scholarly journals The Novel Bivariate Distribution: Statistical Properties and Real Data Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
B. I. Mohammed ◽  
Abdulaziz S. Alghamdi ◽  
Hassan M. Aljohani ◽  
Md. Moyazzem Hossain
Keyword(s):  
Negative Binomial ◽  
Real Data ◽  
Information Criterion ◽  
Mass Function ◽  
Bivariate Distribution ◽  
Statistical Properties ◽  
Model Parameters ◽  
Probability Mass Function ◽  
New Class ◽  
Probability Mass

This article proposes a novel class of bivariate distributions that are completely defined by stating their conditionals as Poisson exponential distributions. Numerous statistical properties of this distribution are also examined here, including the conditional probability mass function (PMF) and moments of the new class. The techniques of maximum likelihood and pseudolikelihood are used to estimate the model parameters. Additionally, the effectiveness of the bivariate Poisson exponential conditional (BPEC) distribution is compared to that of the bivariate Poisson conditional (BPC), the bivariate Poisson (BP), the bivariate Poisson–Lindley (BPL), and the bivariate negative binomial (BNB) distributions using a real-world dataset. The findings of Akaike information criterion (AIC) and Bayesian information criterion (BIC) reveal that the BPEC distribution performs better than the other distributions considered in this study. As a result, the authors claim that this distribution may be used to fit dependent and overspread count data.

scholarly journals A discrete Ramos-Louzada distribution for asymmetric and over-dispersed data with leptokurtic-shaped: Properties and various estimation techniques with inference

2022 ◽  
Vol 7 (2) ◽  
pp. 1726-1741
Author(s):  
Ahmed Sedky Eldeeb ◽  
◽  
Muhammad Ahsan-ul-Haq ◽  
Mohamed. S. Eliwa ◽  
◽  
...  
Keyword(s):  
Count Data ◽  
Discrete Distribution ◽  
Real Data ◽  
Mass Function ◽  
Probability Mass Function ◽  
Data Sets ◽  
Estimation Techniques ◽  
Proposed Model ◽  
Probability Mass ◽  
Von Mises

<abstract> <p>In this paper, a flexible probability mass function is proposed for modeling count data, especially, asymmetric, and over-dispersed observations. Some of its distributional properties are investigated. It is found that all its statistical and reliability properties can be expressed in explicit forms which makes the proposed model useful in time series and regression analysis. Different estimation approaches including maximum likelihood, moments, least squares, Andersonӳ-Darling, Cramer von-Mises, and maximum product of spacing estimator, are derived to get the best estimator for the real data. The estimation performance of these estimation techniques is assessed via a comprehensive simulation study. The flexibility of the new discrete distribution is assessed using four distinctive real data sets ԣoronavirus-flood peaks-forest fire-Leukemia? Finally, the new probabilistic model can serve as an alternative distribution to other competitive distributions available in the literature for modeling count data.</p> </abstract>

Generalized Hypergeometric Factorial Moment Distribution of Order k

2000 ◽  
Vol 50 (1-2) ◽  
pp. 71-78 ◽  
Author(s):  
C. Satheesh Kumar ◽  
T. S. K. Moothathu
Keyword(s):  
Negative Binomial ◽  
Factorial Moment ◽  
Mass Function ◽  
Discrete Distributions ◽  
Probability Mass Function ◽  
Factorial Moments ◽  
Point Distribution ◽  
Moment Distribution ◽  
Special Cases ◽  
Probability Mass

Here we introduce the generalized hypergeometric functional moment distribution of order k (GHFMD (k)) in the distribution of the random sum [Formula: see text] having Hira no's k-point distribution, where N, independent of X j's, has the generalized hypergeomet ric factorial moment distribution. Well-known discrete distributions of order k such as cluster binomial, cluster negative binomial and extended Poisson are shown to be special cases of GHFMD(k). The probability mass function, recurrence relations for probabilities and factorial moments of GHFMD (k) are found out. The beta or the gamma mixture of GHFMD (k) is shown to be a GHFMD (k). Finally GHFMD (k) is obtained as a limit of another GHFMD (k). AMS (2000) Subject Classification: Primary 60E05, 60E10; Secondary 33C20.

scholarly journals Nonparametric estimation for probability mass function with Disake: an R package for discrete associated kernel estimators

2015 ◽  
Vol Volume 19 - 2015 - Special... ◽  
Author(s):  
W.E. Wansouwé ◽  
C.C. Kokonendji ◽  
D.T. Kolyang
Keyword(s):  
Kernel Smoothing ◽  
Simulated Data ◽  
Kernel Estimator ◽  
Real Data ◽  
R Package ◽  
Mass Function ◽  
Kernel Estimators ◽  
Probability Mass Function ◽  
Probability Mass ◽  
Discrete Associated Kernel

