Search Results (15)

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown. Full article

We use house prices (HP) and house price indices (HPI) as a proxy to income distribution. Specifically, we analyze distribution of sale prices in the 1970–2010 window of over 116,000 single-family homes in Hamilton County, Ohio, including Cincinnati metro area of about 2.2 million people. We also analyze distributions of HPI, published by Federal Housing Finance Agency (FHFA), for nearly 18,000 US ZIP codes that cover a period of over 40 years starting in 1980’s. If HP can be viewed as a first derivative of income, HPI can be viewed as its second derivative. We use generalized beta (GB) family of functions to fit distributions of HP and HPI since GB naturally arises from the models of economic exchange described by stochastic differential equations. Our main finding is that HP and multi-year HPI exhibit a negative Dragon King (nDK) behavior, wherein power-law distribution tail gives way to an abrupt decay to a finite upper limit value, which is similar to our recent findings for realized volatility of S&P500 index in the US stock market. This type of tail behavior is best fitted by a modified GB (mGB) distribution. Tails of single-year HPI appear to show more consistency with power-law behavior, which is better described by a GB Prime (GB2) distribution. We supplement full distribution fits by mGB and GB2 with direct linear fits (LF) of the tails. Our numerical procedure relies on evaluation of confidence intervals (CI) of the fits, as well as of p-values that give the likelihood that data come from the fitted distributions. Full article

In this study, we undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of the realized volatility (RV) of the S&P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is, they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with $RV>40$, we observe the appearance of “potential” DK, which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase in the number of days over which the average RV is calculated—here from daily, $n=1$, to “monthly”, $n=21$. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as a Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check whether they can come from the respective distributions. Full article

Inverted distributions, also known as inverse distributions, are essential statistical models for analyzing real-life data in biomedical sciences, engineering, and other fields. In this paper, we use the odd beta prime-G family and the inverted Kumaraswamy distribution to introduce a new inverted distribution called the odd beta prime inverted Kumaraswamy. The new distribution exhibits right-skewed, J-shaped densities and features increasing-constant, concave-convex, and bathtub hazard functions. Some of its statistical properties are explored. The parameters are estimated via the maximum likelihood method. The empirical importance of the new model is proved through its application to COVID-19 mortality data from Italy. Numerical results demonstrate that the proposed model outperforms its competitors. We hope that this proposed distribution can be considered as a viable alternative to some well-established distributions for modeling real-life data across various application areas. Full article

In this article, we pioneer a new Burr X distribution using the odd beta prime generalized (OBP-G) family of distributions called the OBP-Burr X (OBPBX) distribution. The density function of this model is symmetric, left-skewed, right-skewed, and reversed-J, while the hazard function is monotonically increasing, decreasing, bathtub, and N-shaped, making it suitable for modeling skewed data and failure rates. Various statistical properties of the new model are obtained, such as moments, moment-generating function, entropies, quantile function, and limit behavior. The maximum-likelihood-estimation procedure is utilized to determine the parameters of the model. A Monte Carlo simulation study is implemented to ascertain the efficiency of maximum-likelihood estimators. The findings demonstrate the empirical application and flexibility of the OBPBX distribution, as showcased through its analysis of petroleum rock samples and COVID-19 mortality data, along with its superior performance compared to well-known extended versions of the Burr X distribution. We anticipate that the new distribution will attract a wider readership and provide a vital tool for modeling various phenomena in different domains. Full article

Global and local biological motion processing are likely influenced by an observer’s perceptual experience. Skilled athletes anticipating an opponent’s movements use globally distributed motion information, while less skilled athletes focus on single kinematic cues. Published reports have demonstrated that attention can be primed globally or locally before perceptual tasks; such an intervention could highlight motion processing mechanisms used by skilled and less skilled observers. In this study, we examined skill differences in biological motion processing using attentional priming. Skilled (N = 16) and less skilled (N = 16) players anticipated temporally occluded videos of volleyball attacks after being primed using a Navon matching task while parietal EEG was measured. Skilled players were more accurate than less skilled players across priming conditions. Global priming improved performance in both skill groups. Skilled players showed significantly reduced alpha and beta power in the right compared to left parietal region, but brain activity was not affected by the priming interventions. Our findings highlight the importance of right parietal dominance for skilled performers, which may be functional for inhibiting left hemispheric local processing or enhancing visual spatial attention for dynamic visual scenes. Further work is needed to systematically determine the function of this pattern of brain activity during skilled anticipation. Full article

