scholarly journals A note on an extreme left skewed unit distribution: Theory, modelling and data fitting

2021 ◽  
Vol 2 (1) ◽  
pp. 1-23
Author(s):  
Christophe Chesneau
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Power Distribution ◽  
Stochastic Order ◽  
Cumulative Distribution ◽  
Closed Form Expression ◽  
Survival Times ◽  
Data Set ◽  
Wide Range

Abstract In probability and statistics, unit distributions are used to model proportions, rates, and percentages, among other things. This paper is about a new one-parameter unit distribution, whose probability density function is defined by an original ratio of power and logarithmic functions. This function has a wide range of J shapes, some of which are more angular than others. In this sense, the proposed distribution can be thought of as an “extremely left skewed alternative” to the traditional power distribution. We discuss its main characteristics, including other features of the probability density function, some stochastic order results, the closed-form expression of the cumulative distribution function involving special integral functions, the quantile and hazard rate functions, simple expressions for the ordinary moments, skewness, kurtosis, moments generating function, incomplete moments, logarithmic moments and logarithmically weighted moments. Subsequently, a simple example of an application is given by the use of simulated data, with fair comparison to the power model supported by numerical and graphical illustrations. A new modelling strategy beyond the unit domain is also proposed and developed, with an application to a survival times data set.

scholarly journals Bivariate Burr Type III Distribution: Estimation and Prediction

2021 ◽  
pp. 16-36
Author(s):  
A. A. M. Mahmoud ◽  
R. M. Refaey ◽  
G. R. AL-Dayian ◽  
A. A. EL-Helbawy
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Real Life ◽  
Bivariate Distribution ◽  
Cumulative Distribution ◽  
Data Set ◽  
Squared Error Loss Function ◽  
Type Iii ◽  
Estimation And Prediction

In this paper, a bivariate Burr Type III distribution is constructed and some of its statistical properties such as bivariate probability density function and its marginal, joint cumulative distribution and its marginal, reliability and hazard rate functions are studied. The joint probability density function and the joint cumulative distribution are given in closed forms. The joint expectation of this distribution is proposed. The maximum likelihood estimation and prediction for a future observation are derived. Also, Bayesian estimation and prediction are considered under squared error loss function. The performance of the proposed bivariate distribution is examined using a simulation study. Finally, a data set is analyzed under the proposed distribution to illustrate its flexibility for real-life application.

Properties and methods of estimation for a bivariate exponentiated Fréchet distribution

2020 ◽  
Vol 70 (5) ◽  
pp. 1211-1230
Author(s):  
Abdus Saboor ◽  
Hassan S. Bakouch ◽  
Fernando A. Moala ◽  
Sheraz Hussain
Keyword(s):  
Maximum Likelihood ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Superior Performance ◽  
Estimation Methods ◽  
Conditional Probability Density ◽  
Data Set ◽  
Fréchet Distribution ◽  
Frechet Distribution

AbstractIn this paper, a bivariate extension of exponentiated Fréchet distribution is introduced, namely a bivariate exponentiated Fréchet (BvEF) distribution whose marginals are univariate exponentiated Fréchet distribution. Several properties of the proposed distribution are discussed, such as the joint survival function, joint probability density function, marginal probability density function, conditional probability density function, moments, marginal and bivariate moment generating functions. Moreover, the proposed distribution is obtained by the Marshall-Olkin survival copula. Estimation of the parameters is investigated by the maximum likelihood with the observed information matrix. In addition to the maximum likelihood estimation method, we consider the Bayesian inference and least square estimation and compare these three methodologies for the BvEF. A simulation study is carried out to compare the performance of the estimators by the presented estimation methods. The proposed bivariate distribution with other related bivariate distributions are fitted to a real-life paired data set. It is shown that, the BvEF distribution has a superior performance among the compared distributions using several tests of goodness–of–fit.

scholarly journals On a new distribution based on the arccosine function

2021 ◽  
Author(s):  
Christophe Chesneau ◽  
Lishamol Tomy ◽  
Jiju Gillariose
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Power Functions ◽  
New Distribution

AbstractThis note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

scholarly journals A Probability Density Function Generator Based on Deep Learning

2018 ◽  
Author(s):  
Chi-Hua Chen ◽  
Fangying Song ◽  
Feng-Jang Hwang ◽  
Ling Wu
Keyword(s):  
Deep Learning ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Real Data ◽  
Learning Model ◽  
Cumulative Distribution ◽  
Function Generator ◽  
Deep Learning Model

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

Fourier Transforms in Probability, Random Variables and Stochastic Processes

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Fourier Analysis ◽  
Stochastic Processes ◽  
Probability Density ◽  
Density Function ◽  
Fourier Transforms ◽  
Random Variables ◽  
Cumulative Distribution ◽  
The Face

