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In this study, the stress-strength reliability, R = P(Y < X) where Y represents the stress of a component and X represents this component’s strength, is obtained when X and Y have two independents generalized Gompertz distribution with different shape parameters under progressive type-II censoring. The Bayes and maximum likelihood estimators of stress-strength reliability can not be acquired in closed forms. The approximate Bayes estimators under squared error loss function by using Lindley’s approximations for stressstrength reliability are derived. A Monte Carlo simulation study is done to check performances of the approximate Bayes against performances of maximum likelihood estimators and observe the coverage probabilities and the intervals’ average width. In addition, the coverage probabilities of the parametric bootstrap estimates are calculated. Two applications based on real datasets are provided.

Let $V$ be a countably generated right vector space over a field $F$ and $\sigma\in End(V_F )$ be a shift operator. We show that there exist a unit $u$ and an idempotent $e$ such that $1-u,\sigma-u$ are units in $End(V_F)$ and $1-e,\sigma-e$ are idempotents in $End(V_F)$. We also obtain that if $D$ is a division ring and $D\ncong \mathbb Z_2, \mathbb Z_3 $, then $1-u,\alpha-u$ are units in $End(V_D)$ for any $\alpha\in End(V_D)$.

This paper is concerned with the study of stability analysis to a complicated recovered fracturing fluid model (RFFM, for short), which consists of a stationary incompressible Stokes equation involving multivalued and nonmonotone boundary conditions, and a reaction-diffusion equation with Neumann boundary conditions. Firstly, we introduce a family of perturbated problems corresponding to (RFFM) and deliver the variational formulation of perturbated problem which is a hemivariational inequality coupled with a variational equation. Then, we prove that the existence of solutions to perturbated problems and the solution sequence to perturbated problems are uniformly bounded. Finally, via employing Mosco convergent approach and the theory of nonsmooth, a stability result to (RFFM) is established.

The Moore and Bilikam family includes lifetime distributions, hence there is a need for a meticulous investigation of the proposed family. We evaluate different estimation procedures for both parameters and reliability function of the Moore and Bilikam family comprehensively, including the maximum likelihood, Bayesian and E-Bayesian estimation methods. The estimation methods of the Moore and Bilikam family are compared via the simulation data, whereas simulation results of the Burr XII sub-model are reported. Based on the simulation approach, we concluded the estimates of the Moore and Bilikam family are convergent to the corresponding parameters, and the root mean square error values derived by the E-Bayesian method are less than other estimators. The analysis of the time between failures of secondary reactor pumps data set has been represented for illustrative purposes, which confirmed simulation results.

An important bias can occur when estimating coefficients by maximizing the known partial likelihood function in the Cox regression model with the measurement error covariate. We focus here on Bayesian methods in order to adjust measurement error and aim to propose an adjusting Bayesian method. Constructing simulation studies using Markov Chain Monte Carlo simulation techniques to investigate the performance of models. We compare the proposed method with the existing method that used partial likelihood function, Bayesian Cox regression model ignoring measurement error, the adjusted Bayesian Cox regression model that exists in the literature by a simulation study which consists of different sample sizes, censoring rates, reliability levels, and regression coefficients. Simulation studies indicate that the proposed method outperformed others given some scenarios. A real data set is analyzed for an illustration of the findings.

Recently‎, ‎the beta regression model has been used in several fields of science to model data in the form of rate or proportion‎. ‎In this paper‎, ‎we propose some novel and improved methods to estimate parameters in the beta regression model‎. ‎We consider a sub-space on the regression coefficients of the beta regression model and combine the unrestricted and restricted estimators then we present Stein-type and preliminary estimators‎. ‎We develop the expressions for the proposed estimators' asymptotic biases and their quadratic risks‎. ‎Numerical studies through Monte Carlo simulations are used to evaluate the performance of the proposed estimators in terms of their simulated relative efficiency‎. ‎The results show that the proposed estimators outperform the unrestricted estimator when the restrictions hold‎. ‎Finally‎, ‎an empirical application is provided to demonstrate the practical usefulness of the proposed ‎estimators.‎

This article presents two approaches to extend classical MANOVA tests to the unequal covariances case. The first approach is illustrated by extending the classical Wilks test, which is valid only when covariances are equal. Such tests will be based on exact probabilities of certain extreme regions. We will also show how tests numerically equivalent to the parametric bootstrap tests could be easily obtained without using any bootstrap sampling arguments, so that resulting p-values are also based on exact probabilities of well defined extreme regions. Being systematic approaches, by taking similar approaches, re- searchers should be able to derive generalized tests in MANCOVA, higher-way MANOVA, and in RM MANOVA under heteroscedasticity, in which the parametric bootstrap type approaches run into difficulties.

