An intermediate-precision approximation of the inverse cumulative normal distribution

Accurate methods used to evaluate the inverse of the standard normal cumulative distribution function at probability ρ commonly used today are too cumbersome and/or slow to obtain a large number of evaluations reasonably quickly, e.g., as required in certain Monte Carlo applications. Previously reported simple approximations all have a maximum absolute error ε_m > 10^-4 for a ρ-range of practical concern, such as Min[ρ,l?ρ]≥10_?6. An 11-term polynomial-based approximationis presented for which ε_m > 10^-6 in this range.     

The group inverse of finite homogeneous QBD processes

Generalized inverses of I?P, where P is a stochastic matrix, play an important role in the theory of Markov chains. In particular, the group inverse (I?P)^# has a probabilistic interpretation and is well suited for algorithmic implementation. We determine (I?P)^# for finite homogeneous quasi-birth-and-death (QBD) processes by exploiting both the structure of the process and the probabilistic properties of the group inverse.     

Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes

Abstract.  Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter μ explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logarithms of absolute μ -centred moments with respect to the logarithm of time are calculated explicitly for NIG Lévy processes. Computer implementation of the formulae obtained is briefly discussed. Finally, some further insight into the apparent scaling behaviour of NIG Lévy processes is gained.     

Singular weighing designs and estimation of total weight

The problem of estimation of total weight of objects using a singular spring balance weighing design has been studied in this paper. A lower bound of the estimated total weight is obtained and some classes of designs attaining the lower bound are studied.     

Questions & Answers: The Role of Mathematical Models in Clinical Research

Given any generalized inverse (X'X)^? appropriate to normal equations X'Xb ⁰ = X'y for the linear model y = Xb + e, a procedure is given for obtaining from it a generalized inverse appropriate to a restricted model having restrictions P'b = 0 for P'b nonestimable.     

Optimum spring balance weighing designs for estimating the total weight

The problem of estimation of the total weight or objects using a spring balance weighing design has been deait with in this paper Based on a theorem by Dey and Gupta (1977) giving a lower bound for the variance of the estimated total weight, a necessary and sufficient condition for this lower bound to be attained is obtained. A few special cases where the lower bound is attained are enumerated.     

A simplified proof of a theorem on the difference of the moore–penrose inverses of two positive semi–difinte matrices

Simplified proofs are given of a standard result that establishes positive semi–definiteness of the difference of the inverses of two non–singular matrices, and of the extension of this result by Milliken and Akdeniz (1977) to the difference of the Moore–Penrose inverse of two singular matrices.     

Generalized Inverse Gaussian Distributions and their Wishart Connections

The matrix generalized inverse Gaussian distribution (MGIG) is shown to arise as a conditional distribution of components of a Wishart distributio n. In the special scalar case, the characterization refers to members of the class of generalized inverse Gaussian distributions (GIGs) and includes the inverse Gaussian distribution among others     

Goodness-of-fit testing strategies from indirect observations

We consider in this paper a goodness-of-fit testing problem in a density framework. In particular, we deal with an error-in-variables model where each new incoming observation is gathered with a random independent error. It is well known that in such a situation, we are faced with an inverse (deconvolution) problem. Nevertheless, following recent results in the Gaussian white noise model, we prove that using procedures containing a deconvolution step is not always necessary.     

Asa Advisory Committees to Federal Statistical Agencies: A Report of the Committee on Committees of the American Statistical Association

(Editor's Note: The following report on ASA advisory committees was prepared by Edwin D. Goldfield as part of a continuing study of the Association's committees and Sections being conducted by the Committee on Committees. Because of the importance of advisory committees to Federal agencies, the report is included here for the information of the members of the Association.)     

Fitting and comparing seed germination models with a focus on the inverse normal distribution

This paper reviews current methods for fitting a range of models to censored seed germination data and recommends adoption of a probability‐based model for the time to germination. It shows that, provided the probability of a seed eventually germinating is not on the boundary, maximum likelihood estimates, their standard errors and the resultant deviances are identical whether only those seeds which have germinated are used or all seeds (including seeds ungerminated at the end of the experiment). The paper recommends analysis of deviance when exploring whether replicate data are consistent with a hypothesis that the underlying distributions are identical, and when assessing whether data from different treatments have underlying distributions with common parameters. The inverse normal distribution, otherwise known as the inverse Gaussian distribution, is discussed, as a natural distribution for the time to germination (including a parameter to measure the lag time to germination). The paper explores some of the properties of this distribution, evaluates the standard errors of the maximum likelihood estimates of the parameters and suggests an accurate approximation to the cumulative distribution function and the median time to germination. Additional material is on the web, at http://www.agric.usyd.edu.au/staff/oneill/ .     

Estimating the structural dimension of regressions via parametric inverse regression

A new estimation method for the dimension of a regression at the outset of an analysis is proposed. A linear subspace spanned by projections of the regressor vector X , which contains part or all of the modelling information for the regression of a vector Y on X , and its dimension are estimated via the means of parametric inverse regression. Smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model. No restrictions are placed on the distribution of the regressors. The estimate of the dimension of the regression is based on optimal estimation procedures. A simulation study shows the method to be more powerful than sliced inverse regression in some situations.     

