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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.

Long-term dependence is an essential feature for the predictability of time series. Estimating the parameter that describes long memory is essential to describing the behavior of time series models. However, most long memory estimation methods assume that this parameter has a constant value throughout the time series, and do not consider that the parameter may change over time. In this work, we propose an automated methodology that combines the estimation methodologies of the fractional differentiation parameter (and/or Hurst parameter) with its application to Recurrent Neural Networks (RNNs) in order for said networks to learn and predict long memory dependencies from information obtained in nonlinear time series. The proposal combines three methods that allow for better approximation in the prediction of the values of the parameters for each one of the windows obtained, using Recurrent Neural Networks as an adaptive method to learn and predict the dependencies of long memory in Time Series. For the RNNs, we have evaluated four different architectures: the Simple RNN, LSTM, the BiLSTM, and the GRU. These models are built from blocks with gates controlling the cell state and memory. We have evaluated the proposed approach using both synthetic and real-world data sets. We have simulated ARFIMA models for the synthetic data to generate several time series by varying the fractional differentiation parameter. We have evaluated the proposed approach using synthetic and real datasets using Whittle’s estimates of the Hurst parameter classically obtained in each window. We have simulated ARFIMA models in such a way that the synthetic data generate several time series by varying the fractional differentiation parameter. The real-world IPSA stock option index and Tree Ringtime series datasets were evaluated. All of the results show that the proposed approach can predict the Hurst exponent with good performance by selecting the optimal window size and overlap change.

Missing or unavailable data (NA) in multivariate data analysis is often treated with imputation methods and, in some cases, records containing NA are eliminated, leading to the loss of information. This paper addresses the problem of NA in multiple factor analysis (MFA) without resorting to eliminating records or using imputation techniques. For this purpose, the nonlinear iterative partial least squares (NIPALS) algorithm is proposed based on the principle of available data. NIPALS presents a good alternative when data imputation is not feasible. Our proposed method is called MFA-NIPALS and, based on simulation scenarios, we recommend its use until 15% of NAs of total observations. A case of groups of quantitative variables is studied and the proposed NIPALS algorithm is compared with the regularized iterative MFA algorithm for several percentages of NA.

In this paper, a new class of the continuous distributions is established via compounding the arctangent function with a generalized log-logistic class of distributions. Some structural properties of the suggested model such as distribution function, hazard function, quantile function, asymptotics and a useful expansion for the new class are given in a general setting. Two special cases of this new class are considered by employing Weibull and normal distributions as the parent distribution. Further, we derive a survival regression model based on a sub-model with Weibull parent distribution and then estimate the parameters of the proposed regression model making use of Bayesian and frequentist approaches. We consider seven loss functions, namely the squared error, modified squared error, weighted squared error, K-loss, linear exponential, general entropy, and precautionary loss functions for Bayesian discussion. Bayesian numerical results include a Bayes estimator, associated posterior risk, credible and highest posterior density intervals are provided. In order to explore the consistency property of the maximum likelihood estimators, a simulation study is presented via Monte Carlo procedure. The parameters of two sub-models are estimated with maximum likelihood and the usefulness of these sub-models and a proposed survival regression model is examined by means of three real datasets.

There remains a lack of holistic approaches for analyzing how different density-independent and density-dependent (endogenous) mechanisms interact to drive the dynamics of the small pelagic fish populations of the southern Humboldt Current ecosystem. In this study, we analyzed the drivers of the small pelagic fishes (SPF) off the coast of Chile from the late 1980s until the early 2010s. We postulated that climate, fishing, and endogenous effects drove the biomass dynamics of these populations. Per capita growth rates (R models) were used to investigate how these factors regulated the dynamics of three anchovy populations and one population of common sardine (CS) off the Chilean coast. We found that the dynamics of the anchovy populations located off northern Chile were driven by endogenous components and by the effects of the climate, fishing, and the climate–fishing interaction. We proposed that during the study period, the climate conditions favored the population growth of the anchovies in the north; however, fishing had a negative effect on anchovy biomass, which was facilitated by the climate. The dynamics of the SPF off central-southern Chile showed weaker endogenous effects. Indeed, the anchovy population displayed the lowest density-dependent effect, and fishing played the most significant role. The endogenous effect on the CS was slightly higher in comparison to that on the anchovy; however, climate [sea surface temperature (SST)] seemed to be the main driver of the flourishment in the CS biomass following 2006, which supported the previous hypothesis regarding the effect of climate on the species. We discussed that the R models approach could be used to provide a holistic understanding of the drivers of the biomass dynamics of these populations. The approach provided a framework for integrating climate variability in the population dynamics of these species and moving toward an ecosystem approach to fisheries management. Further steps involve exploring the effects of competition and predation on the population dynamics of these species.

