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The three-parameter Logistic model (3PLM) and the four-parameter Logistic model (4PLM) have been proposed to reduce biases in cases of response disturbances, including random guessing and carelessness. However, they could also influence the examinees who do not guess or make careless errors. This paper proposes a new approach to solve this problem, which is a robust estimation based on the 4PLM (4PLM-Robust), involving a critical-probability guessing parameter and a carelessness parameter. This approach is compared with the 2PLM-MLE(two-parameter Logistic model and a maximum likelihood estimator), the 3PLM-MLE, the 4PLM-MLE, the Biweight estimation and the Huber estimation in terms of bias using an example and three simulation studies. The results show that the 4PLM-Robust is an effective method for robust estimation, and its calculation is simpler than the Biweight estimation and the Huber estimation.

In the paper, Ablowitz—Ladik hierarchy with new self-consistent sources is investigated. First the source in the hierarchy is described as nn+1, where n is related to the Ablowitz—Ladik spectral problem, instead of the corresponding adjoint spectral problem. Then by means of the inverse scattering transform, the multi-soliton solutions for the hierarchy are obtained. Two reductions to the discrete mKdV and nonlinear Schrodinger hierarchies with self-consistent sources are considered by using the uniqueness of the Jost functions, as well as their N-soliton solutions.

Generalized solutions in double Casoratian form of the four-potential isospectral Ablowitz–Ladik equation possessing bilinear form are derived through a matrix method for constructing double Casoratian entries. A novel class of explicit solutions, such as soliton, rational-like, Matveev, Complexiton and interaction solutions, are obtained by letting the general matrix be some special cases. Interestingly, a periodic solution is deduced from the Complexiton solution.

The non-isospectral Ablowitz–Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz–Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz–Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.

A hierarchy of the inverse KdV equation is discussed. Through the bilinear form of Lax pairs, we prove a generalized Darboux–Crum theorem of the hierarchy. The Backlund transformation and the generalized Wronskian solutions are presented. The soliton solutions, explicit rational solutions are obtained then.

In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax pair under the Bargmann constraint. It is also shown that the potential u in the stationary Schrodinger equation can be a summation of squares of wave functions from bilinear point of view.

A (2+1)-dimensional breaking soliton equation with self-consistent sources ((2+1)DBSESCS) is derived. The bilinear form of it is obtained and the N -soliton solutions are worked out by means of the Hirota method and Wronskian technique, respectively. By making use of the Laplace expansion formula, a simple method is presented to treat the nonlinear term in the bilinear equation and finish the Wronskian verification. The dynamics and source effects are investigated.

In this paper, two hierarchies of the Geng equations are presented, including positive non-isospectral hierarchy and negative non-isospectral hierarchy. Moreover, integrable couplings of the corresponding Geng hierarchies are also constructed by enlarging the associated matrix spectral problem. Three new integrable decompositions and conservation laws of the isospectral Geng equation are also obtained. The Gauge transformations are used to obtain the associated binary symmetry constraints of the Geng equation at the first time.

In this paper, a non-isospectral differential-difference Kadomtsev–Petviashvilli equation (n-DΔKPE) is presented. Then, the Casoratian solutions of the n-DΔKPE are obtained by generalizing Casoratian conditions of the non-isospectral DΔKPE, single-soliton solution is also derived by using Hiorta's method.

Gauge transform between the first order nonisospectral AKNS spectral problem and isospectral AKNS spectral problem is obtained, corresponding transform operator between the first order nonisospectral AKNS hierarchy and isospectral AKNS hierarchy is given.

The new method for constructing the Wronskian entries is applied to the Boussinesq equation. The novel Wronskian solutions to it are obtained, including solitons, rational solutions, Matveev solutions, and complexitons.

A new method for constructing the Wronskian entries is proposed and applied to the differential-difference Kadomtsev–Petviashvilli (DΔKP) equation. The generalized Wronskian solutions to it are obtained, including rational solutions and Matveev solutions.

A new representation of N -soliton solution of the fifth order KdV equation is obtained by using Backlund transformation method. We show a direct method for constructing the novel N -soliton solution by performing an appropriate limiting procedure on the known soliton solutions.

We discuss variable selection procedures in regression models based on large data sets. Our purpose is to discover the pattern of the association between the dependent variable and the independent variables. The real-valued variable Y is the response (dependent) variable. The variables in the vector X are the predictor (independent) variables, which may be either ordered or categorical. A function d ( X ) is defined on the measure space χ taking on real values. In regression models, predictors have been constructed using a parametric approach under the assumption that E ( Y | X = x ) = d ( x , θ ) , where d has known functional form depending on x and a finite set of parameters θ = ( θ 1 , θ 2 , … , θ m ) . Then θ is estimated by the least squares method. In practical applications, however, the functional form of d is usually unknown. In such a situation, it is difficult to determine the regression function and we use Classification And Regression Trees (CART) methods that integrate the data and the model. We propose selection procedures to select important predictor variables in the regression model based on data. Some criteria for selecting the important variables are discussed. An empirical study based on an annual survey of inbound visitors in Taiwan is provided to illustrate the implementation of our multiple decision procedure.

Positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct three integrable coupling systems of the positive hierarchy through enlarging Lax pair method.

In terms of properties of the known loop algebra A ∼ 1 and difference operators, a new algebraic system χ is constructed. By using the algebraic system χ, a discrete matrix spectral problem is introduced and a hierarchy of nonlinear lattice equations is derived. From the hierarchy the celebrated cubic Volterra lattice equation is engendered. We call the hierarchy a generalized cubic Volterra hierarchy. Then an extended algebraic system χ ˜ of χ is presented, from which the integrable couplings system of the generalized cubic Volterra lattice are obtained.

The bilinear equation of the non-isospectral AKNS equation is derived. The N -soliton solutions and the double Wronskian solution are obtained through the Hirota method and the Wronskian technique, respectively. A non-isospectral Schrodinger equation and its multi-soliton solutions are given by reducing.

In this Letter, a systematic method is presented to construct the Darboux transformation with multi-parameters for the coupled nonlinear Schrodinger equations, from which the solutions of Schrodinger equations are reduced to solving a linear algebraic system and a constraint differential equation. With the aid of symbolic computation, new soliton-like solutions for the nonlinear Schrodinger equations are obtained by using its Darboux transformation.

A bilinear form of a nonisospectral sine-Gordon equation is given. Exact solutions are further obtained through the Hirota method and Wronskian technique, respectively. Some nonisospectral characteristics of the obtained solutions, such as the time-dependent shape, time-dependent speed and some non-propagating solitary waves, are discussed. The conservation laws are derived.

A new simple loop algebra G ∼ M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component WKI hierarchy is obtained. Furthermore, an expanding loop algebra F ∼ M of the loop algebra G ∼ M is presented. Based on F ∼ M , the multi-component integrable couplings system with two arbitrary functions of the multi-component WKI hierarchy is worked out. How to design the isospectral problem of the multi-component hierarchy of equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.

It is shown that a modified Backlund transformation is presented by the dependent variable transformation. Starting from it, a new representation of N -soliton solution and a class of novel multi-soliton solution of the Boussinesq equation have been derived. We also find the novel soliton solution may be deduced by limiting.

In this Letter, a new isospectral problem is established by choosing subalgebra of loop algebra A 2 as its basis. By making use of Tu's scheme, the N Hamiltonian structure is established and it is shown that the new hierarchy is integrable in Liouville sense. As two reductions, the two generalized nonlinear Schrodinger equations are obtained, respectively. Integrable couplings of the new hierarchy are obtained by taking a proper algebraic transformation to turn the loop algebra A 2 into its equivalent loop algebra like the form of A 1 . Finally, the Hamiltonian form of a binary symmetric constrained flows of the system is given.

Using homogenous balance method we obtain Backlund transformation (BT) and a linear partial differential equation of (2+1)-dimensional dispersive long water equations (DLWE). As a result, multisoliton and single soliton and other exact solutions of (2+1)-dimensional DLWE are obtained. By analyzing single soliton solution, we get some dromions solutions. This method, which can be generalized to some (2+1)-dimensional nonlinear evolution equations, is simple and powerful.

Soliton-like solutions for a nonisospectral KdV hierarchy are obtained by using the inverse scattering transform. The interaction of two solitons is investigated.

A new bilinear Backlund transformation is presented for the shallow water waves equation. Starting from the new Backlund transformation, a wide variety of novel soliton solutions of the shallow water waves equation are generated. Moreover, the relationship between these solutions has been discussed.

The KP equation with self-consistent sources is derived through the linear problem of the KP system. The multisoliton solutions for the KP equation with self-consistent sources are obtained by using Hirota method and Wronskian technique. The coincidence of these solutions is shown by direct computation.

The paper proposes an approach to constructing the symmetries and their algebraic structures for isospectral and nonisospectral evolution equations of (1+2)-dimensional systems associated with the linear problem of Sato theory. To do that, we introduce the implicit representations of the isospectral flows {Km} and nonisospectral flows {σn} in the high dimensional cases. Three examples, the Kodomstev–Petviashvili system, BKP system and new CKP system, are considered to demonstrate our method.

A systematic approach to constructing an infinite number of conservation laws for discrete soliton systems is proposed, and three examples are given.

By imposing constraint (Lk)−=q∂−1r∂ on the pseudo-differential operator Lk, the k constrained modified Kadomtsev–Petviashvili (KP) hierarchy and their corresponding Lax pair are obtained from the linear problem and its adjoint of the modified KP system. Especially, the modified KdV system, the GNS system with derivative coupling, the Burgers system, and a new 3×3 integrable system are presented as examples.

