TWO SECTIONAL MODELS INVOLVING TWO WEIBULL DISTRIBUTIONS

1999 ◽  
Vol 06 (02) ◽  
pp. 103-122 ◽  
Author(s):  
RENYAN JIANG ◽  
MING J. ZUO ◽  
D. N. P. MURTHY
Keyword(s):  
Probability Density ◽  
Failure Rate ◽  
Real Data ◽  
Model Parameters ◽  
Weibull Distributions ◽  
Data Set ◽  
Failure Data ◽  
Rate Functions ◽  
Estimation Of Model Parameters

In this paper, we study two sectional models, each involving two Weibull distributions. Characterization of the plot on Weibull plotting paper (WPP) for each model is carried out. We also study the shapes of the probability density and the failure rate functions. These are useful in determining if a given failure data set can be modeled by such a model. We discuss the estimation of model parameters based on the WPP plot and illustrate through two examples involving real data.

scholarly journals Mixture of Inverse Weibull and Lognormal Distributions: Properties, Estimation, and Illustration

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
K. S. Sultan ◽  
A. S. Al-Moisheer
Keyword(s):  
Information Matrix ◽  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Component Mixture ◽  
Data Set ◽  
Hazard Functions ◽  
Lognormal Distributions ◽  
Proposed Model ◽  
Estimation Of Model Parameters

We discuss the two-component mixture of the inverse Weibull and lognormal distributions (MIWLND) as a lifetime model. First, we discuss the properties of the proposed model including the reliability and hazard functions. Next, we discuss the estimation of model parameters by using the maximum likelihood method (MLEs). We also derive expressions for the elements of the Fisher information matrix. Next, we demonstrate the usefulness of the proposed model by fitting it to a real data set. Finally, we draw some concluding remarks.

scholarly journals The Alpha Power Transformation Family: Properties and Applications

2019 ◽  
pp. 525-545 ◽  
Author(s):  
Mohamed E. Mead ◽  
Gauss M. Cordeiro ◽  
Ahmed Z. Afify ◽  
Hazem Al Mofleh
Keyword(s):  
Real Data ◽  
Likelihood Method ◽  
Model Parameters ◽  
Power Transformation ◽  
Data Sets ◽  
Alpha Power ◽  
Weibull Distributions ◽  
Exponentiated Weibull ◽  
Rate Functions ◽  
Modified Weibull

Mahdavi A. and Kundu D. (2017) introduced a family for generating univariate distributions called the alpha power transformation. They studied as a special case the properties of the alpha power transformed exponential distribution. We provide some mathematical properties of this distribution and define a four-parameter lifetime model called the alpha power exponentiated Weibull distribution. It generalizes some well-known lifetime models such as the exponentiated exponential, exponentiated Rayleigh, exponentiated Weibull and Weibull distributions. The importance of the new distribution comes from its ability to model monotone and non-monotone failure rate functions, which are quite common in reliability studies. We derive some basic properties of the proposed distribution including quantile and generating functions, moments and order statistics. The maximum likelihood method is used to estimate the model parameters. Simulation results investigate the performance of the estimates. We illustrate the importance of the proposed distribution over the McDonald Weibull, beta Weibull, modified Weibull, transmuted Weibull and exponentiated Weibull distributions by means of two real data sets.

scholarly journals Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 548
Author(s):  
Yuri S. Popkov
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Maximum Entropy ◽  
Density Function ◽  
Lagrange Multipliers ◽  
Real Data ◽  
Model Parameters ◽  
Lake Area ◽  
Data Set ◽  
Maximum Entropy Estimation

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia.

