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Månsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a remedy, Türkan and Özel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE ($$ \hat{k}_{q4} $$ k ^ q 4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation.

In this paper, we introduce a new family of generalized blending-type bivariate Bernstein operators which depends on four parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ . Approximation properties of these operators are studied, and we obtain the rate of convergence in terms of mixed and partial modulus of continuities. Moreover, we prove a Korovkin- and a Voronovskaja-type theorems for these operators. The last part of the paper is devoted to the associated GBS operators. In this part, we study degree of approximation of the GBS operators in terms of mixed modulus of continuity. GBS operators obtained here give better approximation than the original operators to the function f(x, y). Finally, approximation properties of the suggested operators and their associated GBS operators are discussed on graphs, for some numerical examples to show how GBS operator gives better approximation to f(x, y). Also, approximation properties of the suggested operators for different values of parameters $$s_{1}$$ , $$s_{2}$$ , $$\alpha _{1}$$ and $$\alpha _{2}$$ are illustrated on graphs. It should be mentioned that any increase in $$\alpha _{i}$$ values or any decrease in $$s_{i}$$ values gives better approximation of the suggested operators to f(x, y).

In their remarkable paper, Fazli et al. (Int J Comput Math, 2019. https://doi.org/10.1080/00207160.2019.1658870 ) have recently established existence results for fractional Basset–Boussinesq–Oseen equation in an appropriate partially ordered Banach space. In this paper, we improve this study by proposing different approach that provides existence criterion for the addressed equation. The main assumptions are less restrictive and easily verifiable. Examples with graphical representations are constructed to demonstrate consistency to theoretical findings. Besides, we construct an iterative sequence that converges to the unique solution. We end the paper by a conclusion that demonstrates the advantage of our theorem compared to the previous results.

In the present paper, our boundary value problem is Sturm–Liouville problem with retarded argument and transmission conditions at the finitely many points of discontinuity. This is the first work containing several points of discontinuity in the theory of differential equations with retarded argument. The main purpose of this paper is to obtain asymptotic formulas for the eigenvalues and corresponding eigenfunctions of this boundary value problem. The simplicity of the eigenvalues is proved. In special case when our problem with retarded argument and transmission conditions at one point of discontinuity, the obtained results coincide with the corresponding results in Bayramov et al. (Appl Math Comput 191, 592–600, 2007).

This paper resolves the difficulty in using Almon’s technique in the estimation of distributed lag coefficients when the researchers run into the problem of multicollinearity. Multicollinearity arising in the analysis leads us to the biased estimation methods, one of which is Almon ridge estimator. In consideration of two-parameter ridge solution of Lipovetsky and Conklin (Appl Stoch Mod Bus Ind 21:525–540, 2005), we offer Almon two-parameter ridge estimator, which has advantage over Almon estimator as well as Almon ridge estimator with regard to mean square error criterion. Besides, an improvement in coefficient of multiple determination is observed for the new estimator relative to the Almon ridge estimator. An illustrative application of the mentioned estimators is examined by widely used Almon data to support the theoretical results. Additionally, a Monte Carlo experiment is performed to investigate the performance of the estimators in detail. © 2018, Shiraz University. Firat University Scientific Research Projects Management Unit: FBA-2018-10169 This paper was supported by Çukurova University Scientific Research Projects Unit Project Number: FBA-2018-10169.

The purpose of this paper is to give some new Karush–Kuhn–Tucker-type necessary optimality conditions for nonsmooth semi-infinite problems. Moreover, we present some suitable examples for our results. The paper is organized by Frechet and Mordukhovich subdifferentials.

In this paper, we propose a meshless method using radial basis functions (RBFs) method based on finite-difference method to solve a nonlinear inverse convection–reaction–diffusion problem with an unknown source function. Indeed, the aim of this paper is to show that the meshless method based on the radial basis functions approach is also suitable for the treatment of the inverse nonlinear partial differential equations. Usually, the matrices obtained from the discretization of the equations are ill-conditioned, especially in higher-dimensional problems. To overcome such difficulties, we use Tikhonov regularization method. We prove that the time discrete scheme is stable using the energy method. The effectiveness of the proposed method is illustrated by numerical examples.