Many authors have studied variable selection in multiple linear regression models. In this paper, we derive some generalized selection procedures for the linear models. An approximation of noncentral F distribution has also been obtained.

The analysis of residuals may reveal various functional forms suitable for the regression model. In this paper, we investigate some selection criteria for selecting important regression variables. In doing so, we use statistical selection and ranking procedures. Thus, we derive an appropriate criterion to measure the influence and bias for the reduced models. We show that the reduced models are based on some noncentrality parameters which provide a measure of goodness of fit for the fitted models. In this paper, we also discuss the relationships of influence diagnostics and the statistic proposed earlier by Gupta and Huang (J. Statist. Plann. Inference 20 (1988) 155–167). We introduce a new measure for detecting influential data as an alternative to Cook's measure.

Some novel solutions of the KdV equation are obtained through the modified bilinear Backlund transformation.

A bilinear form of a nonisospectral differential-difference equation related to the Ablowitz–Ladik (AL) spectral problem is derived by a transformation of dependent variables. Exact solutions to the resulting bilinear equation are found. The [Formula: see text]-soliton-like solutions and the double Casoratian solutions are derived by means of Hirota’s direct method and the double Casoratian technique, respectively. Moreover, the connection between those two classes of solutions is explored.

A popular capability index Cpm is defined by C pm = USL − LSL 6γ , where USL and LSL are the upper specification limit and the lower specification limit on the characteristic X of each item from the process, respectively. γ2 = E(X − T)2 = σ2 + (μ − T)2. T is a given target value. μ and σ2 are the mean and the process variance of X, respectively. γ2 expresses the deviation of X from the target T by σ2 and the process centering (μ − T) (Boyles, J. Quality Technology 23 (1991) 17–26). In the situation of the manufacturing process being under control, we assume that X is normally distributed. Besides, USL and LSL are usually fixed and determined in advance. Under the circumstance, to search the largest Cpm which are used to provide unitless measure of the process performance is equivalent to looking for the smallest γ2. The main purpose of the present paper is to select a subset containing the process associated with the smallest γ2 from k given independent processes (populations). The results are also related to usual selection problems in analysis of variance.

In this paper, we propose a two-stage variable selection procedure for multivariate linear regression models. We select appropriate models under a guaranteed probability by using the summation of noncentralities in the first stage. In the second stage, we exclude those models with large individual noncentrality, and then select the best model with the minimum Akaike's information criterion (AIC). Empirical study is provided to show how to achieve our goal in variable selection and to demonstrate the efficiency and usefulness of the procedure in practical applications. In addition, we have built a reasonable model to explain and predict the earnings and productivity in Taiwan area.

A negative KdV-mKdV hierarchy is presented through the KdV-mKdV operator. The generalized Wronskian solution to the negative KdV-mKdV equation is obtained. Some soliton-like solutions and a complexiton solution are presented explicitly as examples.

Direct linearization method is used to solve the non-isospectral KdV equation. The corresponding singular linear integral equation and the time dependence of measure in the singular linear integral equation are proposed. Furthermore, the solutions to the non-isospectral modified KdV equation are also derived by using the singular linear integral equation of the non-isospectral KdV equation.

A predator-prey interaction of both populations was treated by the Painleve analysis method. The analytic solutions for the related equations was obtained using the truncated Painleve expansion.

A large body of literature exists on the techniques for selecting the important variables in linear regression analysis. Many of these techniques are ad hoc in nature and have not been studied from a theoretical viewpoint. In this paper we discuss some of the more commonly used techniques and propose a selection procedure based on the statistical selection and ranking approach. This procedure is easy to compute and apply. The procedure depends on the goodness of fit of the model and the total error associated with it.

Let π0,π1,…,πk be k+1 independent populations. For i=0,1,…,k,πi has the densit f(x,θi), where the (unknown) parameter θi belongs to an interval of the real line. Our goal is to select from π1,… πk (experimental treatments) those populations, if any, that are better (suitably defined) than π0 which is the control population. A locally optimal rule is derived in the class of rules for which Pr(πi is selected)γi, i=1,…,k, when θ0=θ1=⋯=θk. The criterion used for local optimality amounts to maximizing the efficiency in a certain sense of the rule in picking out the superior populations for specific configurations of θ=(θ0,…,θk) in a neighborhood of an equiparameter configuration. The general result is then applied to the following special cases: (a) normal means comparison — common known variance, (b) normal means comparison — common unknown variance, (c) gamma scale parameters comparison — known (unequal) shape parameters, and (d) comparison of regression slopes. In all these cases, the rule is obtained based on samples of unequal sizes.

It is very important to study the monotonicity properties of distributions in order to obtain inequalities useful in statistical inference. Some monotonicity properties of distributions are well known and have proved to be very useful. During the last decade, more concepts have been introduced and used by several authors in multiple decision problems.