When a regression model is applied as an approximation of underlying model of data, the model checking is important and relevant. In this paper, we investigate the lack-of-fit test for a polynomial error-invariables model. As the ordinary residuals are biased when there exist measurement errors in covariables, we correct them and then construct a residual-based test of score type. The constructed test is asymptotically chi-squared under null hypotheses. Simulation study shows that the test can maintain the significance level well. The choice of weight functions involved in the test statistic and the related power study are also investigated. The application to two examples is illustrated. The approach can be readily extended to handle more general models.
When a regression model is applied as an approximation of underlying model of data, the model checking is important and relevant. In this paper, we investigate the lack-of-fit test for a polynomial error-invariables model. As the ordinary residuals are biased when there exist measurement errors in covariables, we correct them and then construct a residual-based test of score type. The constructed test is asymptotically chi-squared under null hypotheses. Simulation study shows that the test can maintain the significance level well. The choice of weight functions involved in the test statistic and the related power study are also investigated. The application to two examples is illustrated. The approach can be readily extended to handle more general models.
Consider the partly linear regression model y_i = x'_iβ + g(t_i) + ε_i, 1 ≤ i ≤ n, where y_i's are responses, x_i = (x_i1,x_i2,…,x_ip)＇ and t_i ∈ Τ are known and nonrandom design Τ is a compact set in the real line R, β = (β_1, …, β_p)' is an unknown parameter vector, g(·) is an unknown function and {ε_i} is a linear process, i.e., ε_i = ∑ from j = 0 to ∞ of ψ_je_i-j, ψ_0 = 1, ∑ from j = 0 to ∞ of |ψ_j| < ∞, where e_j are i.i.d. random variables with zero mean and variance σ_e~2. Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε_i}. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε_i} are an ARMA process, out result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.
Consider the partly linear regression model y_i = x'_iβ + g(t_i) + ε_i, 1 ≤ i ≤ n, where y_i's are responses, x_i = (x_i1,x_i2,…,x_ip)＇ and t_i ∈ Τ are known and nonrandom design Τ is a compact set in the real line R, β = (β_1, …, β_p)' is an unknown parameter vector, g(·) is an unknown function and {ε_i} is a linear process, i.e., ε_i = ∑ from j = 0 to ∞ of ψ_je_i-j, ψ_0 = 1, ∑ from j = 0 to ∞ of |ψ_j| < ∞, where e_j are i.i.d. random variables with zero mean and variance σ_e~2. Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε_i}. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε_i} are an ARMA process, out result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.
We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.
We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.
Point-wise confidence intervals for a nonparametric regression function with random design points are considered. The confidence intervals are those based on the traditional normal approximation and the empirical likelihood. Their coverage accuracy is assessed by developing the Edgeworth expansions for the coverage probabilities. It is shown that the empirical likelihood confidence intervals are Bartlett correctable.
Point-wise confidence intervals for a nonparametric regression function with random design points are considered. The confidence intervals are those based on the traditional normal approximation and the empirical likelihood. Their coverage accuracy is assessed by developing the Edgeworth expansions for the coverage probabilities. It is shown that the empirical likelihood confidence intervals are Bartlett correctable.
A unified approach is proposed for making a continuity adjustment on some control charts for attributes, e.g., np-chart and c-chart, through adding a uniform (0, 1) random observation to the conventional sample statistic (e.g., (npi)-circumflex and c_i). The adjusted sample statistic then has a continuous distribution. Consequently, given any Type I risk a (the probability that the sample statistic is on or beyond the control limits), control charts achieving the exact value of a can be readily constructed. Guidelines are given for when to use the continuity adjustment control chart, the conventional Shewhart control chart (with ±3 standard deviations control limits), and the control chart based on the exact distribution of the sample statistic before adjustment.
A unified approach is proposed for making a continuity adjustment on some control charts for attributes, e.g., np-chart and c-chart, through adding a uniform (0, 1) random observation to the conventional sample statistic (e.g., (npi)-circumflex and c_i). The adjusted sample statistic then has a continuous distribution. Consequently, given any Type I risk a (the probability that the sample statistic is on or beyond the control limits), control charts achieving the exact value of a can be readily constructed. Guidelines are given for when to use the continuity adjustment control chart, the conventional Shewhart control chart (with ±3 standard deviations control limits), and the control chart based on the exact distribution of the sample statistic before adjustment.