In parametric statistical modeling, it is important to construct new extensions of existing probability distributions (PDs) that can make modeling data more flexible and help stakeholders make better decisions. In the present study, a new family of probability distributions (FPDs) called the odd beta prime generalized (OBP-G) FPDs is proposed to improve the traditional PDs. A new PD called the odd beta prime-logistic (OBP-logistic) distribution has been developed based on the developed OBP-G FPDs. Some desirable mathematical properties of the proposed OBP-logistic distribution, including the moments, moment-generating function, information-generating function, quantile function, stress–strength, order statistics, and entropies, are studied and derived. The proposed OBP-logistic distribution’s parameters are determined by adopting the maximum likelihood estimation (MLE) method. The applicability of the new PD was demonstrated by employing three data sets and these were compared by the known extended logistic distributions, such as the gamma generalized logistic distribution, new modified exponential logistic distribution, gamma-logistic distribution, exponential modified Weibull logistic distribution, exponentiated Weibull logistic distribution, and transmuted Weibull logistic distribution. The findings reveal that the studied distribution provides better results than the competing PDs. The empirical results showed that the new OBP-logistic distribution performs better than the other PDs based on several statistical metrics. We hoped that the newly constructed OBP-logistic distribution would be an alternative to other well-known extended logistic distributions for the statistical modeling of symmetric and skewed data sets. Full article

In this work, a novel generalized family of distributions called the odd beta prime is introduced. The linear representations of the proposed family are obtained. The expressions for the moments, the moment-generating function, and entropy are derived. A three-parameter special sub-model of the proposed family called the odd beta prime exponential distribution is proposed. Finally, two real data sets are used to illustrate the usefulness and flexibility of the proposed distribution. Full article

This manuscript presents three families of distributions, namely the Beta, Beta Prime and Beta Exponential families of distributions. From all the distributions of these families, 14 statistical distributions of three, four and five parameters are presented that have applicability in the analysis of extreme phenomena in hydrology. These families of distributions were analyzed regarding the improvement of the existing legislation for the determination of extreme events, specifically the elaboration of a norm regarding frequency analysis in hydrology. To estimate the parameters of the analyzed distributions, the method of ordinary moments and the method of linear moments were used; the latter conforms to the current trend for estimating the parameters of statistical distributions. The main purpose of the manuscript was to identify other distributions from these three families with applicability in flood frequency analysis compared to the distributions already used in the literature from these families, such as the Log–logistic distribution, the Dagum distribution and the Kumaraswamy distribution. The manuscript does not exclude the applicability of other distributions from other families in the frequency analysis of extreme values, especially since these families were also analyzed within the research carried out in the Faculty of Hydrotechnics and presented in other materials. All the necessary elements for their use are presented, including the probability density functions, the complementary cumulative distribution functions, the quantile functions and the exact and approximate relations for estimating parameters. A flood frequency analysis case study was carried out for the Prigor RiverRiver, to numerically present the proposed distributions. The performance of this distributions were evaluated using the relative mean error, the relative absolute error and the L-skewness–L-kurtosis diagram. The best fit distributions are the Kumaraswamy, the Generalized Beta Exponential and the Generalized Beta distributions, which presented a stability related to both the length of the data and the presence of outliers. Full article

Motivated by the recent popularization of the beta prime distribution, a more flexible generalization is presented to fit symmetrical or asymmetrical and bimodal data, and a non-monotonic failure rate. Thus, the Weibull-beta prime distribution is defined, and some of its structural properties are obtained. The parameters are estimated by maximum likelihood, and a new regression model is proposed. Some simulations reveal that the estimators are consistent, and applications to censored COVID-19 data show the adequacy of the models. Full article