International audience Kernel smoothing is one of the most widely used nonparametric data smoothing techniques. We introduce a new R package, Disake, for computing discrete associated kernel estimators for probability mass function. When working with a kernel estimator, two choices must be made: the kernel function and the smoothing parameter. The Disake package focuses on discrete associated kernels and also on cross-validation and local Bayesian techniques to select the appropriate bandwidth. Applications on simulated data and real data show that the binomial kernel is appropriate for small or moderate count data while the empirical estimator or the discrete triangular kernel is indicated for large samples.

scholarly journals Bivariate exponentiated discrete Weibull distribution: statistical properties, estimation, simulation and applications

2019 ◽  
Vol 14 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. El- Morshedy ◽  
M. S. Eliwa ◽  
A. El-Gohary ◽  
A. A. Khalil
Keyword(s):  
Weibull Distribution ◽  
Joint Probability ◽  
Discrete Distribution ◽  
Real Data ◽  
Reliability Function ◽  
Statistical Properties ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Model Parameters ◽  
Probability Mass Function

AbstractIn this paper, a new bivariate discrete distribution is defined and studied in-detail, in the so-called the bivariate exponentiated discrete Weibull distribution. Several of its statistical properties including the joint cumulative distribution function, joint probability mass function, joint hazard rate function, joint moment generating function, mathematical expectation and reliability function for stress–strength model are derived. Its marginals are exponentiated discrete Weibull distributions. Hence, these marginals can be used to analyze the hazard rates in the discrete cases. The model parameters are estimated using the maximum likelihood method. Simulation study is performed to discuss the bias and mean square error of the estimators. Finally, two real data sets are analyzed to illustrate the flexibility of the proposed model.

scholarly journals COMPOUND RANDOM VARIABLES

2004 ◽  
Vol 18 (4) ◽  
pp. 473-484 ◽  
Author(s):  
Erol Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Arbitrary Function ◽  
Negative Binomial ◽  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Probabilistic Proof ◽  
Recursive Formulas ◽  
Probability Mass ◽  
Positive Compound

We give a probabilistic proof of an identity concerning the expectation of an arbitrary function of a compound random variable and then use this identity to obtain recursive formulas for the probability mass function of compound random variables when the compounding distribution is Poisson, binomial, negative binomial random, hypergeometric, logarithmic, or negative hypergeometric. We then show how to use simulation to efficiently estimate both the probability that a positive compound random variable is greater than a specified constant and the expected amount by which it exceeds that constant.

scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

Quantification of Low-Cycle Fatigue Life for a Gas Turbine Compressor Vane Carrier Under Varying Operating Conditions

2020 ◽  
Author(s):  
Zixi Han ◽  
Zixian Jiang ◽  
Sophie Ehrt ◽  
Mian Li
Keyword(s):  
Low Cycle Fatigue ◽  
Computational Cost ◽  
Mass Function ◽  
Operating Conditions ◽  
Probability Mass Function ◽  
Assessment Approach ◽  
Probability Mass ◽  
Gas Turbine Compressor ◽  
Operating Cycles ◽  
Operational Data

Abstract The design of a gas turbine compressor vane carrier (CVC) should meet mechanical integrity requirements on, among others, low-cycle fatigue (LCF). The number of cycles to the LCF failure is the result of cyclic mechanical and thermal strain effects caused by operating conditions on the components. The conventional LCF assessment is usually based on the assumption on standard operating cycles — supplemented by the consideration of predefined extreme operations and safety factors to compensate a potential underestimate on the LCF damage caused by multiple reasons such as non-standard operating cycles. However, real operating cycles can vary significantly from those standard ones considered in the conventional methods. The conventional prediction of LCF life can be very different from real cases, due to the included safety margins. This work presents a probabilistic method to estimate the distributions of the LCF life under varying operating conditions using operational fleet data. Finite element analysis (FEA) results indicate that the first ramp-up loading in each cycle and the turning time before hot-restart cycles are two predominant contributors to the LCF damage. A surrogate model of LCF damage has been built with regard to these two features to reduce the computational cost of FEA. Miner’s rule is applied to calculate the accumulated LCF damage on the component and then obtain the LCF life. The proposed LCF assessment approach has two special points. First, a new data processing technique inspired by the cumulative sum (CUSUM) control chart is proposed to identify the first ramp-up period of each cycle from noised operational data. Second, the probability mass function of the LCF life for a CVC is estimated using the sequential convolution of the single-cycle damage distribution obtained from operational data. The result from the proposed method shows that the mean value of the LCF life at a critical location of the CVC is significantly larger than the calculated result from the deterministic assessment, and the LCF lives for different gas turbines of the same class are also very different. Finally, to avoid high computational cost of sequential convolution, a quick approximation approach for the probability mass function of the LCF life is given. With the capability of dealing with varying operating conditions and noises in the operational data, the enhanced LCF assessment approach proposed in this work provides a probabilistic reference both for reliability analysis in CVC design, and for predictive maintenance in after-sales service.

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

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