In this paper, we provide a new bivariate distribution obtained from a Kibble-type bivariate gamma distribution. The stochastic representation was obtained by the sum of a Kibble-type bivariate random vector and a bivariate random vector builded by two independent gamma random variables. In addition, the resulting bivariate density considers an infinite series of products of two confluent hypergeometric functions. In particular, we derive the probability and cumulative distribution functions, the moment generation and characteristic functions, the Hazard, Bonferroni and Lorenz functions, and an approximation for the differential entropy and mutual information index. Numerical examples showed the behavior of exact and approximated expressions.

In several applications, the assumption of normality is often violated in data with some level of skewness, so skewness affects the mean’s estimation. The class of skew–normal distributions is considered, given their flexibility for modeling data with asymmetry parameter. In this paper, we considered two location parameter (μ) estimation methods in the skew–normal setting, where the coefficient of variation and the skewness parameter are known. Specifically, the least square estimator (LSE) and the best unbiased estimator (BUE) for μ are considered. The properties for BUE (which dominates LSE) using classic theorems of information theory are explored, which provides a way to measure the uncertainty of location parameter estimations. Specifically, inequalities based on convexity property enable obtaining lower and upper bounds for differential entropy and Fisher information. Some simulations illustrate the behavior of differential entropy and Fisher information bounds.

We show statistical evidence that pension fund withdrawals and the Emergency Family Income (EFI) increased the likelihood that a laid off construction worker would reject a proposal for a formal employment contract. This favors the hypothesis that pension fund withdrawals and government subsidies related to the health crisis have, to some extent, contributed to the shortage of formal labor in the construction sector. Based on estimations of the logit model, we found that rejection probability increased with work experience (approximated by the worker’s age). For example, the probability of not accepting a formal contract for a highly experienced worker, who withdrew funds from their mandatory private Pension Fund Administrator and received the EFI, increased by 28%. The figure is approximately 2.5 times the rejection probability of a worker with the same experience but without having received this additional income.

In this paper, we approached the concept of real estate bubble, analyzing the risk its bursting could generate for the Chilean financial market. Specifically, we analyzed the relationship between real housing prices, the economic activity index, and mortgage interest rates denominated in inflation-linked units from 1994 to 2020. The analysis was based on a second order Markov switching model with the predetermined variables mentioned later, whose parameters were obtained through the expectation–maximization algorithm. Then, we built a probability index as early warning indicator for potential imbalances in the real estate price that could put financial market stability at risk. The indicator is important to evaluate economic policy calibrations in time. A main finding was that the real housing price had a non-linear relationship with economic activity and the mortgage interest rate. Therefore, the evolution of the real estate price has been consistent with fundamental macroeconomic variables, even under a high growth regime, with increases above 12% per year. About 92% of housing price variability derived from changing macrofinancial conditions, suggesting a low margin of speculative behavior.

The copper price is a leading indicator of real estate activity. Price increases are statistically related to increasing numbers of applications for residential building permits. However, this reciprocity is not instantaneous as permit numbers lag price rises by 9 to 10 months. This dynamic is implicit in various transmission channels: from the first effects on investment plans and demand for durable goods due to better expectations from investors and consumers to the real impact of higher copper revenues on the economy’s aggregate production and demand (multiplier or second-round effect). In this paper, we proposed the impulse-response functions of a vector autoregressive model to capture the dynamic between the copper price and house building permits. Therefore, it would be expected that the recent copper price increase will boost construction and real estate activity. The effects could materialize this year and extend into early 2022.

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented.

The southern Humboldt Current ecosystem is an important topic among researchers working on the drivers of pelagic species’ biological indicators. While sea surface temperature is believed to be a major driver in anchovies’ (Engraulis ringens) reproductive and body condition indicators, this paper shows that regional drivers such as Pacific decadal oscillation anomalies also influence these biological processes. In addition, a warm condition could trigger increased gonad development of anchovies and synchronization of body condition dynamics with local environmental conditions stemming from sea turbulence and Ekman transport. To test the statistical significance of causality between two time series and determine the direction of causality, the frequency-domain Granger-causality method is considered. Therefore, this study provides additional predictive information, derived from past data on anchovy reproductive and feeding activities. The study could be useful for researchers working on relationships of environmental conditions and pelagic species to predict biological processes’ maximum and minimum peak movements and anchovy abundance in the southern Humboldt Current ecosystem.

