Asymptotic Correlations Between Rotated Solutions in Factor Analysis

The asymptotic correlations between differently rotated solutions in exploratory factor analysis are derived. The solutions are orthogonally or obliquely rotated for unstandardized or standardized manifest variables. To obtain the asymptotic correlations between different solutions, the covariance models for manifest variables have been constructed so that two sets of solutions are involved in a single covariance structure. The asymptotic correlations can be used for the statistical test of the differences of rotated solutions. The correlation matrix between the rotated factors of the first solution and those of the second is also introduced in the models with appropriate restrictions to identify the models. The asymptotic standard errors of the estimates of the correlations between the factors in different solutions are simultaneously provided. A numerical example is presented with simulated values.

     

Standard Errors for the Harris-Kaiser Case II Orthoblique Solution

Asymptotic standard errors of the estimates of the obliquely rotated parameters by the Harris-Kaiser Case II orthoblique method are derived under the assumption of the multivariate normal distribution for observed variables. A covariance structure model for observed variables is constructed such that both unrotated and orthogonally rotated parameters are involved in the model. The asymptotic standard errors for the final oblique solution (orthoblique solution) are derived by a stepwise method. First, the asymptotic variance-covariance matrix for the estimates of the unrotated and orthogonally rotated parameters is derived. Second, the delta method is used to obtain the asymptotic variances of the estimates of the obliquely rotated parameters. Results by simulation indicate that the theoretical values of the asymptotic standard errors are close to simulated ones.

     

Analysis of left-truncated right-censored or doubly censored data with linear transformation models

We analyze left-truncated right-censored (LTRC) data or doubly censored data using semiparametric transformation models. It is demonstrated that the extended estimating equations of both?Cheng et al. (Biometrika 82:835?C845,?1995) and?Chen et al. (Biometrika 89:659?C668,?2002) can be used to analyze LTRC data or doubly censored data when left-censored variables are always observed. The asymptotic properties of the proposed estimators are derived. A simulation study is conducted to investigate the performance of the proposed estimators.     

A Test Statistic in Canonical Correlation Analysis for Categorical Variables

In canonical correlation analysis (CCA), it is important to estimate the number of nonzero canonical correlations in the population. One way to estimate the number is to consider the dimensionality testing problem. In CCA for continuous variables, some test statistics for the problem have been derived under the normality assumption. However, there are only a few papers on test statistics in CCA for categorical variables.

In this article, a test statistic in CCA for categorical variables is suggested. The test statistic is derived from the rational used for continuous variables. Some properties of the test statistic are examined through mathematical investigations and numerical simulations.

     

Multicollinearity and Avoidance in Regression Analysis

For the multiple linear regression problem, a number of alternative estimators to ordinary least squares (OLS) have been proposed for situations in which multicollinearity is present among the explanatory variables. Multicollinearity may have several adverse effects on estimated coefficients in a multiple regression analysis.

This paper investigates the relative efficiency of these 12 alternative estimators from the point of view of mean squared error (MSE) by the Monte Carlo simulation, and discusses the practical implication of the use of such estimators. The results of this study are that OLS, Ridges, BYS and ITR estimators are more efficient than the others, when multicollinearity is not present. However, when multicollinearity is present, Ridges, GRB, BYS, ITR and PCA estimators are more efficient than OLS for almost all values of σ. Ridges have uniformly smaller MSEs than OLS. Relative efficiencies of these estimators vary with the value of σ. In the interval of small σ. Ridges are more efficient than the others, but, for large σ, each of GRB, BYS, ITR, PCA and LAT is more efficient.

From our experiment in which these 12 estimators are applied to the economic data of France, we find that, while OLS has the negative coefficient, some of these alternative have positive appropriate values, where regression coefficient must have the positive sign from the point of view of Economics. Therefore, we can conclude that these alternative estimators are effective for the practical regression problem with multicollinearity.

     

A study of two estimation approaches for parameters of Weibull distribution based on WPP

Least-squares estimation (LSE) based on Weibull probability plot (WPP) is the most basic method for estimating the Weibull parameters. The common procedure of this method is using the least-squares regression of Y on X, i.e. minimizing the sum of squares of the vertical residuals, to fit a straight line to the data points on WPP and then calculate the LS estimators. This method is known to be biased. In the existing literature the least-squares regression of X on Y, i.e. minimizing the sum of squares of the horizontal residuals, has been used by the Weibull researchers. This motivated us to carry out this comparison between the estimators of the two LS regression methods using intensive Monte Carlo simulations. Both complete and censored data are examined. Surprisingly, the result shows that LS Y on X performs better for small, complete samples, while the LS X on Y performs better in other cases in view of bias of the estimators. The two methods are also compared in terms of other model statistics. In general, when the shape parameter is less than one, LS Y on X provides a better model; otherwise, LS X on Y tends to be better.     

Asymptotic Expansions of the Distribution of the Estimator for the Generalized Partial Correlation Under Nonnormality

The generalized partial correlation is denned as a correlation between two variables, where the linear effects of common and unique third variables are partialed out from the two variables. The generalized partial correlation includes simple, partial, part/semipartial and bipartial correlations as special cases. The Edgeworth expansion of the distribution of the standardized sample coefficient for the generalized partial correlation is obtained up to order O(1/n) under nonnormality. Also asymptotic expansions of the distribution of the Studentized estimator are obtained using the Edgeworth expansion, Cornish-Fisher expansion and Hall’s method with variable transformation. As extensions, the results of multivariate cases or generalized partial set-correlations are given.

