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This paper attempts to develop a repetitive group sampling (RGS) plan by variables inspection for controlling the process fraction defective or the number of nonconformities when the quality characteristic follows a normal distribution and has only the lower or upper specification limit. The proposed sampling plan is derived by the exact sampling distribution rather than the approximation approach. The plan parameters are solved by a nonlinear optimization model which minimizes the average sample number required for inspection and fulfills the classical two-point conditions on the operating characteristic (OC) curve. The efficiency of the proposed variables RGS is examined and also compared with the existing variables single sampling plan in terms of the sample size required for inspection. The results indicate that the proposed variables RGS plan could significantly reduce samples required for inspection compared to the traditional variables single sampling plan. [ABSTRACT FROM AUTHOR]

Let (X1, X2) be a bivariate Lp-norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α1,α2. If p=α1=α2=2, then (X1, X2) is a spherical random vector. The estimation of the conditional distribution of Zu*:=X2 | X1>u for u large is of some interest in statistical applications. When (X1, X2) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Zu* can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Zu:=X2| X1=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Zu* and Zu can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function. [ABSTRACT FROM AUTHOR]

The phase space characteristics of a quantum state are best captured by the Wigner distribution. This displays transparently the diagonality information of the density matrix. The complementary function offering transparently the off-diagonal elements is captured by a function called the S-function, or the ambiguity. In carrying the maximal information about the quantum coherences it represents the uncertainties or ambiguity of the diagonal information. Mathematically this is manifested in its role as the phase space moment generating function. Formally it complements the information in the Wigner function. These formal relations provide the starting point for the present investigations. As a measure of quantum uncertainties, ambiguity may be used to define a probability measure on the off-diagonality. The mathematical and physical consistency of this view is presented in this paper. For a pure state, we find the extraordinary result that such distributions are their own Fourier transforms. The physical interpretation of this distribution as a carrier of classical signal fuzziness suggests the introduction of heuristic approximations to the observational uncertainties. We illustrate the properties and interpretation of the ambiguity function by some specific examples. We find that for smooth, 'Gaussian-like' distributions, the heuristic considerations provide good approximations. On the other hand, representing quantum interferences, the ambiguity serves as the most positive probe for the ultimate quantum structures which have been called sub-Planckian. They are interesting because it has been argued that such structures are physically observable. [ABSTRACT FROM AUTHOR]

In the growing literature of factor analysis, little is done to understand the finite sample properties of an approximate factor model solution. In empirical applications with relatively small samples, the asymptotic theory might be a poor approximation and the resulting distortions might affect the estimation (bias in the point estimate and the standard errors) and the statistical inference. The present paper uses the estimation method of Bai and Ng [Bai, J. and Ng, S., 2002, Determining the number of factors in approximate factor models. Econometrica , 70, 191–221.] and assesses the sampling behavior of the estimated common components, common factors and factor loadings. The study compares the empirical distributions to the asymptotic theory of Bai [Bai, J., 2003, Inference on factor models of large dimension. Econometrica , 71, 135–171.]. Simulation results suggest that the point estimates have a Gaussian distribution for panels with relatively small dimensions. However, these estimates have a significant finite sample bias and the dispersion of their sampling distribution is severely underestimated by the asymptotic theory. [ABSTRACT FROM AUTHOR]

This paper proposes several methods for approximating the expected value of an eigenvalue of a random matrix. A comparative study assesses the accuracy of these methods. The proposed methods for approximating the mean value of an eigenvalue of a random matrix consist of a regression approach using the dimensions of the data and position of the eigenvalue as predictors, two methods using expected values of order statistics for the normal distribution, and another approach using percentiles of the normal distribution. These methods provide researchers who desire to readily determine the dimensionality of data by using parallel analysis, with easy-to-implement alternatives to calculate the mean eigenvalues of a random matrix. [ABSTRACT FROM AUTHOR]