This article introduces exponentiated transmuted-inverted beta (ET-IB) distribution, supported by a continuous positive real line, as a synthetic aperture radar (SAR) imagery descriptor. It is an extension of the inverted beta distribution, an important texture model for SAR imagery. The considered distribution extension approach increases the flexibility of the baseline distribution, and is a new probabilistic model useful in SAR image applications. Besides introducing the new model, the maximum likelihood method is discussed for parameter estimation. Numerical experiments are performed to validate the use of the ET-IB distribution as a SAR amplitude image descriptor. Finally, three measured SAR images referring to forest, ocean, and urban regions are considered, and the performance of the proposed distribution is compared to distributions usually considered in this field. The proposed distribution outperforms the competitor models for modeling SAR images in terms of some selected goodness-of-fit measures. The results show that the ET-IB distribution is suitable as a SAR descriptor and can be used to develop image-processing tools in remote sensing applications. Full article

In this paper, we propose a new family of probability laws based on the Generalized Beta Prime distribution to evaluate the relative accuracy between two Lagrange finite elements $P_{k_1}$ and $P_{k_2},(k_1<k_2)$. Usually, the relative finite element accuracy is based on the comparison of the asymptotic speed of convergence, when the mesh size h goes to zero. The new probability laws we propose here highlight that there exists, depending on h, cases where the $P_{k_1}$ finite element is more likely accurate than the $P_{k_2}$ element. To confirm this assertion, we highlight, using numerical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities, as determined by the probability law. This illustrates that, when h goes away from zero, a finite element $P_{k_1}$ may produce more precise results than a finite element $P_{k_2}$, since the probability of the event “$P_{k_1}$is more accurate than$P_{k_2}$” becomes greater than 0.5. In these cases, finite element $P_{k_2}$ is more likely overqualified. Full article

Despite the enormous literature documenting the importance of amyloid beta (Ab) protein in Alzheimer's disease, we do not know how Ab aggregation is initiated and why it has its unique distribution in the brain. In vivo and in vitro evidence has been developed to suggest that functional microbial amyloid proteins produced in the gut may cross-seed Ab aggregation and prime the innate immune system to have an enhanced and pathogenic response to neuronal amyloids. In this commentary, we summarize the molecular mechanisms by which the microbiota may initiate and sustain the pathogenic processes of neurodegeneration in aging. Full article

There exist several estimators of the regression line in the simple linear regression: Least Squares, Least Absolute Deviation, Right Median, Theil–Sen, Weighted Balance, and Least Trimmed Squares. Their performance for heavy tails is compared below on the basis of a quadratic loss function. The case where the explanatory variable is the inverse of a standard uniform variable and where the error has a Cauchy distribution plays a central role, but heavier and lighter tails are also considered. Tables list the empirical sd and bias for ten batches of one hundred thousand simulations when the explanatory variable has a Pareto distribution and the error has a symmetric Student distribution or a one-sided Pareto distribution for various tail indices. The results in the tables may be used as benchmarks. The sample size is $n=100$ but results for $n=\infty$ are also presented. The error in the estimate of the slope tneed not be asymptotically normal. For symmetric errors, the symmetric generalized beta prime densities often give a good fit. Full article

Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and \(q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2\). For the binomial distribution of \(n\) one obtains \(q=1-1/k\), for the negative binomial \(q=1+1/(k+1)\). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion \(\omega \ll E\). For general systems the average phase-space volume ratio \(\langle e^{S(E-\omega)}/e^{S(E)}\rangle\) to second order delivers \(q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2\) with \(\beta=S^{\prime}(E)\) and \(C=dE/dT\) heat capacity. However, \(q \ne 1\) leads to non-additivity of the Boltzmann–Gibbs entropy, \(S\). We demonstrate that a deformed entropy, \(K(S)\), can be constructed and used for demanding additivity, i.e., \(q_K=1\). This requirement leads to a second order differential equation for \(K(S)\). Finally, the generalized \(q\)-entropy formula, \(K(S)=\sum p_i K(-\ln p_i)\), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, \(\Delta\beta^2\) we obtain a novel entropy formula. Full article