Search Results (56)

We propose a new distribution which is the sum of two independent Akash (AK) random variables with the same parameter. We refer to this distribution as the AKS distribution. We study the density and some of its properties. We derive the method of moments (MM) and the maximum likelihood (ML) estimator, along with the Fisher information. The performance of the ML estimator is evaluated through a simulation study. In addition, we present two real data applications, demonstrating in these cases the superiority of the AKS distribution compared to two other distributions. Full article

The inversion of the ground stress field in rock masses is critical for accurate tunnel and underground engineering design. This study addresses the challenge of accurately capturing both the primary and secondary stress field components in rock masses. The ground stress field consists of the primary stress field, generated by applied tectonic loads, and a secondary stress field, which cannot be fully explained by these loads and is attributed to long-term tectonic processes. This unexplained secondary stress field is often non-random in nature. To improve the accuracy of the ground stress field inversion, we propose prioritizing the use of a regression model with a constant term. This model better accounts for the secondary stress field by capturing long-term tectonic influences. The constant term in the regression model is shown to represent the non-random secondary stress field, which cannot be explained by applied tectonic loads. Furthermore, we define two key conditions for applying this regression model: (1) the constant term should not exceed the maximum measured stress and preferably should not surpass the minimum measured stress, and (2) the residual sum of squares of the regression model with a constant term should be smaller than that of the model without a constant term. By incorporating the constant term, the model improves the representation of both primary and secondary stress fields, offering a more accurate inversion of the ground stress field, especially when the stress field contribution from independent variables is incomplete. Full article

In this work, we present a new distribution, which is a slash extension of the distribution of the sum of two independent Lindley random variables. This new distribution is developed using the slash methodology, resulting in a distribution with more flexible kurtosis, i.e., the ability to model atypical data. We study the density function of the new model and some of its properties, such as the cumulative distribution function, moments, and its asymmetry and kurtosis coefficients. The parameters are estimated by the maximum likelihood method with the EM algorithm. Finally, we apply the proposed model to two real datasets with high kurtosis, showing that it provides a better fit than two distributions known in the literature. Full article

The exponentiated exponential distribution has received great attention from many statisticians due to its popularity, many applications, and the fact that it is an efficient alternative to many famous distributions such as Weibull and gamma distributions. Many statisticians have studied the mathematical properties of this distribution and estimated its parameters under different censoring schemes. However, it seems that the distribution of the random sum, the distribution of the linear combination, and the value of the reliability index, $R=P(X_2<X_1)$, in the case of unequal scale parameters, were not known for this distribution. Therefore, in this article, we present the saddlepoint approximation to the distribution of the random sum, the distribution of linear combination, and the value of the reliability index $R=P(X_2<X_1)$ for exponentiated exponential variates. These saddlepoint approximations are computationally appealing, and numerical studies confirm their accuracy. In addition to the accuracy provided by the saddlepoint approximation method, it saves time compared to the simulation method, which requires a lot of time. Therefore, the saddlepoint approximation method provides an outstanding balance between precision and computational efficiency. Full article

Inequalities are obtained which connect the probability tails and moments of functions of the nth partial sums of independent random variables taking values in a separable Banach space and those for the accompanying infinitely divisible laws. Some applications to empirical processes are studied. Full article

Accurately assessing tree mortality probability in the context of global climate changes is important for formulating scientific and reasonable forest management scenarios. In this study, we developed a climate-sensitive individual tree mortality model for Masson pine using data from the seventh (2004), eighth (2009), and ninth (2014) Chinese National Forest Inventory (CNFI) in Hunan Province, South–Central China. A generalized linear mixed-effects model with plots as random effects based on logistic regression was applied. Additionally, a hierarchical partitioning analysis was used to disentangle the relative contributions of the variables. Among the various candidate predictors, the diameter (DBH), Gini coefficient (GC), sum of basal area for all trees larger than the subject tree (BAL), mean coldest monthly temperature (MCMT), and mean summer (May–September) precipitation (MSP) contributed significantly to changes in Masson pine mortality. The relative contribution of climate variables (MCMT and MSP) was 44.78%, larger than tree size (DBH, 32.74%), competition (BAL, 16.09%), and structure variables (GC, 6.39%). The model validation results based on independent data showed that the model performed well and suggested an influencing mechanism of tree mortality, which could improve the accuracy of forest management decisions under a changing climate. Full article