In this Chapter, we present application of Fourier analysis to probability, random variables and stochastic processes [1089, 1097, 1387, 1329]. Arandom variable, X, is the assignment of a number to the outcome of a random experiment. We can, for example, flip a coin and assign an outcome of a heads as X = 1 and a tails X = 0. Often the number is equated to the numerical outcome of the experiment, such as the number of dots on the face of a rolled die or the measurement of a voltage in a noisy circuit. The cumulative distribution function is defined by FX(x) = Pr[X ≤ x]. (4.1) The probability density function is the derivative fX(x) = d /dxFX(x). Our treatment of random variables focuses on use of Fourier analysis. Due to this viewpoint, the development we use is unconventional and begins immediately in the next section with discussion of properties of the probability density function.

scholarly journals Alpha Power Transformed Lindley Distribution

2021 ◽  
pp. 130-142
Author(s):  
L A Rosa ◽  
S Nurrohmah ◽  
I Fithriani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Alpha Power ◽  
Lindley Distribution ◽  
Modelling Data

The one parameter Lindley distribustion (theta) has been widely used in various field such as biology, technique, medical, and industries. Lindley distribution is capable for modelling data with monotone increasing hazard rate. However, in real life, there are situations where the hazard rate is not monotone. Therefore, to enhance the Lindley distribution capabilitiesfor modelling data, a modification can be used by using Alpha Power Transformed method. The result of the modification of Lindley distribution is commonly called Alpha Power Transformed Lindley distribution (APTL) distribution that has two parameters (alpha, theta). This new APTL distribution is appropriate in modelling data with decreasing or unimodal shaped of probability density function, and has hazard rates with increasing, decreasing, and upside-down bathtub shaped. The properties of the proposed distribution are discussed include probability density function, cumulative distribution function, survival function, hazard rate function, moment generating function, and rth moment. Themodel parameters are obtained using maximum likelihood method. The waiting time data is used as an illustration to describe the utility of APTL distribution.

The joint asymptotic distribution of the k-smallest sample spacings

1969 ◽  
Vol 6 (02) ◽  
pp. 442-448
Author(s):  
Lionel Weiss
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Open Interval ◽  
Cumulative Distribution ◽  
The Right ◽  
Sample Spacings ◽  
Independent Identically Distributed

Suppose Q 1 ⋆, … Q n ⋆ are independent, identically distributed random variables, each with probability density function f(x), cumulative distribution function F(x), where F(1) – F(0) = 1, f(x) is continuous in the open interval (0, 1) and continuous on the right at x = 0 and on the left at x = 1, and there exists a positive C such that f(x) > C for all x in (0, l). f(0) is defined as f(0+), f(1) is defined as f(1–).

An Efficient Approach for Incremental Association Rule Mining through Histogram Matching Technique

2012 ◽  
Vol 2 (2) ◽  
pp. 29-42 ◽  
Author(s):  
Ajay Kumar ◽  
Shishir Kumar ◽  
Sakshi Saxena
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Association Rule ◽  
Clustering Algorithm ◽  
Synthetic Data ◽  
Rule Mining ◽  
Data Set ◽  
Matching Technique ◽  
Burst Data

The objective of the work being presented is to propose an approach for obtaining appropriate association rules when the data set is being incrementally updated. During this process raw data is clustered by K-mean Clustering Algorithm and appropriate rules are generated for each cluster. Further, a histogram and probability density function are also generated for each cluster. When Burst data set is coming to the system, initially the histogram and probability density function of this new data set are obtained. The new data set has to be added to the cluster whose histogram and probability density functions are almost similar. The proposed method is evaluated and explained on synthetic data.

Empirical Analyses of Income: Finland (2009) and Australia (1967-1968)

2021 ◽  
pp. 35-53
Author(s):  
Johan Fellman
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Lorenz Curve ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Income Data ◽  
Difference Quotients ◽  
The Lorenz Curve

Analyses of income data are often based on assumptions concerning theoretical distributions. In this study, we apply statistical analyses, but ignore specific distribution models. The main income data sets considered in this study are taxable income in Finland (2009) and household income in Australia (1967-1968). Our intention is to compare statistical analyses performed without assumptions of the theoretical models with earlier results based on specific models. We have presented the central objects, probability density function, cumulative distribution function, the Lorenz curve, the derivative of the Lorenz curve, the Gini index and the Pietra index. The trapezium rule, Simpson´s rule, the regression model and the difference quotients yield comparable results for the Finnish data, but for the Australian data the differences are marked. For the Australian data, the discrepancies are caused by limited data. JEL classification numbers: D31, D63, E64. Keywords: Cumulative distribution function, Probability density function, Mean, quantiles, Lorenz curve, Gini coefficient, Pietra index, Robin Hood index, Trapezium rule, Simpson´s rule, Regression models, Difference quotients, Derivative of Lorenz curve

scholarly journals A NOTE CONCERNING THE DISTANCES OF UNIFORMLY DISTRIBUTED POINTS FROM THE CENTRE OF A RECTANGLE

2012 ◽  
Vol 87 (1) ◽  
pp. 115-119 ◽  
Author(s):  
ROBERT STEWART ◽  
HONG ZHANG
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Cumulative Distribution Function ◽  
Average Distance ◽  
Cumulative Distribution

AbstractGiven a rectangle containing uniformly distributed random points, how far are the points from the rectangle’s centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.

Sign in / Sign up

Export Citation Format

Share Document