Alzalg, Baha/0000-0002-1839-8083; Momani, Shaher M./0000-0002-6326-8456; WOS: 000413785300005 In a recent paper (Zeb et al. in Appl Math Model 37(7):5326-5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.

Let N be a module in category $$\sigma [M]$$ . N is called generalized $$\delta$$ –M-small (briefly G– $$\delta$$ –M) if, $$N \subseteq \delta (L)$$ for some $$L \in \sigma [M]$$ . In this paper we characterize G– $$\delta$$ –M modules and get some suitable results related to this kind of modules. We will show that a module $$N \in \sigma [M]$$ is G– $$\delta$$ –M if and only if $$N \subseteq \delta (\hat{N})$$ , where $$\hat{N}$$ is the M-injective envelope of N in $$\sigma [M]$$ . Also we prove that if there is no non-zero G– $$\delta$$ –M module, then M is cosemisimple and by giving an example we show that the converse need not be true. The relation between G– $$\delta$$ –M modules and some other classes of modules would be investigated in this paper. The torsion theory cogenerated by this class of modules will be introduced and studied in this paper. For a module $$N \in \sigma [M]$$ we show that $$N = \mathrm{{Re}}_{GD[M]}(N)$$ iff for every nonzero homomorphism $$f : N \longrightarrow K$$ in $$\sigma [M],$$ $$\mathrm{{Im}}(f) \not \subseteq \delta (K)$$ iff $$\Delta _{\delta }(N,A) = 0,$$ for all $$A \in \sigma [M]$$ .

In this paper, Modified Crank–Nicolson method is combined with Richardson extrapolation to solve the 1D heat equation. The method is found to be unconditionally stable, consistent and hence the convergence of the method is guaranteed. The method is also found to be second-order convergent both in space and time variables. When combined with the Richardson extrapolation, the order of the method is improved from second-to fourth-order. To validate the proposed method, two model examples are considered and solved for different values of spatial and temporal step lengths.

In this paper, the fractional pantograph equations in quantum calculus are studied. We propose a q-fractional variant of the nonlinear pantograph equation with a given initial condition and a nonlinear neutral q-fractional pantograph equation. Based on the fixed-point theory, the main theorems provide some new results regarding the existence and uniqueness of solutions for q-fractional pantograph equations.

In this paper, some characterizations for the boundedness, compactness and essential norm of a class of product-type operators on the Zygmund space are given.

In this paper, we first provide a better estimate of the second inequality in Hermite–Hadamard inequality. Next, we study the reverse of the celebrated Davis–Choi–Jensen’s inequality. Our results are employed to establish a new bound for the operator Kantorovich inequality.

In this paper, we give a new lower bound for the eigenvalues of the Dirac operator defined on the Spin Riemannian hypersurface manifold endowed with 2-tensor, in terms of the Energy-Momentum tensor, scalar curvature and extrinsic curvature. Then this estimate is improved in two different ways by considering the conformal invariance of the Dirac operator. The first is given in term of the first eigenvalue of the Yamabe operator. The latter, is given in terms of the the area of a topological 2-sphere., TUBITAK The Scientific and Technological Research Council of Turkey [120F109]; TUBITAK (The Scientific and Technological Research Council of Turkey), This study was supported by TUBITAK The Scientific and Technological Research Council of Turkey (Project Number: 120F109). Also, The authors would like to appreciate TUBITAK (The Scientific and Technological Research Council of Turkey) for supporting this research.

This paper considers the estimation of parameters for the inverse Gaussian regression model in the presence of multicollinearity. To mitigate this issue, we propose the inverse Gaussian James–Stein estimator (IGJSE) and compare its performance with other estimation methods i.e. maximum likelihood estimator (MLE), ridge and Liu estimators. Theoretical comparisons between the usual MLE, ridge, Liu and IGJSE are derived using matrix mean − squared error and scalar mean squared error criterions. The statistical properties of these estimators are systematically scrutinized via a Monte Carlo simulation. The findings of the simulation study clearly demonstrate that the proposed IGJSE showed a consistent behaviour as compared to the other estimation methods in all the evaluated situations. Finally, we evaluate the performance of IGJSE with the help of real example. The simulation and real example results show that the performance of IGJSE is better as compared to other estimation methods with correlated explanatory variables.

