proposed distribution over already existing distributions. Methods: In this study, a new Beta Lehmann-2 Power function distribution (BL2PFD) is proposed. We suggest a new generator that will modify the Power function distribution called Beta Lehmann-2 generator (BL2-G). Findings: The various properties of the new distribution have been discussed in detail such as moments, vitality function, conditional moments and order statistics etc. We have also characterized the BL2PFD based on conditional variance. This distribution can be used for approximately symmetric data (normal data), positive and negative skewed data. Application: The application of this distribution is illustrated by using data sets from medical and engineering sources. The shape of the new distribution has been studied for applied sciences. After analyzing data, we conclude that the proposed model BL2PFD perform better in all the data sets while compared to different competitor models. Keywords: Beta Lehmann-2 Power function distribution; Characterization of truncated distribution; Lehmann alternatives; Percentile estimator; Power function distribution

Chance constrained programming with some non-normal continuous random variables

Stochastic or probabilistic programming is a branch of mathematical programming that deals with some situations in which an optimal decision is desired under random uncertainty of some parameters. In this paper, we consider some chance constrained linear programming problems where the right hand side parameters of the chance-constraints follow some non-normal continuous distributions such as power function distribution, triangular distribution and trapezoidal distribution. To find the solution of the stated problems, we first convert the problems in to equivalent deterministic models. Then standard linear programming techniques are used to solve the equivalent deterministic models. Some numerical examples are presented to illustrate the methodology.

Exponentiated Power Function Distribution: Properties and Applications

In this study, we have focused to propose a flexible model that demonstrates increasing, decreasing and upside-down bathtubshaped density and failure rate functions. The proposed model refers to as the exponentiated power function (EPF) distribution. Some mathematical and reliability measures are developed and derived. We develop explicit expressions for the moments, quantile function and order statistics. Some shapes of the density and the reliability functions are sketched out and discussed. We suggest the method to estimate the unknown parameters of EPF by the maximum likelihood estimation. Two suitable lifetime datasets from engineering sector are used to explore the dominance of the EPF distribution.

A nonlinear mixed-effects tree height prediction model: Application to Pinus pinaster Ait in Northwest Spain

This study introduced a new height prediction model based on the modification of power function distribution. The model was fitted to the Pinus pinaster Ait (Maritime pine) data set comprised of 14339 trees measured from 155 permanent sample plots in northwest Spain. Nonlinear mixed-effect was to refit the model and calibrated with the random effects predicted from one to four sample trees per plot using validation data set (3621 trees from 29 plots). The models were evaluated based on different indices including root mean square error (RMSE), critical error (Ecrit) and adjusted coefficient of determination (R¯2). The results showed that the performance of the model was improved by the inclusion of random parameter with RMSE, Ecrit and R¯2of 1.143, 0.700 and 0.910, respectively. The calibration response of the mixed-effect model involved the selection of four trees per sample plot – each close to the 25th, 50th, 75th quartiles and maximum diameter. It resulted in about 40% reduction of the RMSE compared to other alternatives.