Parametric Study of Competing Risk Model Involving Two Weibull Distributions

1997 ◽  
Vol 04 (01) ◽  
pp. 17-34 ◽  
Author(s):  
R. Jiang ◽  
D. N. P. Murthy
Keyword(s):  
Failure Rate ◽  
Parametric Study ◽  
Risk Model ◽  
Competing Risk ◽  
Weibull Distributions ◽  
Competing Risk Model ◽  
Rate Functions ◽  
Parameter Values ◽  
Independent Parameters

The competing risk model involving two Weibull distributions is characterized by four independent parameters. The shapes of the density and failure rate functions for the model depend on the parameter values. In this paper a complete parametric characterization of these shapes is presented.

PARAMETER ESTIMATE WITH ONLY ONE COMPLETE FAILURE OBSERVATION USING MONTE CARLO EM ALGORITHM

2001 ◽  
Vol 08 (02) ◽  
pp. 109-122 ◽  
Author(s):  
WAN-KAI PANG ◽  
PING-KEI LEUNG ◽  
XIAO-LONG PU ◽  
SHI-SONG MAO
Keyword(s):  
Monte Carlo ◽  
Weibull Distribution ◽  
Real Data ◽  
Model Parameters ◽  
Reliability Model ◽  
Estimation Of Parameters ◽  
Data Set ◽  
Failure Data ◽  
Mcem Algorithm ◽  
Monte Carlo Em

In reliability studies, often we only have one failure data recorded in a life testing experiment. If there are two parameters in the reliability model, such as the model using Weibull distribution, then maximum likelihood estimation of parameters becomes a difficult problem. Mao and Chen published a real data set of the lifetime of a certain type of bearings which only contains one failure data. They used a Bayesian method to analyze the data and obtained some results for model parameter estimation. However, in their method the choice of prior distribution will affect heavily the final results. In this paper, we propose a Monte Carlo EM (MCEM) algorithm to estimate reliability model parameters using the Weibull distribution. Based on the same data set of Mao and Chen, we obtain some results using the MCEM algorithm. Our results do not depend on the choice of arbitrary prior distributions.

Type II Half Logistic Linear Failure Rate Distribution with Application

2020 ◽  
Vol 17 (7) ◽  
pp. 2912-2917
Author(s):  
Maha A. Aldahlan
Keyword(s):  
Failure Rate ◽  
Probability Distributions ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Model Parameters ◽  
Type Ii ◽  
Data Set ◽  
Rate Distribution ◽  
New Distribution

In recent years, several of new improved probability distributions have been discovered from the current distributions to facilitate their applications in various areas. A new three-parameter model extended from the linear failure rate model, the so called the type II half logistic linear failure rate distribution. Some mathematical properties of the new distribution are proposed. Explicit expressions for the moments, probability weighted moments and order statistics are calculated. Maximum likelihood estimation method is assessed to estimate the model parameters are presented. The superiority of the new distribution is illustrated with an application to one real data set.

scholarly journals A New Generating Family of Distributions: Properties and Applications to the Weibull Exponential Model

2021 ◽  
Vol 19 (1) ◽  
Author(s):  
El-Sayed A. El-Sherpieny ◽  
Salwa Assar ◽  
Tamer Helal
Keyword(s):  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Exponential Model ◽  
Rate Function ◽  
Likelihood Method ◽  
Model Parameters ◽  
Data Set ◽  
Rate Functions ◽  
Family Of Distributions

A new method for generating family of distributions was proposed. Some fundamental properties of the new proposed family include the quantile, survival function, hazard rate function, reversed hazard and cumulative hazard rate functions are provided. This family contains several new models as sub models, such as the Weibull exponential model which was defined and discussed its properties. The maximum likelihood method of estimation is using to estimate the model parameters of the new proposed family. The flexibility and the importance of the Weibull-exponential model is assessed by applying it to a real data set and comparing it with other known models.