In this paper, we obtain the mixture representations for the extropy of a mixed used system and the extropy of the inactivity time of a system. These representations demonstrate the extropy of conditional mixed systems as a mixture of a conditional coefficients vector, left (or right) truncated beta distribution, and the baseline distribution of the system components. Based on these representations, the comparisons of the extropy of conditional mixed systems are carried out with respect to the stochastic orders and stochastically ordered conditional coefficients vector between systems. Moreover, some bounds for the extropy of theses systems are derived. The utility of these representations is illustrated in two examples in which the extropy of a mixed used system and the extropy of the inactivity time of a system are computed and compared.

This paper shall be concerned with the notion of von-Neumann regular acts. First, the notions of semiretraction homomorphisms and fully semiretractable (FSR) S-acts are introduced and some properties of them are investigated. Finally, using semiretraction notion, we investigate locally projective and von-Neumann regular S-acts.

The aim of this paper is to present a reproducing kernel method for solving system of integro-differential equations with nonlocal boundary conditions. The solution obtained by using the present method takes the form of a convergent series with easily computable components. The analysis of convergence shows that the approximate solution and its derivatives converge to the exact solution and its derivatives, respectively. The numerical examples are given to demonstrate the accuracy of the present method. Results obtained by using the present method are compared with those of the exact solution of each example and are found to be in good agreement.

In this paper, two subclasses of close-to-star functions and two subclasses of close-to-convex functions are studied. By using the coefficient estimates of the known Caratheodory function classes, the upper bounds of the Zalcman functional $${J}_{3,4}$$ , Hankel determinant $${{H}_{2}}(3)$$ and Toeplitz determinant of given function classes are estimated. Some sharp results are obtained.

A hom-Lie algebroid is a vector bundle together with a Lie algebroid-like structure which is twisted by a homomorphism and a VB-hom algebroid is essentially defined as a hom-Lie algebroid object in the category of vector bundles. In this paper, we show that, there exists a correspondence between the VB-hom algebroids and two term representations up to homotopy of hom-Lie algebroid.

The purpose of this paper is to present an appropriate numerical method for solving linear two-dimensional weakly singular stochastic integral equations. This method is based on bicubic B-spline interpolation, Gauss–Legendre quadrature formula, and two-dimensional Ito approximation. The proposed method is used to transform the solution of mentioned problem to the linear system of algebraic equations which can be solved by using an appropriate method. Convergence analysis of the approach has been investigated. In the end, two examples are given to show efficiency and accuracy of the method and their numerical results are reported.

Generalized Sylvester quaternion matrix equation with some constrictions is explored in this paper. A novel expression of the general solution of the constraint system is given with some necessary and sufficient conditions. Its Cramer’s rule is derived within the framework of the theory of row-column quaternion determinants. An example is given to justify obtained theoretical results.

Let S be a unital ring in which 2 is invertible, and let $$R=H(S)$$ be the quaternion ring over S. In this paper, we characterize the generalized derivations of R and show that every generalized Jordan derivation on R is a generalized derivation. We also consider the question when a derivation (generalized derivation) of a quaternion ring is an inner derivation (generalized inner derivation). In addition, we show that the structures of the center, ideals, and the above-mentioned derivations of the quaternion rings and matrix rings are similar.

