ON GENERALISATION OF GOMPERTZ-MAKEHAM DISTRIBUTION

In this paper an exponentiated generalised Gompertz-Makeham distribution. An exponentiated generalised family was introduced by Codeiro, et. al., which allows greater flexibility in the analysis of data. Some Mathematical and Statistical properties including cumulative distribution function, hazard function and survival function of the distribution are derived. The estimation of model parameters are derived via the maximum likelihood estimate method.

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Estimasi Fungsi Survival dan Fungsi Hazard Kumulatif Pada Data Survival Pederita Multiple Myeloma Serta Faktor-faktor yang Mempengaruhi Waktu Survivalnya

Intersections ◽

10.47200/intersections.v4i2.507 ◽

2019 ◽

Vol 4 (2) ◽

pp. 33-43

Author(s):

Toto Hermawan ◽

Dwi Nurrohmah ◽

Ismi Fathul Jannah

Keyword(s):

Multiple Myeloma ◽

Distribution Function ◽

Data Analysis ◽

Probability Density Function ◽

Weibull Distribution ◽

Hazard Function ◽

Cumulative Distribution Function ◽

Survival Function ◽

Random Variable ◽

Cumulative Distribution

Multiple myeloma is an infectious disease characterized by the accumulation of abnormal plasma cells, a type of white blood cell, in the bone marrow. The main objective of this data analysis is to investigate the effect of Bun, Ca, Pcells and Protein risk factors on the survival time of multiple myeloma patients from diagnosis to death. In the survival data analysis, the observed random variable T is the time needed to achieve success. To explain a random variable, the cumulative distribution function or the probability density function can be used. In survival analysis, the function of the random variable that becomes important is the survival function and the hazard function which can be derived using the cumulative distribution function or the probability density function. In general, it is difficult to determine the survival function or hazard function of a population group with certainty. However, the survival function or hazard function can still be approximated by certain estimation methods. The Kaplan-Meier method can be used to find estimators of the survival function of a population. Meanwhile, to find the estimator of the cumuative hazard function, the Nelson-Aalen method can be used. From the variables studied, it turned out that the one that gave the most significant effect was the Bun variable, namely blood urea nitrogen levels using both the exponential and weibull distribution. However, by using the weibull distribution, the presence of Bence Jones Protein in urine also has a quite real effect

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On a Special Weighted Version of the Odd Weibull-Generated Class of Distributions

Mathematical and Computational Applications ◽

10.3390/mca26030062 ◽

2021 ◽

Vol 26 (3) ◽

pp. 62

Author(s):

Zichuan Mi ◽

Saddam Hussain ◽

Christophe Chesneau

Keyword(s):

Distribution Function ◽

Weibull Distribution ◽

Probability Density ◽

Hazard Rate ◽

Cumulative Distribution Function ◽

Data Fitting ◽

Environmental Data ◽

Cumulative Distribution ◽

Model Parameters ◽

New Class

In recent advances in distribution theory, the Weibull distribution has often been used to generate new classes of univariate continuous distributions. They find many applications in important disciplines such as medicine, biology, engineering, economics, informatics, and finance; their usefulness is synonymous with success. In this study, a new Weibull-generated-type class is presented, called the weighted odd Weibull generated class. Its definition is based on a cumulative distribution function, which combines a specific weighted odd function with the cumulative distribution function of the Weibull distribution. This weighted function was chosen to make the new class a real alternative in the first-order stochastic sense to two of the most famous existing Weibull generated classes: the Weibull-G and Weibull-H classes. Its mathematical properties are provided, leading to the study of various probabilistic functions and measures of interest. In a consequent part of the study, the focus is on a special three-parameter survival distribution of the new class defined with the standard exponential distribution as a reference. The exploratory analysis reveals a high level of adaptability of the corresponding probability density and hazard rate functions; the curves of the probability density function can be decreasing, reversed N shaped, and unimodal with heterogeneous skewness and tail weight properties, and the curves of the hazard rate function demonstrate increasing, decreasing, almost constant, and bathtub shapes. These qualities are often required for diverse data fitting purposes. In light of the above, the corresponding data fitting methodology has been developed; we estimate the model parameters via the likelihood function maximization method, the efficiency of which is proven by a detailed simulation study. Then, the new model is applied to engineering and environmental data, surpassing several generalizations or extensions of the exponential model, including some derived from established Weibull-generated classes; the Weibull-G and Weibull-H classes are considered. Standard criteria give credit to the proposed model; for the considered data, it is considered the best.