Section 3.3 of “Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy2016, 18, 382” contains errors. Therefore, this section is retracted. However, these changes do not influence the conclusions and the other results of the paper.

Sea urchin (Loxechinus albus) is one of the most economically important species in the northeast of Chilean Patagonia, forming part of the highly diverse benthic community. This resource is being harvested under selective fishing pressure, which suppresses growth rates. In response, the National Standards Institute established Regulation 44 as a quality control measure, based on a sampling approach considering the percentage of sea urchin under the minimum legal size (UMLS) through an acceptance sampling plan. In 2019, the Chilean Undersecretary of Fisheries and Aquaculture (SUBPESCA), under Exempt Resolution No. 910, ruled that any vessel that landed 15% with sea urchin of 6.5 cm UMLS of the total catch weight will be sanctioned. In this paper, a sampling design is presented to establish criteria regarding compliance with this provision. The plan provides a statistical methodology for the evaluation of the percentage of UMLS sea urchin, which corroborates and certifies the current legislation that sanctions and fines the capture of sea urchin that does not comply with UMLS values outlined in Regulation 44. Assuming a systematic sampling plan and under assumed UMLS values of 6.0, 6.5, and 7.0 cm, it is determined that 15% of total landings (margin of tolerance) of sea urchin under 6.5 cm is the optimal level. Thus, a correct specification of the UMLS allows protecting the most vulnerable fraction of the population from overexploitation for growth.

The aim of this work was to extend the results of Perez et al. (Physica A (2006), 365 (2), 282−288) to the two-dimensional (2D) fractional Brownian field. In particular, we defined Shannon entropy using the wavelet spectrum from which the Hurst exponent is estimated by the regression of the logarithm of the square coefficients over the levels of resolutions. Using the same methodology. we also defined two other entropies in 2D: Tsallis and the Rényi entropies. A simulation study was performed for showing the ability of the method to characterize 2D (in this case, α = 2 ) self-similar processes.

The Skew-Reflected-Gompertz (SRG) distribution, introduced by Hosseinzadeh et al. (J. Comput. Appl. Math. (2019) 349, 132–141), produces two-piece asymmetric behavior of the Gompertz (GZ) distribution, which extends the positive to a whole dominion by an extra parameter. The SRG distribution also permits a better fit than its well-known classical competitors, namely the skew-normal and epsilon-skew-normal distributions, for data with a high presence of skewness. In this paper, we study information quantifiers such as Shannon and Rényi entropies, and Kullback–Leibler divergence in terms of exact expressions of GZ information measures. We find the asymptotic test useful to compare two SRG-distributed samples. Finally, as a real-world data example, we apply these results to South Pacific sea surface temperature records.

In this paper, we examine the finite mixture (FM) model with a flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family components. This family allows the development of a robust estimation of FM models. The TP-SMN is a rich class of distributions that covers symmetric/asymmetric and light/heavy tailed distributions. It represents an alternative family to the well-known scale mixtures of the skew normal (SMSN) family studied by Branco and Dey (2001). Also, the TP-SMN covers the SMN (normal, t, slash, and contaminated normal distributions) as the symmetric members and two-piece versions of them as asymmetric members. A key feature of this study is using a suitable hierarchical representation of the family to obtain maximum likelihood estimates of model parameters via an EM-type algorithm. The performances of the proposed robust model are demonstrated using simulated and real data, and then compared to other finite mixture of SMSN models.

The aim of this work is to provide the tools to compute the well-known Kullback–Leibler divergence measure for the flexible family of multivariate skew-normal distributions. In particular, we use the Jeffreys divergence measure to compare the multivariate normal distribution with the skew-multivariate normal distribution, showing that this is equivalent to comparing univariate versions of these distributions. Finally, we applied our results on a seismological catalogue data set related to the 2010 Maule earthquake. Specifically, we compare the distributions of the local magnitudes of the regions formed by the aftershocks.