In this paper, we study the estimation of the stress–strength reliability model when the stress and the strength variables are modeled by two independent but not identically distributed random variables from the generalized inverted exponential distributions. The stress strength reliability estimator is considered using, mainly, median ranked set sampling (MRSS) compared to ranked set sampling (RSS) and simple random sampling (SRS). Firstly, we discussed the reliability estimation when the data of the strength and stress random variables are in the form of the MRSS with even or odd set sizes. Secondly, we derived the reliability estimator when the data of the strength variable are selected using the MRSS with even or odd set sizes. In contrast, the data of the stress variable are drawn using RSS and vice versa. Thirdly, we obtained the reliability estimator when the data of the strength variable are selected using the MRSS with even or odd set sizes, while the data of the stress variable are chosen using the SRS and vice versa. Monte Carlo simulation study is performed to compare the behavior of different estimates with respect to the MRSS scheme. This study indicated that the reliability estimates based on MRSS are more efficient than the corresponding estimates based on the RSS and SRS in most of the situations. One real data analysis has been performed for illustrative purposes.

This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order $$\alpha $$ ( $$0

In this paper, we study the localization operator associated to quaternionic continuous wavelet transform. We show that the quaternionic localization operator is bounded if the continuous wavelet transform is real valued. Also, we investigate the relation between the localization operators in the setting of quaternion and complex Hilbert spaces. Furthermore, we prove that the quaternionic two-wavelet localization operator is bounded under some conditions.

In this paper, we propose an efficient method to approximate bivariate functions by Szasz–Mirakjan operators via Dunkl generalization. More precisely, we generalize the Dunkl formation of Szasz–Mirakjan operators by introducing the exponential functions and obtain the approximation by means of well-known Korovkin’s theorem, modulus of continuity, Lipschitz functions and Peetre’s K-functional. Next, we also discuss approaches of approximation by the operators in space of Bogel continuous functions.

In the present paper, we describe cubic spline combined with high accuracy compact formulation for computing approximate solution values to nonlinear convection-dominated diffusion equations, and apply the method to several variants of Burger’s type parabolic partial differential equations. A combination of cubic spline interpolating polynomial and quasi-variable grids sequence yields a two-level implicit compact computational scheme that falls in the range of increased accuracy. The grid stretching parameter plays an important role while considering the boundary layer phenomenon to convection-dominated parabolic equations. The discretization on the quasi-variable grid network yields a more accurate and precise solution than those obtained on a uniform grid network. It is easy to extend the proposed scheme to the singular equation with compact operators, preserving the order and accuracy. It is shown through stability analysis that the new method is stable with an accuracy of order four and two in the spatial and temporal direction, respectively. Numerical illustrations with generalized Burgers–Fisher equation, Burgers–Huxley equation, and some physically relevant parabolic partial differential equations corroborate the theoretical analysis.

In this paper, we propose the numerical scheme for solving nonlinear higher multi-order Caputo fractional differential equations with integral and antiperiodic boundary conditions, establish the existence of an approximate solution to our problem, and prove the convergence of our numerical method. The operational matrix of fractional integration based on hat functions is used to solve the given problem. Also, we present a numerical example to illustrate our main results.

In this paper, approximate duals and Q-approximate duals of standard Bessel sequences are studied, where Q is a bounded (not necessarily adjointable) operator on a Hilbert $$C^*$$ -module. Particularly, we consider the stability of Q-approximate duals under morphisms of Hilbert $$C^*$$ -modules and also their stability under small perturbations is discussed. Then, we get some properties of (a, m)-approximate duals in Hilbert $$C^*$$ -modules. These approximate duals are defined using the distance between the identity operator and the operator defined by multiplying the Bessel multiplier with symbol m by an element a in the center of the $$C^*$$ -algebra. Moreover, using multipliers for pair Bessel sequences, the notion of (a, m)-approximate duality is generalized for the cases that the sequences are non-Bessel or m is not necessarily a symbol.

This paper introduces a new hybrid fractional model for image denoising. This proposed model is a combination of two models Rudin–Osher–Fatemi and fractional-order total variation. We try to use the advantages of two mentioned models. In this regard, after introducing an appropriate norm space, we prove the existence and uniqueness of the presented model. Furthermore, finite difference method is employed for numerically solving the obtained equation. Finally, the results illustrate the efficiency of the proposed model that yields good visual effects and a better signal-to-noise ratio.