The cause-specific hazard function plays an important role in developing the regression models for competing risks survival data. Proportional hazards and additive hazards are the commonly used regression approaches in survival analysis. Mostly, in literature, the proportional hazards model was used for parametric regression modelling of survival data. In this article, we introduce a parametric additive hazards regression model for survival analysis with competing risks. For employing a parametric model we consider the modified Weibull distribution as a baseline model which is capable to model survival data with non-monotonic behaviour of hazard rate. The estimation process is carried out via maximum likelihood and Bayesian approaches. In addition to Bayesian methods, a class of non-informative types of prior is introduced with squared error (symmetric) and linear-exponential (asymmetric) loss functions. The relative performance of the different estimators is assessed using Monte Carlo simulation. Finally, using the proposed methodology, a real data analysis is performed.

Most statistical methods require likelihood evaluation to draw a statistical inference. However, in some situations, likelihood evaluation becomes difficult analytically or computationally. Different likelihood-free methods are available that eliminate the need to compute the likelihood function. Approximate Bayesian Computation (ABC) is a framework that implements likelihood-free inference and replaces the likelihood evaluation with simulations by using forward modeling. The goal of ABC methods is to approximate the posterior distribution. However, posterior approximation via ABC methods is still considerably expensive for high dimensions. ABC requires many simulations that become computationally infeasible for complex models. Here, a technique is proposed that combines a somewhat more efficient form of ABC (Population Monte Carlo, PMC) with a Conditional Density Estimation (CDE) approach. The proposed framework provides an estimation of the posterior distribution which is referred to as PMC-CDE. A simulation study is performed that provides empirical evidence to show the efficiency of PMC-CDE in terms of integrated squared error loss. Furthermore, real-life datasets manifest the application of the proposed method.

The increase of the product variety in the financial markets requires a clear understanding of the dependence between such instruments for the decision-makers. For a few decades, such dependence structures were often modeled with symmetric copula families. However, financial data may reveal an asymmetric structure, which can be determined via directional dependence measures in the context of copulas. Previously, some asymmetric copula models were proposed in different ways using Khoudraji’s device. But they are merely used for financial time series data in a broader sense. In this study, a new set of asymmetric copulas were defined by using one parameter of Archimedean copula families. For this aim, widely used copula families were studied and the corresponding directional dependence measures were analyzed. To illustrate the efficiency of the parameter estimation method, a small simulation scenario consisting of an asymmetric dependence pattern was carried out. Thereafter, the proposed asymmetric bi-variate copulas with directional dependence coefficients were investigated for two different stock market data. The study’s primary findings suggested that the newly generated asymmetric models might be useful for directional dependence. Especially, the estimated directional dependence coefficients can serve as an indicator to explain the variability of one stock in terms of the other.

Characterizing the wind speed distribution properly is essential for the satisfactory production of potential energy in wind farms, being the mixture models usually employed in the description of such data. However, some mixture models commonly have the undesirable property of non-identifiability. In this work, we present an alternative distribution which is able to fit the wind speed data decently. The new model, called Normal-Weibull-Weibull, is identifiable and its cumulative distribution function is written as a composition of two baseline functions. We discuss structural properties of the class that generates the proposed model, such as the linear representation of the probability density function, moments and moment generating function. We perform a Monte Carlo simulation study to investigate the behavior of the maximum likelihood estimates of the parameters. Finally, we present applications of the new distribution for modelling wind speed data measured in five different cities of the Northeastern Region of Brazil.