This paper investigates the randomly stopped sums, minima and maxima of heavy- and light-tailed random variables. The conditions on the primary random variables, which are independent but generally not identically distributed, and counting random variable are given in order that the randomly stopped sum, random minimum and maximum is heavy/light tailed. The results generalize some existing ones in the literature. The examples illustrating the results are provided. Full article

The sum of independent circular uniformly distributed random variables is also circular uniformly distributed. In this study, it is shown that a family of circular distributions based on nonnegative trigonometric sums (NNTS) is also closed under summation. Given the flexibility of NNTS circular distributions to model multimodality and skewness, these are good candidates for use as alternative models to test for circular uniformity to detect different deviations from the null hypothesis of circular uniformity. The circular uniform distribution is a member of the NNTS family, but in the NNTS parameter space, it corresponds to a point on the boundary of the parameter space, implying that the regularity conditions are not satisfied when the parameters are estimated by using the maximum likelihood method. Two NNTS tests for circular uniformity were developed by considering the standardised maximum likelihood estimator and the generalised likelihood ratio. Given the nonregularity condition, the critical values of the proposed NNTS circular uniformity tests were obtained via simulation and interpolated for any sample size by the fitting of regression models. The validity of the proposed NNTS circular uniformity tests was evaluated by generating NNTS models close to the circular uniformity null hypothesis. Full article

The abilities of quantitative description of noise are restricted due to its origin, and only statistical and spectral analysis methods can be applied, while an exact time evolution cannot be defined or predicted. This emphasizes the challenges faced in many applications, including communication systems, where noise can play, on the one hand, a vital role in impacting the signal-to-noise ratio, but possesses, on the other hand, unique properties such as an infinite entropy (infinite information capacity), an exponentially decaying correlation function, and so on. Despite the deterministic nature of chaotic systems, the predictability of chaotic signals is limited for a short time window, putting them close to random noise. In this article, we propose and experimentally verify an approach to achieve Gaussian-distributed chaotic signals by processing the outputs of chaotic systems. The mathematical criterion on which the main idea of this study is based on is the central limit theorem, which states that the sum of a large number of independent random variables with similar variances approaches a Gaussian distribution. This study involves more than 40 mostly three-dimensional continuous-time chaotic systems (Chua’s, Lorenz’s, Sprott’s, memristor-based, etc.), whose output signals are analyzed according to criteria that encompass the probability density functions of the chaotic signal itself, its envelope, and its phase and statistical and entropy-based metrics such as skewness, kurtosis, and entropy power. We found that two chaotic signals of Chua’s and Lorenz’s systems exhibited superior performance across the chosen metrics. Furthermore, our focus extended to determining the minimum number of independent chaotic signals necessary to yield a Gaussian-distributed combined signal. Thus, a statistical-characteristic-based algorithm, which includes a series of tests, was developed for a Gaussian-like signal assessment. Following the algorithm, the analytic and experimental results indicate that the sum of at least three non-Gaussian chaotic signals closely approximates a Gaussian distribution. This allows for the generation of reproducible Gaussian-distributed deterministic chaos by modeling simple chaotic systems. Full article

We study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an optimal strategy to maximise the expected reward. In our previous work, we presented a methodology, borrowing techniques from applied mathematics, for obtaining asymptotic expressions for the expectation duration of the optimal stopping time where one stop is permitted. In this study, we generalise further to the case where more than one stop is permitted, with an updated objective function of maximising the expected sum of the variables chosen. We formulate a complete generalisation for an exponential family as well as the uniform distribution by utilising an inductive approach in the formulation of the stopping rule. Explicit examples are shown for common probability functions as well as simulations to verify the asymptotic calculations. Full article