Let H be a compact subgroup of a locally compact group G. In this paper we define a convolution on M(G/H) , the space of all bounded complex Radon measures on the homogeneous space G/H. Then we prove that the measure space M(G/H) with the newly well-defined convolution is a non-unital Banach algebra that possesses an approximate identity. Finally, it is shown that this Banach algebra is not involutive and also $$L^1(G/H)$$ with the new convolution is a two-sided ideal of it.

In this paper, we calculate the characters of two classes of P-polynomial table algebras. To this end, we apply some linear algebra methods and tridiagonal matrices to obtain the characters of these table algebras.

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G which is in a good situation with respect to a homogeneous function algebra $$(A,\Vert .\Vert _{A})$$ on G. Feichtinger showed that there exists a minimal Banach space $$B_{\rm{min}}$$ in the family of all homogeneous Banach spaces C on G containing all elements of B with compact support. In this paper, we study the amenability and super-amenability of some Feichtinger algebras such as $$L^{p}(G)_{\rm{min}}, C_{0}(G)_{\rm{min}}$$ , $$A(G)_{\rm{min}}$$ , and $$B(G)_{\rm{min}}$$ .

In this paper, we obtain some Hermite-Hadamard type inequalities for h-convex functions. Also, some applications to special means of real numbers are given. The results presented here provide extensions of those given in earlier works. WOS:000537926700002 2-s2.0-85086095154

A Roman $$\{2\}$$-dominating function (R2DF) $$f:V\longrightarrow \{0,1,2\}$$ of a graph $$G=(V, E)$$ has the property that for every vertex $$v\in V$$ with $$f(v)=0$$ either there is a vertex $$u\in N(v)$$ with $$f(u)=2$$ or there are two vertices $$x,y\in N(v)$$ with $$f(x)=f(y)=1$$. The weight of f is the sum $$f(V)=\sum _{v\in V}f (v)$$. The minimum weight of an R2DF on G is the Roman $$\{2\}$$-domination number of G. In this paper, we first show that the associated decision problem for Roman $$\{2\}$$-domination is NP-complete even when restricted to planar graphs. Then, we give a linear algorithm that computes the Roman $$\{2\}$$-domination number of a given unicyclic graph.

A peacock graph $$PG=(i, j, k\,; b_1, b_2,\ldots ,b_s)$$, where $$i,j,k\ge 3$$ is a graph consisting of three cycles $$C_i$$, $$C_j$$, $$C_k$$ and $$s\,(\ge 1)$$ paths $$P_{b_1+1}, P_{b_2+1},\ldots , P_{b_s+1}$$ intersecting in a single vertex that all meet in one vertex. A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS, for short) if every graph with the same Laplacian spectrum is isomorphic to G. If $$s=1$$, then the peacock graph is called clover graph. In Wang and Wang (Linear Multilinear Algebra 63(12):2396–2405, 2015), it was proved that all clover graphs are DLS. In this paper, we generalize this result and show that all peacock graphs are DLS.

The current paper is devoted to introducing a semi-analytical technique for the numerical solution of nonlinear oscillators under the damping effect by using the reproducing kernel Hilbert space method. The proposed scheme is a hybrid of the reproducing kernel Hilbert space and the perturbation methods. This method finds solutions to periodic and nonlinear problems by converting to a system of linear differential equations. Finally, some test problems are given to illustrate the validity of the novel algorithm. Also, the presented numerical results using the method are compared with the numerical solutions of the fourth-order Runge–Kutta method. The numerical results reported show that the present method can provide very accurate approximate solutions.

The prime objective of this paper is to construct the Kantorovich variant of Ismail–May operators depending on a non-negative parameter $$\lambda $$. We estimate the rate of convergence of the proposed operators for functions in Lipschitz-type space. Further, an improved quantitative Voronovskaya-type estimate in terms of the second-order modulus of continuity and a direct approximation theorem using Ditzian–Totik modulus of smoothness is also given. The last section is dedicated to the bivariate generalisation of the proposed operators and estimation of their rate of convergence in terms of partial and total modulus of continuity and Peetre’s K-functional. A Voronovskaya-type result is also obtained. Finally, some graphs and error estimation table to illustrate the convergence of the proposed operators are presented using Mathematica software.