Geometric process (GP) was introduced by Lam, it is defined as a stochastic process {X_n, n = 1, 2, …} for which there exists a real number a > 0, such that {a~(n-1)X_n, n = 1, 2, …} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S_n with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.
Geometric process (GP) was introduced by Lam, it is defined as a stochastic process {X_n, n = 1, 2, …} for which there exists a real number a > 0, such that {a~(n-1)X_n, n = 1, 2, …} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for S_n with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.
We propose the concept of a coalitional power value for set games, and present its axiomatic characterization of global efficiency, equal treatment property and coalitional power monotonicity. The coalitional power value is a generalization of the marginalistic value introduced by Aarts et al.
We propose the concept of a coalitional power value for set games, and present its axiomatic characterization of global efficiency, equal treatment property and coalitional power monotonicity. The coalitional power value is a generalization of the marginalistic value introduced by Aarts et al.
In this note we first prove a fixed point theorem in H-spaces which unities and extends the corresponding results in [6] and [9]. Then, by applying the fixed point theorem, we prove an existence theorem of an equilibrium point of an abstract economy in H-spaces which improves and generalizes similar result in [4].
In this note we first prove a fixed point theorem in H-spaces which unities and extends the corresponding results in [6] and [9]. Then, by applying the fixed point theorem, we prove an existence theorem of an equilibrium point of an abstract economy in H-spaces which improves and generalizes similar result in [4].
The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover. Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.
The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover. Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.
Among a variety of adaptive designs, stage-wise design, especially, two-stage design is an important one because patient responses are not available immediately but are available in batches or intermittently in some situations. In this paper, by Bayesian method, the general formula of asymptotical optimal worth is given, meanwhile the length of some optimal designs at first stage concerning two-stage trials in several important cases has been obtained.
Among a variety of adaptive designs, stage-wise design, especially, two-stage design is an important one because patient responses are not available immediately but are available in batches or intermittently in some situations. In this paper, by Bayesian method, the general formula of asymptotical optimal worth is given, meanwhile the length of some optimal designs at first stage concerning two-stage trials in several important cases has been obtained.
We prove the approxomate controllability and finite dimensional exact controllability of semilinear heat equation in R~N with the same control by introducing the weighted Soblev spaces.
We prove the approxomate controllability and finite dimensional exact controllability of semilinear heat equation in R~N with the same control by introducing the weighted Soblev spaces.
A non-autonomous single species dispersal model is considered, in which individual member of the population has a life history that goes through two stages, immature and mature. By applying the theory of monotone and concave operators to functional differential equations, we establish conditions under which the system admits a positive periodic solution which attracts all other positive solutions.
A non-autonomous single species dispersal model is considered, in which individual member of the population has a life history that goes through two stages, immature and mature. By applying the theory of monotone and concave operators to functional differential equations, we establish conditions under which the system admits a positive periodic solution which attracts all other positive solutions.
In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method
In this paper, the global well-posedness of initial-boundary value problem to the nonlinear Kirchhoff equation with source and damping term is established by energy method
This paper studies the existence and uniqueness of solution of infinite interval backward stochastic differential equation (BSDE) in the plane driven by a Brownian sheet.
This paper studies the existence and uniqueness of solution of infinite interval backward stochastic differential equation (BSDE) in the plane driven by a Brownian sheet.
By using the continuation theorem of Mawhin's coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition model {u'(t) = u(t)[r_1(t)-a_1(t)u(t)-b_1(t) ∫from x = -T to x = 0 of L_1(s)u(t+s)ds-c_1(t)∫from x = -T to x = 0 of K_1(s)v(t+s)ds], v'(t) = u(t)[r_2(t)-a_2(t)v(t)-b_2(t) ∫from x = -T to x = 0 of L_2(s)v(t+s)ds-c_2(t)∫from x = -T to x = 0 of K_2(s)u(t+s)ds], where r_1 and r_2 are continuous ω-periodic functions in R_+ = [0,∞), b_i (i = 1,2) is nonnegative continuous ω-periodic function in R_+ = [0,∞), b_i (i = 1,2) is nonnegative continuous ω-periodic function in R_+ = [0,∞), ω and T are positive constants, K_i, L_i ∈ C([-T,0], (0,∞)) and ∫from x = -T to x = 0 of K_i(s)ds = 1, ∫from x = -T to x = 0 of L_i(s)ds = 1, i = 1,2. Some known results are improved and extended.