We consider $n\times n$ real-valued matrices $A = (a_{ij})$ satisfying $a_{ii} \geq a_{i,i+1} \geq \dots \geq a_{in} \geq a_{i1} \geq \dots \geq a_{i,i-1}$ for $i = 1,\dots,n$. With such a matrix $A$ we associate a directed graph $G(A)$. We prove that the solutions to the system $A^T x = \lambda e$, with $\lambda \in \mathbb{R}$ and $e$ the vector of all ones, are linear combinations of 'fundamental' solutions to $A^T x=e$ and vectors in $\ker A^T$, each of which is associated with a closed strongly connected component (SCC) of $G(A)$. This allows us to characterize the sign of $\det A$ in terms of the number of closed SCCs and the solutions to $A^T x = e$. In addition, we provide conditions for $A$ to be a $P$-matrix., Comment: 17 pages, 2 figures; minor changes in introduction, added Figure 1, corrected typos

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type $I$. In our framework, each theta relation is controlled by combinatorial properties of a pair $(T,P)$ of a regular matrix $T$ over an imaginary quadratic field and a positive-definite Hermitian matrix $P$ over the complex number field., Comment: 16 pages, typos are corrected, unclear descriptions are corrected, abstract is revised

In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral versions of Mantel's theorem., 14 pages, 4 figures

The Kato's decomposition \cite[Theorem 4]{kato} is generalized to semi-B-Fredholm operators.

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$, called linearization coefficients, that satisfy $$ p_n(t) p_m(t)=\sum_{k=0}^{n+m} d(n,m,k) p_k(t).$$ For any pair of polynomial sequences $\{u_k(t)\}$ and $\{p_k(t)\}$ we find infinite matrices whose entries are the coefficients $e(n,m,k)$ that satisfy $$p_n(t) p_m(t)=\sum_{k=0}^{n+m} e(n,m,k) u_k(t).$$ Such results are obtained using a matrix approach. We also obtain recurrence relations for the linearization coefficients, apply the general results to general orthogonal polynomial sequences and to particular families of orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families., 14 pages

We study a relation between Hadamard powers and polynomial kernel perceptrons. The rank of Hadamard powers for the special case of a Boolean matrix and for the generic case of a real matrix is computed explicitly. These results are interpreted in terms of the classification capacities of perceptrons., Comment: 12 pages

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$ for which $Q_q$ is positive semidefinite is fundamental in asymptotic spectral analysis of graphs from the viewpoint of quantum probability theory. Assume that $G$ has at least two vertices. Then $\pi(G)$ is easily seen to be a nonempty closed subset of the interval $[-1,1]$. In this note, we show that $\pi(G)=[-1,1]$ if and only if $G$ is isometrically embeddable into a hypercube (infinite-dimensional if $G$ is infinite) if and only if $G$ is bipartite and does not possess certain five-vertex configurations, an example of which is an induced $K_{2,3}$., Comment: 6 pages

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $$\alpha(\phi(X))= \alpha(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=\phi(J_n-I_n)$ with all off-diagonal entries positive.

We say that an algebra is zero-product balanced if $ab\otimes c$ and $a\otimes bc$ agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra of Bre\v{s}ar, Gra\v{s}i\v{c} and Ortega. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements. We show that a semiprime, commutative algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements. We deduce a dichotomy for unital, zero-product balanced algebras: They either admit a character or are generated by nilpotents., Comment: 25 pages. This is the published version

We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to ±1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching.

We focus on two problems relating to the question of when the product of two posinormal operators is posinormal, giving (1) necessary conditions and sufficient conditions for posinormal operators to have closed range, and (2) sufficient conditions for the product of commuting closed-range posinormal operators to be posinormal with closed range. We also discuss the relationship between posinormal operators and EP operators (as well as hypo-EP operators), concluding with a new proof of the Hartwig-Katz Theorem, which characterizes when the product of posinormal operators on $\CC^n$ is posinormal.

We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach. We also establish a criterion for the positivity and log-convexity of such sequences., 13 pages

A group $G$ is called to be acceptable (due to M. Larsen) if for any finite group $H$, two element-conjugate homomorphisms are globally conjugate. We answer the acceptability question for general linear, special linear, unitary, symplectic and all orthogonal groups over an algebraically closed field of non-zero characteristics., Comment: Preliminary version, comments are welcome