In this paper, we find conditions under which distribution functions of randomly stopped minimum, maximum, minimum of sums and maximum of sums belong to the class of generalized subexponential distributions. The results presented in this article complement the closure properties of randomly stopped sums considered in the authors’ previous work. In this work, as in the previous one, the primary random variables are supposed to be independent and real-valued, but not necessarily identically distributed. The counting random variable describing the stopping moment of random structures is supposed to be nonnegative, integer-valued and not degenerate at zero. In addition, it is supposed that counting random variable and the sequence of the primary random variables are independent. At the end of the paper, it is demonstrated how randomly stopped structures can be applied to the construction of new generalized subexponential distributions. Full article

The central limit theorem states that, in the limits of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to attain a stable distribution. The condition of independence, however, only holds in real systems as an approximation. To extend the theorem to more general situations, previous studies have derived a version of the central limit theorem that also holds for variables that are not independent. Here, we present numerical results that characterize how convergence is attained when the variables being summed are deterministically related to one another through the recurrent application of an ergodic mapping. In all the explored cases, the convergence to the limit distribution is slower than for random sampling. Yet, the speed at which convergence is attained varies substantially from system to system, and these variations imply differences in the way information about the deterministic nature of the dynamics is progressively lost as the number of summands increases. Some of the identified factors in shaping the convergence process are the strength of mixing induced by the mapping and the shape of the marginal distribution of each variable, most particularly, the presence of divergences or fat tails. Full article

From the perspective of wireless communication, as communication systems become more complex, order statistics have gained increasing importance, particularly in evaluating the performance of advanced technologies in fading channels. However, existing analytical methods are often too complex for practical use. In this research paper, we introduce innovative statistical findings concerning the sum of ordered gamma-distributed random variables. We examine various channel scenarios where these variables are independent but not-identically distributed. To demonstrate the practical applicability of our results, we provide a comprehensive closed-form expression for the statistics of the signal-to-interference-plus-noise ratio in a multiuser scheduling system. We also present numerical examples to illustrate the effectiveness of our approach. To ensure the accuracy of our analysis, we validate our analytical results through Monte Carlo simulations. Full article

In the paper, quasi-exponentiated normal distributions are introduced for any real power (exponent) no less than two. With natural exponents, the quasi-exponentiated normal distributions coincide with the distributions of the corresponding powers of normal random variables with zero mean. Their representability as scale mixtures of normal and exponential distributions is proved. The mixing distributions are written out in the closed form. Two approaches to the construction of asymmetric quasi-exponentiated normal distributions are described. A limit theorem is proved for sums of a random number of independent random variables in which the asymmetric quasi-exponentiated normal distribution is the limit law. Full article

When there is uncertainty in the value of parameters of the input random components of a stochastic simulation model, two-level nested simulation algorithms are used to estimate the expectation of performance variables of interest. In the outer level of the algorithm n observations are generated for the parameters, and in the inner level m observations of the simulation model are generated with the values of parameters fixed at the values generated in the outer level. In this article, we consider the case in which the observations at both levels of the algorithm are independent and show how the variance of the observations can be decomposed into the sum of a parametric variance and a stochastic variance. Next, we derive central limit theorems that allow us to compute asymptotic confidence intervals to assess the accuracy of the simulation-based estimators for the point forecast and the variance components. Under this framework, we derive analytical expressions for the point forecast and the variance components of a Bayesian model to forecast sporadic demand, and we use these expressions to illustrate the validity of our theoretical results by performing simulation experiments with this forecast model. We found that, given a fixed number of total observations $nm$, the choice of only one replication in the inner level ($m=1$) is recommended to obtain a more accurate estimator for the expectation of a performance variable. Full article