Odd powers of even-indexed Chebyshev polynomials of the second kind and odd powers of odd-indexed Chebyshev polynomials of the first kind were computed. In this paper, we evaluate all of the rest kinds of power sums of the Chebyshev polynomials. We present the relationships between the Chebyshev polynomials and general Fibonacci, Lucas sequences.

In this paper, we give some necessary conditions of the density of periodic elements for cosine operator functions generated by group actions. The sufficient condition for such cosine operator functions to be chaotic is given as well. The obtained results can be applied for quotient spaces, double coset spaces and hypergroups.

A star edge coloring of a graph G is a proper edge coloring of G such that no path or cycle of length 4 is bicolored. The star chromatic index of G, denoted by $$\chi ^{\prime }_{s}(G)$$ , is the minimum k such that G admits a star edge coloring with k colors. Bezegova et al. (J Graph Theory 81(1):73–82, 2016) conjectured that the star chromatic index of outerplanar graphs with maximum degree $$\Delta$$ is at most $$\left\lfloor \frac{3\Delta }{2}\right\rfloor +1$$ . In this paper, we prove this conjecture for a class of outerplanar graphs, namely Cactus graphs, wherein every edge belongs to at most one cycle.

In this paper, we give the necessary and sufficient conditions for the existence of strong (1, 1, 2)-kernels in the corona of graphs. Moreover, we consider lower and upper strong (1, 1, 2)-kernel numbers and we prove that the difference between these parameters can be arbitrarily large.

Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where $$a\in R.$$ In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where $$m\in M$$ ($$r\in R)$$. In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension $$R\ltimes M$$.

In this paper, we attempt to indicate the links between representations over a Lie algebra and line bundles over a scheme. Our hypothesis is that we can make a correspondence between representations of a solvable Lie algebra and line bundles. Finally, we study the decomposition of these bundles by using the representation theory over solvable Lie algebras.

In this paper, the concept of farthest orthogonality, distance orthogonality and $$*$$-farthest orthogonality in Banach spaces is introduced and the relation between these concepts with the dual space is found. Also, the weakly $$\phi $$-contraction and farthest continuous maps and their relationship are studied. Then, some best proximity and farthest point theorems are proved in Banach spaces. Some examples are given to illustrate the results.

In this paper, we introduce Banach and Kannan integral type contractions in partially ordered metric spaces and investigate the existence and uniqueness of best proximity points for them. Additionally, we pose two non-trivial examples for our main results.

The coefficient of variation is a variability measure used in manufacturing industries when the process mean fluctuates with time and process standard deviation is a linear function of process mean. In this paper, we proposed an exponential weighted moving coefficient of variation control chart by using three parametric log-normal transformations under ranked set sampling with different sample sizes. The average run length and standard deviation of run lengths are computed with an extensive Monte Carlo simulation study to measure the performance of the proposed chart. The comparison of the proposed control chart is made with considered CV control charts and concluded that the proposed control chart is efficient as compared to considered CV control charts to detect the small or moderate shifts in the process CV. An example is presented to demonstrate the implementation of the proposed control chart on the real data set.

In the present paper, we prove some results on rate of convergence for Dunkl analogue of Stancu type q-Szasz–Mirakjan–Kantorovich operators in terms of second-order modulus of continuity and Lipschitz functions. Further, we construct the bivariate extension of these operators and obtain some approximation results.

Log-normal distribution is used widely in application fields such as economics and finance. This paper considers confidence interval estimates for common signal-to-noise ratio of log-normal distributions based on generalized confidence interval (GCI), adjusted method of variance estimates recovery, and computational approaches. A simulation study is conducted to compare the performance of these confidence intervals. A Monte Carlo simulation is applied to report coverage probability and average length of the confidence intervals. Based on the simulation study, for $$k=$$ 3, the GCI can be used. For $$k=$$ 6, the results of GCI approach perform similarly to the results of computational approach. For $$k=$$ 10, the computational approach can be considered as an alternative to estimate the confidence interval. A numerical example based on real data is presented to illustrate the proposed approaches.