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THE EXACT CUMULATIVE DISTRIBUTION FUNCTION OF A RATIO OF QUADRATIC FORMS IN NORMAL VARIABLES, WITH APPLICATION TO THE AR(1) MODEL

Econometric Theory ◽

10.1017/s0266466602184015 ◽

2002 ◽

Vol 18 (4) ◽

pp. 823-852 ◽

Cited By ~ 11

Author(s):

G. Forchini

Keyword(s):

Distribution Function ◽

Maximum Likelihood ◽

Maximum Likelihood Estimator ◽

Quadratic Forms ◽

Cumulative Distribution Function ◽

Closed Form Solution ◽

Form Solution ◽

Cumulative Distribution ◽

Likelihood Estimator ◽

Start Up

Often neither the exact density nor the exact cumulative distribution function (c.d.f.) of a statistic of interest is available in the statistics and econometrics literature (e.g., the maximum likelihood estimator of the autocorrelation coefficient in a simple Gaussian AR(1) model with zero start-up value). In other cases the exact c.d.f. of a statistic of interest is very complicated despite the statistic being “simple” (e.g., the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere). The first part of the paper tries to explain why this is the case by studying the analytic properties of the c.d.f. of a statistic under very general assumptions. Differential geometric considerations show that there can be points where the c.d.f. of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself. The second part of the paper derives the exact c.d.f. of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found. These results are then specialized to the maximum likelihood estimator of the autoregressive parameter in a Gaussian AR(1) model with zero start-up value, which is shown to have precisely those properties highlighted in the first part of the paper.

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A New Approach to Assess Wind Potential

Global NEST Journal ◽

10.30955/gnj.004021 ◽

2021 ◽

Keyword(s):

Distribution Function ◽

Maximum Likelihood ◽

Weibull Distribution ◽

Wind Power ◽

Cumulative Distribution Function ◽

Maximum Likelihood Method ◽

New Method ◽

Cumulative Distribution ◽

Likelihood Method ◽

Wind Distribution

<p>Weibull Cumulative Distribution Function (C.D.F.) has been employed to assess and compare wind potentials of two wind stations Europlatform and Stavenisse of The Netherland. Weibull distribution has been used for accurate estimation of wind energy potential for a long time. The Weibull distribution with two parameters is suitable for modeling wind data if wind distribution is unimodal. Whereas wind distribution is generally unimodal, random weather changes can make the distribution bimodal. It is always desirable to find a method that accurately represents actual statistical data. Some well-known statistical methods are Method of Moment (MoM), Linear Least Square Method (LLSM), Maximum Likelihood Method (M.L.M.), Modified Maximum Likelihood Method (MMLM), Energy Pattern Factor Method (EPFM), and Empirical Method (E.M.), etc. All these methods employ Probability Density Function (PDF) of Weibull distribution, except LLSM, which uses Cumulative Distribution Function (C.D.F.). In this communication, we are presenting a newly proposed method of evaluating Weibull parameters. Unlike most methods, this new method employs a cumulative distribution function. A MATLAB® GUI-based simulation is developed to estimate Weibull parameters using the C.D.F. approach. It is found that the Mean Square Error (M.S.E.) is the lowest when using the new method. The new method, therefore, estimates wind power density with reasonable accuracy. Wind Power (W.P.) is estimated by considering four different Wind Turbine (W.T.) models for two sites, and maximum W.P. is found using Evance R9000.</p>

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ANALISIS KUMULATIF COVID-19 PROVINSI PAPUA TAHUN 2020 MENGGUNAKAN MODEL DISTRIBUSI JOHNSON SB

STATISTIKA: Journal of Theoretical Statistics and Its Applications ◽

10.29313/jstat.v21i1.7820 ◽

2021 ◽

Vol 21 (1) ◽

pp. 21-28

Author(s):

Felix Reba ◽

Alvian Sroyer

Keyword(s):

Distribution Function ◽

Maximum Likelihood ◽

Cumulative Distribution Function ◽

Data Distribution ◽

Secondary Data ◽

Cumulative Distribution ◽

Distribution Model ◽

Cumulative Probability ◽

Number Of Patients ◽

Kolmogorov Smirnov

Coronavirus belongs to the coronaviridae family. The coronavirus family groups are alpha (α), beta (β), gamma (γ) and delta (δ) coronavirus. Although research related to covid-19 in several provinces in Indonesia has been conducted by several researchers so far there has been no research related to the Covid-19 model in Papua province. One of the obstacles faced by some researchers is related to the Covid-19 data parameters which are difficult to estimate, so that the model formulated could not describe the outbreak well. Therefore the aim of this study is to conduct a cumulative analysis of the 2020 Papua province Covid-19 using the Johnson SB distribution model. The methods used to perform the analysis are Kolmogorov Smirnov for testing the suitability of the Covid-19 data to the model, Johnson SB to show the data distribution model, Maximum Likelihood to estimate the parameters and the Johnson SB cumulative distribution function to describe the probability of Covid-19 data. 19 Papua Province in 2020. The secondary data on the number of Covid-19 cases in Papua, obtained from the Papua Provincial Health Office is used in this research. The results showed that, the highest increase in the number of patients every day, starting from September 1 2020 to October 31, 2020 for infected cases was on 16-17 September, by 274 patients. Meanwhile, most recovery (308 patients) happened to be on 30-31 October and the highest death (5 people) was on 27-28 September. The highest cumulative probability for cases of infection, recovery and death were (Confirmed <4965) = 0.3, Prob(Cured <6408) = 0.9 and Prob(died <91) = 0.4 respectively.