EXPONENTIATED HALF-LOGISTIC LOMAX DISTRIBUTION WITH PROPERTIES AND APPLICATION

2019 ◽  
Vol XVI (2) ◽  
pp. 1-11
Author(s):  
Farrukh Jamal ◽  
Hesham Mohammed Reyad ◽  
Soha Othman Ahmed ◽  
Muhammad Akbar Ali Shah ◽  
Emrah Altun
Keyword(s):  
Real Data ◽  
Continuous Model ◽  
Model Parameters ◽  
Data Set ◽  
Lomax Distribution ◽  
Mathematical Properties ◽  
Proposed Model ◽  
Probability Weighted Moment ◽  
Record Statistics ◽  
Maximum Likelihood Criterion

A new three-parameter continuous model called the exponentiated half-logistic Lomax distribution is introduced in this paper. Basic mathematical properties for the proposed model were investigated which include raw and incomplete moments, skewness, kurtosis, generating functions, Rényi entropy, Lorenz, Bonferroni and Zenga curves, probability weighted moment, stress strength model, order statistics, and record statistics. The model parameters were estimated by using the maximum likelihood criterion and the behaviours of these estimates were examined by conducting a simulation study. The applicability of the new model is illustrated by applying it on a real data set.

A new parameter set for anisotropic multiparameter full-waveform inversion and application to a North Sea data set

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. U25-U38 ◽  
Author(s):  
Nuno V. da Silva ◽  
Andrew Ratcliffe ◽  
Vetle Vinje ◽  
Graham Conroy
Keyword(s):  
North Sea ◽  
Waveform Inversion ◽  
Real Data ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Radiation Patterns ◽  
Data Set ◽  
Full Waveform ◽  
Seismic Image ◽  
Central North Sea

Parameterization lies at the center of anisotropic full-waveform inversion (FWI) with multiparameter updates. This is because FWI aims to update the long and short wavelengths of the perturbations. Thus, it is important that the parameterization accommodates this. Recently, there has been an intensive effort to determine the optimal parameterization, centering the fundamental discussion mainly on the analysis of radiation patterns for each one of these parameterizations, and aiming to determine which is best suited for multiparameter inversion. We have developed a new parameterization in the scope of FWI, based on the concept of kinematically equivalent media, as originally proposed in other areas of seismic data analysis. Our analysis is also based on radiation patterns, as well as the relation between the perturbation of this set of parameters and perturbation in traveltime. The radiation pattern reveals that this parameterization combines some of the characteristics of parameterizations with one velocity and two Thomsen’s parameters and parameterizations using two velocities and one Thomsen’s parameter. The study of perturbation of traveltime with perturbation of model parameters shows that the new parameterization is less ambiguous when relating these quantities in comparison with other more commonly used parameterizations. We have concluded that our new parameterization is well-suited for inverting diving waves, which are of paramount importance to carry out practical FWI successfully. We have demonstrated that the new parameterization produces good inversion results with synthetic and real data examples. In the latter case of the real data example from the Central North Sea, the inverted models show good agreement with the geologic structures, leading to an improvement of the seismic image and flatness of the common image gathers.

A GAMMA MOVING AVERAGE PROCESS FOR MODELLING DEPENDENCE ACROSS DEVELOPMENT YEARS IN RUN-OFF TRIANGLES

2020 ◽  
pp. 1-22
Author(s):  
Luis E. Nieto-Barajas ◽  
Rodrigo S. Targino
Keyword(s):  
Latent Variables ◽  
Moving Average ◽  
Real Data ◽  
Model Parameters ◽  
Average Process ◽  
Moving Average Process ◽  
Data Set ◽  
Run Off ◽  
Average Form ◽  
Reserve Estimates

ABSTRACT We propose a stochastic model for claims reserving that captures dependence along development years within a single triangle. This dependence is based on a gamma process with a moving average form of order $p \ge 0$ which is achieved through the use of poisson latent variables. We carry out Bayesian inference on model parameters and borrow strength across several triangles, coming from different lines of businesses or companies, through the use of hierarchical priors. We carry out a simulation study as well as a real data analysis. Results show that reserve estimates, for the real data set studied, are more accurate with our gamma dependence model as compared to the benchmark over-dispersed poisson that assumes independence.

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