By using the continuation theorem of Mawhin's coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition model {u'(t) = u(t)[r_1(t)-a_1(t)u(t)-b_1(t) ∫from x = -T to x = 0 of L_1(s)u(t+s)ds-c_1(t)∫from x = -T to x = 0 of K_1(s)v(t+s)ds], v'(t) = u(t)[r_2(t)-a_2(t)v(t)-b_2(t) ∫from x = -T to x = 0 of L_2(s)v(t+s)ds-c_2(t)∫from x = -T to x = 0 of K_2(s)u(t+s)ds], where r_1 and r_2 are continuous ω-periodic functions in R_+ = [0,∞), b_i (i = 1,2) is nonnegative continuous ω-periodic function in R_+ = [0,∞), b_i (i = 1,2) is nonnegative continuous ω-periodic function in R_+ = [0,∞), ω and T are positive constants, K_i, L_i ∈ C([-T,0], (0,∞)) and ∫from x = -T to x = 0 of K_i(s)ds = 1, ∫from x = -T to x = 0 of L_i(s)ds = 1, i = 1,2. Some known results are improved and extended.
The effect of dispersal on the permanence of population in a polluted patch is studied in this paper. The authors constructed a single-species dispersal model with stage-structure in two patches. The analysis focuses on the case that the toxicant input in the polluted patch has a limit value. The authors derived the conditions under which the population will be either permanent, or extinct.
The effect of dispersal on the permanence of population in a polluted patch is studied in this paper. The authors constructed a single-species dispersal model with stage-structure in two patches. The analysis focuses on the case that the toxicant input in the polluted patch has a limit value. The authors derived the conditions under which the population will be either permanent, or extinct.
An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ_m is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ψ(X) denote the number of edges with one end in X and the other not in X and ξ_m = min{ψ(x) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2 + 2, then λ_m ≤ ξ_m. The upper bound of λ_m is sharp.
An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ_m is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ψ(X) denote the number of edges with one end in X and the other not in X and ξ_m = min{ψ(x) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2 + 2, then λ_m ≤ ξ_m. The upper bound of λ_m is sharp.
Recently, Gijbels and Rousson suggested a new approach, called nonparametric least-squares test, to check polynomial regression relationships. Although this test procedure is not only simple but also powerful in most cases, there are several other parameters to be chosen in addition to the kernel and bandwidth. As shown in their paper, choice of these parameters is crucial but sometimes intractable. We propose in this paper a new statistic which is based on sample variance of the locally estimated pth derivative of the regression function at each design point. The resulting test is still simple but includes no extra parameters to be determined besides the kernel and bandwidth that are necessary for nonparametric smoothing techniques. Comparison by simulations demonstrates that our test performs as well as or even better than Gijbels and Rousson's approach. Furthermore, a real-life data set is analyzed by our method and the results obtained are satisfactory.
Recently, Gijbels and Rousson suggested a new approach, called nonparametric least-squares test, to check polynomial regression relationships. Although this test procedure is not only simple but also powerful in most cases, there are several other parameters to be chosen in addition to the kernel and bandwidth. As shown in their paper, choice of these parameters is crucial but sometimes intractable. We propose in this paper a new statistic which is based on sample variance of the locally estimated pth derivative of the regression function at each design point. The resulting test is still simple but includes no extra parameters to be determined besides the kernel and bandwidth that are necessary for nonparametric smoothing techniques. Comparison by simulations demonstrates that our test performs as well as or even better than Gijbels and Rousson's approach. Furthermore, a real-life data set is analyzed by our method and the results obtained are satisfactory.
In this paper we consider the risk process described by a piecewise deterministic Markov processes (PDMP). We mainly discuss the distribution of the deficit at ruin for the risk process. We derive the integro-differential equation satisfied by this distribution. We obtain the explicit expressions for it for certain choices of the claim amount distribution.
In this paper we consider the risk process described by a piecewise deterministic Markov processes (PDMP). We mainly discuss the distribution of the deficit at ruin for the risk process. We derive the integro-differential equation satisfied by this distribution. We obtain the explicit expressions for it for certain choices of the claim amount distribution.