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An Asymmetric Bimodal Distribution with Application to Quantile Regression

Symmetry ◽

10.3390/sym11070899 ◽

2019 ◽

Vol 11 (7) ◽

pp. 899 ◽

Cited By ~ 1

Author(s):

Yolanda M. Gómez ◽

Emilio Gómez-Déniz ◽

Osvaldo Venegas ◽

Diego I. Gallardo ◽

Héctor W. Gómez

Keyword(s):

Distribution Function ◽

Maximum Likelihood ◽

Quantile Regression ◽

Simulation Study ◽

Cumulative Distribution Function ◽

Maximum Likelihood Estimators ◽

Real Data ◽

Bimodal Distribution ◽

Cumulative Distribution ◽

Cauchy Model

In this article, we study an extension of the sinh Cauchy model in order to obtain asymmetric bimodality. The behavior of the distribution may be either unimodal or bimodal. We calculate its cumulative distribution function and use it to carry out quantile regression. We calculate the maximum likelihood estimators and carry out a simulation study. Two applications are analyzed based on real data to illustrate the flexibility of the distribution for modeling unimodal and bimodal data.

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Towards a conceptual approach to predetermining long-return-period avalanche run-out distances

Journal of Glaciology ◽

10.3189/172756504781830178 ◽

2004 ◽

Vol 50 (169) ◽

pp. 268-278 ◽

Cited By ~ 27

Author(s):

Maurice Meunier ◽

Christophe Ancey

Keyword(s):

Distribution Function ◽

Field Data ◽

Cumulative Distribution Function ◽

Extreme Event ◽

A Priori ◽

Rheological Behaviour ◽

Cumulative Distribution ◽

Model Parameters ◽

Conceptual Approach ◽

Deterministic Models

AbstractInvestigating snow avalanches using a purely statistical approach raises several issues. First, even in the heavily populated areas of the Alps, there are few data on avalanche motion or extension. Second, most of the field data are related to the point of furthest reach in the avalanche path (run-out distance or altitude). As data of this kind are tightly dependent on the avalanche path profile, it is a priori not permissible to extrapolate the cumulative distribution function fitted to these data without severe restrictions or further assumptions. Using deterministic models is also problematic, as these are not really physically based models. For instance, they do not include all the phenomena occurring in the avalanche movement, and the rheological behaviour of the snow is not known. Consequently, it is not easy to predetermine extreme-event extensions. Here, in order to overcome this problem, we propose to use a conceptual approach. First, using an avalanche-dynamics numerical model, we fitted the model parameters (friction coefficients and the volume of snow involved in the avalanches) to the field data. Then, using these parameters as random variables, we adjusted appropriate statistical distributions. The last steps involved simulating a large number of (fictitious) avalanches using the Monte Carlo approach. Thus, the cumulative distribution function of the run-out distance can be computed over a much broader range than was initially possible with the historical data. In this paper, we develop the proposed method through a complete case study, comparing two different models.

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On a New Result on the Ratio Exponentiated General Family of Distributions with Applications

Mathematics ◽

10.3390/math8040598 ◽

2020 ◽

Vol 8 (4) ◽

pp. 598 ◽

Cited By ~ 2

Author(s):

Rashad A. R. Bantan ◽

Farrukh Jamal ◽

Christophe Chesneau ◽

Mohammed Elgarhy

Keyword(s):

Distribution Function ◽

Cumulative Distribution Function ◽

Natural Extension ◽

Continuous Distribution ◽

Cumulative Distribution ◽

Likelihood Method ◽

Model Parameters ◽

Mathematical Treatment ◽

General Family ◽

Family Of Distributions

In this paper, we first show a new probability result which can be concisely formulated as follows: the function 2 G β / ( 1 + G α ) , where G denotes a baseline cumulative distribution function of a continuous distribution, can have the properties of a cumulative distribution function beyond the standard assumptions on α and β (possibly different and negative, among others). Then, we provide a complete mathematical treatment of the corresponding family of distributions, called the ratio exponentiated general family. To link it with the existing literature, it constitutes a natural extension of the type II half logistic-G family or, from another point of view, a compromise between the so-called exponentiated-G and Marshall-Olkin-G families. We show that it possesses tractable probability functions, desirable stochastic ordering properties and simple analytical expressions for the moments, among others. Also, it reaches high levels of flexibility in a wide statistical sense, mainly thanks to the wide ranges of possible values for α and β and thus, can be used quite effectively for the real data analysis. We illustrate this last point by considering the Weibull distribution as baseline and three practical data sets, with estimation of the model parameters by the maximum likelihood method.

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