On the Normal Inverse Gaussian Stochastic Volatility Model

In this article, the normal inverse Gaussian stochastic volatility model of Barndorff-Nielsen is extended. The resulting model has a more flexible lag structure than the original one. In addition, the second-and fourth-order moments, important properties of a volatility model, are derived. The model can be considered either as a generalized autoregressive conditional heteroscedasticity model with nonnormal errors or as a stochastic volatility model with an inverse Gaussian distributed conditional variance. A simulation study is made to investigate the performance of the maximum likelihood estimator of the model. Finally, the model is applied to stock returns and exchange-rate movements. Its fit to two stylized facts and its forecasting performance is compared with two other volatility models.     

Inverse power Maxwell distribution: statistical properties,estimation and application

In this paper, we introduced a new probability distribution named as inverse power Maxwell distribution. The proposal distribution can be seen as an extension of the Maxwell distribution with more flexibility in modeling upside-down lifetime data. Some statistical properties of this distribution are derived. In estimation viewpoint, five methods are used for estimating the unknown parameters of the distribution and these methods are performed through the simulation study. Finally, two real data sets were analyzed to illustrate the applicability of the proposed distribution, proving that it fits each real data set much better than some other existing distributions.     

Quantile regression for competing risks analysis under case-cohort design

The case-cohort design brings cost reduction in large cohort studies. In this paper, we consider a nonlinear quantile regression model for censored competing risks under the case-cohort design. Two different estimation equations are constructed with or without the covariates information of other risks included, respectively. The large sample properties of the estimators are obtained. The asymptotic covariances are estimated by using a fast resampling method, which is useful to consider further inferences. The finite sample performance of the proposed estimators is assessed by simulation studies. Also a real example is used to demonstrate the application of the proposed methods.     

New simulation results for the calibration and inverse median estimation problems

This paper gives the results of a new simulation study for the familiar calibration problem and the less familiar inverse median estimation problem. The latter arises when one wishes to estimate from a linear regression analysis the value of the independent variable corresponding to a specified value of the median of the dependent variable. For example, from the results of a regression analysis between stress and time to failure, one might wish to estimate the stress at which the median time to failure is 10,000 hours. In the study, the mean square error, Pitman closeness, and probability of overestimation are compared for both the calibration problem and the inverse median estimation problem for (1) the classical estimator, (2) the inverse estimator, and (3) a modified version of an estimator proposed by Naszodi (1978) for both a small sample and a moderately large sample situation.     

Application of the Bootstrap Approach to the Choice of Dimension and the α Parameter in the SIRα Method

To reduce the dimensionality of regression problems, sliced inverse regression approaches make it possible to determine linear combinations of a set of explanatory variables X related to the response variable Y in general semiparametric regression context. From a practical point of view, the determination of a suitable dimension (number of the linear combination of X) is important. In the literature, statistical tests based on the nullity of some eigenvalues have been proposed. Another approach is to consider the quality of the estimation of the effective dimension reduction (EDR) space. The square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. In this article, we focus on the SIR_α method and propose a naïve bootstrap estimation of the square trace correlation criterion. Moreover, this criterion could also select the α parameter in the SIR_α method. We indicate how it can be used in practice. A simulation study is performed to illustrate the behavior of this approach.     

A generalized inverse regression estimator in multi-univariate linear calibration

A multivariate linear calibration problem, in which response variable is multivariate and explanatory variable is univariate, is considered. In this paper a class of generalized inverse regression estimators is proposed in multi-univariate linear calibration. It includes the classical estimator and the inverse regression one (or Krutchkoff estimator). For the proposed estimator we derive the expressions of bias and mean square error (MSE). Furthermore the behavior of these characteristics is investigated through an analytical method. In addition through a numerical study we confirm the existence of a generalized inverse regression estimator to improve both the classical and the inverse regression estimators on the MSE criterion.     

On asymptotic approximation of inverse moments for a class of nonnegative random variables

Sung [On inverse moments for a class of nonnegative random variables. J Inequal Appl. 2010;2010:1–13. Article ID 823767, doi:10.1155/2010/823767] obtained the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite second moments and satisfying a Rosenthal-type inequality. In the paper, we further study the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite first moments, which generalizes and improves the corresponding ones of Wu et al. [Asymptotic approximation of inverse moments of nonnegative random variables. Statist Probab Lett. 2009;79:1366–1371], Wang et al. [Exponential inequalities and inverse moment for NOD sequence. Statist Probab Lett. 2010;80:452–461; On complete convergence for weighted sums of ? mixing random variables. J Inequal Appl. 2010;2010:1–13, Article ID 372390, doi:10.1155/2010/372390], Sung (2010) and Hu et al. [A note on the inverse moment for the nonnegative random variables. Commun Statist Theory Methods. 2012. Article ID 673677, doi:10.1080/03610926.2012.673677].