This study addresses the problem of age determination of the southern king crab (Lithodes santolla). Given that recapture is difficult for this species and, thus, age cannot be directly determined with the help of the annual marks on the shell, the von Bertalanffy growth function (vBGF) cannot be used to directly model length-frequency data (LFD). To determine age classes, some researchers have proposed using the MIX algorithm that consists of sampling realization of a finite mixture of normal (FMN) distributions for each LFD. However, normality assumption in age-length data has been questioned in several works related to fish growth analysis. For this study, we considered the biological information of the southern king crab for the period 2007⁻2015 and localization between 50 ∘ 06 ′ ⁻ 53 ∘ 15 ′ S and 76 ∘ 36 ′ ⁻ 72 ∘ 18 ′ W. We assumed that LFD could be modelled by the novel class of finite mixture of skew-t (FMST). Assigned age classes were used to estimate the vBGF parameters. The estimated vBGF parameters were L ∞ = 176.756 cm, K = 0.151 year − 1 , t 0 = − 1.678 year for males, and L ∞ = 134.799 cm, K = 0.220 year − 1 , t 0 = − 1.302 year for females. This study concludes that (a) FMST modal decomposition can detect a group of younger individuals at age 2, given that those individuals have LFD with a left heavy-tail and asymmetry; (b) FMST produces a better representation of LFD than the FMN model; (c) males have bigger L ∞ but grow slower than females; and (d) as expected, a high correlation exists among the vBGF estimates.

In the last century, the growing evidence that global fisheries are depleting natural resources much faster than they can recover has led to negative processes, like overfishing, being addressed with increasingly complex models and thus mitigating or regulating actions that aim to protect stocks. Said negative processes contain two components: (i) they can diminish the reproductive potential of fish stocks, called over exploitation by recruitment, and (ii) the effect of early capture prevents the full realization of the growth potential, called overfishing by growth. In this article, the structure of common sardine sizes is analyzed. Due to the precise moment in which pre-recruits are incorporated into the exploited phase of the stock is unknown, the estimation of a recruitment size is a hard problem. This problem is addressed by modeling the mean size via a stochastic process, applying models of structural change. A time series (2001–2015) was analyzed on a weekly time scale based on the size structure of the pelagic fishery landings in the central-southern zone of Chile (32∘10′–40∘14′ LS), from the V. to the XIV. Region. Specifically, the evolution of sizes according to macro-zones was studied for the conglomerates identified in two sub-zones, the V.–VIII. and IX.–XIV. Regions. In this context, the reference size for juveniles to cautiously allow the recruitment process of the common sardine from the central-southern zone of Chile was estimated, and the behavior of these sizes was spatially analyzed. Finally, a statistical inferential criterion was established that confirms the mean size of juveniles with a certain margin of error, which allows nonetheless later on to define a fraction that could be protected to avoid overfishing by growth.

This note analyzes the effects forest fires in Chile have on vegetation and subsequent ecological restoration. We analyze why forest fires have been a main factor that affects the environment and causes the ecosystem to deteriorate, leading to loss of native forests, species extinction, damage to the urban population, and others. The data examined are derived from fire hotspots in Chile’s central and central-south zones ( 33 ∘ 00 ′ S– 41 ∘ 57 ′ S) between 1985 and 2017. We also analyze some key aspects for restoration priorities such as studying affected areas and posterior consequences. Finally, we evaluate actions the country has already taken, and propose further appropriate preventive and restoration measures.

The problem of measuring the disparity of a particular probability density function from a normal one has been addressed in several recent studies. The most used technique to deal with the problem has been exact expressions using information measures over particular distributions. In this paper, we consider a class of asymmetric distributions with a normal kernel, called Generalized Skew-Normal (GSN) distributions. We measure the degrees of disparity of these distributions from the normal distribution by using exact expressions for the GSN negentropy in terms of cumulants. Specifically, we focus on skew-normal and modified skew-normal distributions. Then, we establish the Kullback–Leibler divergences between each GSN distribution and the normal one in terms of their negentropies to develop hypothesis testing for normality. Finally, we apply this result to condition factor time series of anchovies off northern Chile.

Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, we consider a novel class of skew-normal mixture models, whose components capture skewness due to their flexibility. We find upper and lower bounds for Shannon and Rényi entropies for this model. Using such a pair of bounds, a confidence interval for the approximate entropy value can be calculated. In addition, an asymptotic expression for Rényi entropy by Stirling’s approximation is given, and upper and lower bounds are reported using multinomial coefficients and some properties and inequalities of L p metric spaces. Simulation studies are then applied to a swordfish (Xiphias gladius Linnaeus) length dataset.