Bayesian single and double variable sampling plans for the Weibull distribution with censoring

The sampling inspection problem is one of the main research topics in quality control. In this paper, we employ Bayesian decision theory to study single and double variable sampling plans, for the Weibull distribution, with Type II censoring. A general loss function which includes the sampling cost, the time-consuming cost, the salvage value, and the after-sales cost is proposed to determine the Bayes risk and the corresponding optimal sampling plan. Explicit expressions for the Bayes risks for both single and double sampling plans are derived, respectively. Numerical examples are given to illustrate the effectiveness of the proposed method. Comparisons between single and double sampling plans are made, and sensitivity analysis is performed.     

An optimum multivariate stratified double sampling design in presence of non-response

In stratified sampling when strata weights are unknown double sampling technique may be used to estimate them. At first a large simple random sample from the population without considering the stratification is drawn and sampled units belonging to each stratum are recorded to estimate the unknown strata weights. A stratified random sample is then obtained comprising of simple random subsamples out of the previously selected units of the strata. If the problem of non-response is there, then these subsamples may be divided into classes of respondents and non-respondents. A second subsample is then drawn out of non-respondents and an attempt is made to obtain the information. This procedure is called Double Sampling for Stratification (DSS). Okafor (Aligarh J Statist 14:13–23, 1994) derived DSS estimators based on the subsampling of non-respondents. Najmussehar and Bari (Aligarh J Statist 22:27–41, 2002) discussed an optimum double sampling design by formulating the problem as a mathematical programming problem and used the dynamic programming technique to solve it. In the present paper a multivariate stratified population is considered with unknown strata weights and an optimum sampling design is proposed in the presence of non-response to estimate the unknown population means using DSS strategy. The problem turns out to be a multiobjective integer nonlinear programming problem. A solution procedure is developed using Goal Programming technique. A numerical example is presented to illustrate the computational details.     

The double deflating technique for irreducible singular M-matrix algebraic Riccati equations in the critical case

As is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective.     

Determining the Optimum Stratum Boundaries Using Mathematical Programming

The method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation, is known as optimum stratification. In order to make the strata internally homogeneous, the strata are constructed in such a way that the strata variances should be as small as possible for the characteristic under study. In this paper the problem of determining optimum strata boundaries (OSB) is discussed when strata are formed based on a single auxiliary variable with a varying measurement cost per units across strata. The auxiliary variable considered in the problem is a size variable that holds a common model for a whole population. The OSB are achieved effectively by assuming a suitable distribution of the auxiliary variable and creating strata by cutting the range of the distribution at optimum points. The problem of finding the OSB, which minimizes the variance of the estimated population mean under a weighted stratified balanced sampling, is formulated as a mathematical programming problem (MPP). Treating the formulated MPP as a multistage decision problem, a solution procedure using dynamic programming technique is developed. A numerical example using a hospital population data is presented to illustrate the computational details of the solution procedure. A software program coded in JAVA is written to carry out the computation. The distribution of the auxiliary variable in this example is considered to be continuous with an exponential density function.     

A generalized stochastic goal programming model

In this paper we show how one can get stochastic solutions of Stochastic Multi-objective Problem (SMOP) using goal programming models. In literature it is well known that one can reduce a SMOP to deterministic equivalent problems and reduce the analysis of a stochastic problem to a collection of deterministic problems. The first sections of this paper will be devoted to the introduction of deterministic equivalent problems when the feasible set is a random set and we show how to solve them using goal programming technique. In the second part we try to go more in depth on notion of SMOP solution and we suppose that it has to be a random variable. We will present stochastic goal programming model for finding stochastic solutions of SMOP. Our approach requires more computational time than the one based on deterministic equivalent problems due to the fact that several optimization programs (which depend on the number of experiments to be run) needed to be solved. On the other hand, since in our approach we suppose that a SMOP solution is a random variable, according to the Central Limit Theorem the larger will be the sample size and the more precise will be the estimation of the statistical moments of a SMOP solution. The developed model will be illustrated through numerical examples.     

The double pivot simplex method

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time as an iteration of the simplex method. Computational experiments demonstrate that DPSM decreases the average number of pivots by approximately 41% on a small set of benchmark instances.     

On a confidence interval of given length for the parameter of the binomial and the Poisson distributions

Summary The length of the confidence interval which is constructed by one stage sampling is a random variable, whereas we can construct a confidence interval of given length and of given confidence coefficient by a sequential sampling. Of cource, the sample size required by this method is a random variable. This type of problem has already been discussed in several papers when the population is normal (see references [2], [3], [4]). In this paper we shall discuss this subject in the binomial and Poisson cases.     

A parameter uniform difference scheme for parabolic partial differential equation with a retarded argument

The paper deals with the analysis of a non-stationary parabolic partial differential equation with a time delay. The highest order derivative term is affected by the small parameter. This is precisely the case when the magnitude of the convective term becomes much larger compare to that of diffusion term. The solution of problem exhibits steep gradients in the narrow intervals of space and short interval of times. In these cases a dissipative loss turned out to be more complex. Even for the one spatial dimension and one temporal variable, not all difference scheme can capture these steep variation. Although the analysis is restricted to the model in one space dimension, the technique and comparison principles developed should prove useful in assessing the merits of numerical solution of other nonlinear model equations too.     

Optimal stochastic single-machine-tardiness scheduling by stochastic branch-and-bound

A stochastic branch-and-bound technique for the solution of stochastic single-machine-tardiness problems with job weights is presented. The technique relies on partitioning the solution space and estimating lower and upper bounds by sampling. For the lower bound estimation, two different types of sampling (“within” and “without” the minimization) are combined. Convergence to the optimal solution (with probability one) can be demonstrated. The approach is generalizable to other discrete stochastic optimization problems. In computational experiments with the single-machine-tardiness problem, the technique worked well for problem instances with a relatively small number of jobs; due to the enormous complexity of the problem, only approximate solutions can be expected for a larger number of jobs. Furthermore, a general precedence rule for the single-machine scheduling of jobs with uncertain processing times has been derived, essentially saying that “safe” jobs are to be scheduled before “unsafe” jobs.     

An Inverse Problem for a Nonlinear Diffusion-Convection Equation

A method for constructing the Dirichlet-to-Neumann map for a nonlinear diffusion-convection equation is presented. The problem is reduced to the solution of a nonlinear integral equation in one independent variable. Existence and uniqueness of the solution may be proven for small times via a contraction mapping technique.     

The equimaterial approach for the numerical solution of the one dimensional transient diffusion equation with zero flux boundary conditions

A new approach is proposed for the grid motion for the numerical solution of a general transient diffusion equation in one spatial dimension with zero flux boundary conditions. The new criterion for grid motion is that the solute amount contained in each discretization section should be a pre-described fraction of the total solute amount at each time step. This requirement is not explicitly enforced to the solution technique but it is implicitly included in the equation through the appropriate variable transformation. The results showed that although the technique leads to the required grid motion the numerical results are of pure quality due to the appearance of singularities during the variable transformation procedure. Nevertheless, it is shown that by appropriate numerical handling of the solution at the singularity region the technique can lead to accurate results and potentially can replace the existing moving grid algorithms at least for the particular problem at hand.     

Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions

In this article, we consider the multilevel sequential Monte Carlo (MLSMC) method of Beskos et al. (Stoch. Proc. Appl. [to appear]). This is a technique designed to approximate expectations w.r.t. probability laws associated to a discretization. For instance, in the context of inverse problems, where one discretizes the solution of a partial differential equation. The MLSMC approach is especially useful when independent, coupled sampling is not possible. Beskos et al. show that for MLSMC the computational effort to achieve a given error, can be less than independent sampling. In this article we significantly weaken the assumptions of Beskos et al., extending the proofs to non-compact state-spaces. The assumptions are based upon multiplicative drift conditions as in Kontoyiannis and Meyn (Electron. J. Probab. 10 [2005]: 61–123). The assumptions are verified for an example.     

The first and second fundamental boundary-value problems of elasticity theory for a transversally isotropic paraboloid of revolution

We give an exact solution of the interior and exterior problems of elasticity theory for a transversally isotropic paraboloid of revolution in the case when the stresses prescribed on its surface or the displacements along one variable can be represented by a Hankel integral and those along the other variable can be expanded in a trigonometric series. It is assumed that the roots of the characteristic equation are of multiplicity greater than one.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 60–70.     

Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods

The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of fixed step size Nordsieck methods requires weaker quasi-consistency than classical Runge-Kutta formulas, which require consistency up to a certain order. In other words, quasi-consistent Nordsieck methods on fixed step size meshes have a higher order of convergence than on variable step size ones. This fact creates certain difficulties in the automatic error control of these methods. It is shown how quasi-consistent methods can be modified so that the high order of convergence is preserved on variable step size meshes. The regular techniques proposed can be applied to any quasi-consistent Nordsieck methods. Specifically, it is shown how this technique performs for Nordsieck methods based on the multistep Adams-Moulton formulas, which are the most popular quasi-consistent methods. The theoretical conclusions of this paper are confirmed by the numerical results obtained for a test problem with a known solution.     

A Consistent and Stable Approach to Generalized Sampling

We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from a finite number of samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique or as solutions of an overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-conditioned in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and well-conditioned when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, stable and consistent with the original measurements. We also provide analysis in the case of recovering wavelets coefficients from Fourier samples. We show that for compactly supported wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples available.     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

Adaptive multiquadric collocation for boundary layer problems

An adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution.     

Nonlinear interpolation of functions of very many variables

The difficulty is first shown in the nonlinear interpolation of functions defined in a space of very many dimensions. There is a method using a sampling technique [J. ACM, Vol. 17, pp. 420–425, July 1970] that works fairly well in the regime ofk~10?20 (k=number of dimensions). The sampling errors, however, increase exponentially with increasingk, so that fork greater than the above values the computation is no more feasible. This is due to the subtraction between two large sums of about the same magnitude, each of which suffers stochastic fluctuations accompanying the samplings. To avoid this difficulty, a “pairwise” sampling method is devised where one draws two samples at a time when required, one from each of these two sums of terms. With the use of this technique, standard errors are reduced by orders of magnitude. Some details of algorithm are given together with typical computed examples.     

On analytical solutions of vibrations of rods with variable cross sections

Analytical solutions of several rods whose cross-sections vary in the axial direction are considered. The analysis in this paper uses two transformations that exist in the literature to help transform the equation of motion of the rod into a form similar to that of the one-dimensional Schroedinger equation where the shape of the cross-section of the rod is governed by a potential function that satisfies a second order differential equation. By solving this second order differential equation, it is possible to obtain a class of shapes that have a common form of solution that are determined from the well known solution of the Schroedinger equation. There are very few analytical solutions available in the literature especially when the area of cross-section varies. In this paper, it is shown that the set of available solutions for variable cross-sectional rods are particular cases of the general analysis. In addition, analytical solutions to several new variable cross-sections are considered.This paper also considers an existing transformation that transforms the Sturm-Liouville equation to a particular form of the equation of motion under consideration. So, tracing the analysis backwards, for any second order differential equation for which existence of solutions are guaranteed, and that can be reduced in the above manner, it is possible to determine the form of the cross-section of the rod. In this paper, several shapes of the cross-section are investigated when the equation of motion is described by special functions, the Legendre, Hermite and Laugurre functions. The analysis given here is a general one and is applicable when the equation of motion is described by other special functions.     

Analytical and computational solution of adaptive inventory processes

H. Scarf (Ann. Math. Statist.30 (2) (1959)) has discussed a technique for the computation of optimal inventory level in the case where the demand distribution contains an unknown statistical parameter. It was assumed that initially the parameter could be described by a priori distribution, which would be subsequently revised on the basis of additional demand information. By assuming that the demand distribution of a cumulative observed demand is a sufficient statistic for the unknown parameter, it was shown that the optimal inventory levels could be obtained by the recursive computation of a sequence of function of two variables. He also showed (N.R.L.A.7 (8) (1960)) that if the demand distributions are gamme distributions and if the holding and penalty costs are linear, then the sequence of function of two variables may be solved by a related set of equations requiring tabulation of functions of only one variable. The main part of the argument presented in this paper is that an analytical consideration and numerical examples of equation requiring tabulation of function of only one variable given by H. Scarf were given with special choices of the factors, and were compared with solution when demand distribution is assumed exactly known and unknown.