     

An Estimation Method for the Total Quantity of Goods in Distribution by the Use of a Regression Curve

A method for estimating the total quantity of goods distributed in a market is presented by the use of a polynomial regression curve on the plane where the y-coordinate consists of the rates of goods for their product numbers.

Two kinds of estimators for the above-mentioned total quantity are proposed with investigating their asymptotic properties.

As a practical example, estimating the total quantity of beer cases in a market is considered.

     

Hodograph transformation for crack-tip fields in hyperelastic sheets: higher order eigenmodes and asymptotic path-independent integrals

Hodograph transformations can be used to linearize a nonlinear partial differential equation by judicious use of physical quantities (e.g. velocities or displacement gradients) as coordinate variables in the hodograph plane. This approach has been found useful for obtaining the leading order terms of eigenproblems that govern asymptotic singular crack fields in nonlinear materials. There is little work on the use of the hodograph transformation for obtaining higher order terms in the asymptotic expansion of the crack tip fields. In this paper, we develop a framework to obtain such higher order terms using the hodograph transformation. The method relies heavily on the representation of physical quantities of interest in terms of hodograph plane variables. We demonstrate the method via application to a generalized neo-Hookean material. In addition, asymptotic path-independent J-integrals are expressed in terms of either physical or hodograph variables and are used to compute the leading-order amplitude coefficients. A relationship between the asymptotic J-integrals and the energy release rate is established for a mixed crack mode. The asymptotic results are compared with numerical results from finite element computation and excellent agreement is obtained.

     

Asymptotic approximations for probability integrals

This paper considers the asymptotic evaluation of probability integrals. The usual methods require that all random variables are transformed into standard normal variables. The method described here does not use such transformations. Asymptotic approximations are derived in the original space of the random variables. In this way it is also possible to obtain simple formulas for the sensitivity of the failure probability to changes in the distribution parameters.     

Semi-parametric second-order reduced-bias high quantile estimation

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.     

Parameter Estimation for a Generalized Gamma Distribution

It is fairly commonplace in reliability analyses to encounter data which is incompatible with the exponential, Weibull, and other familiar probability models. Such data motivates research to enlarge the group of probability distributions which are useful to the reliability analyst.

In this paper, we examine a three-parameter generalization of the gamma distribution and derive parameter estimation techniques for that distribution. Those techniques, in the general case, depend upon method of moments considerations which lead to simultaneous equations for which closed form solutions are not available. Graphic solution is proposed and aids to the computations are provided. Major concepts in the paper are summarized by means of a numerical example.

Details are given for the special case in which only the scale parameter is unknown. Three unbiased estimators for that parameter are derived along with their variance formulas. Minimum variance considerations are discussed by application of the Cramér-Rao Theorem.     

A weighted mean excess function approach to the estimation of Weibull-type tails

A new class of estimators for the Weibull-tail coefficient is proposed. The estimators are based on linear combinations of log-spacings of the mean excess function evaluated at high levels. The asymptotic distribution of this new class of estimators is derived under some mild conditions on the weight function and a second order condition on the tail behavior. The finite sample properties of some estimators obtained with important special cases of the weight function are examined with a small simulation experiment.     

Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

Asymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric. Explicit formulas for other singular circular edges such as a circumferential crack, an external crack and a 3π/2 reentrant corner are also derived. The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation.     

Inferences for odds ratio with dependent pairs

In familial data analyses quantifying familial association is scientifically important. As analogies of the intraclass and interclass correlations of a normally distributed trait, we study intraclass and interclass (log) odds ratios for a binary trait. We propose non-parametric estimators of the odds ratios and derive the asymptotic variances of the estimators under the assumptions of exchangeability and closure of multivariate binary distributions under marginals. These estimators are straightforward, except for the consideration of how to weight by family size. The relative efficiencies of the non-parametric estimators are studied for some parametric models. It shows that our estimators are highly efficient, and that weighting by family size is recommended for the intraclass odds ratio. The computations of the estimators and their standard errors are illustrated with two examples. This research is part of the PhD thesis of the first author. Support from US Army grant NF990038 and an NSERC grant is gratefully acknowledged.     

Asymptotic behavior of robust estimators in partially linear models with missing responses: the effect of estimating the missing probability on the simplified marginal estimators

In this paper, we consider a semiparametric partially linear regression model where missing data occur in the response. We derive the asymptotic behavior of the robust estimators for the regression parameter and of the weighted simplified location estimator introduced in Bianco et al. (Comput. Stat. Data Anal. 54:546–564, 2010a). For the latter, consistency results and the asymptotic distribution are derived when the missing probability is known and also when it is estimated.     

Uniform integrability of the OLS estimators,and the convergence of their moments

The problem of convergence of moments of a sequence of random variables to the moments of its asymptotic distribution is important in many applications. These include the determination of the optimal training sample size in the cross-validation estimation of the generalization error of computer algorithms, and in the construction of graphical methods for studying dependence patterns between two biomarkers. In this paper, we prove the uniform integrability of the ordinary least squares estimators of a linear regression model, under suitable assumptions on the design matrix and the moments of the errors. Further, we prove the convergence of the moments of the estimators to the corresponding moments of their asymptotic distribution, and study the rate of the moment convergence. The canonical central limit theorem corresponds to the simplest linear regression model. We investigate the rate of the moment convergence in canonical central limit theorem proving a sharp improvement of von Bahr’s (Ann Math Stat 36:808–818, 1965) theorem.     

Field data analyses with additional after-warranty failure data

This paper proposes methods of estimating the lifetime distribution for situations where additional field data can be gathered after the warranty expires in a parametric time to failure distribution. For satisfactory inference about parameters involved, it is desirable to incorporate these after-warranty data in the analysis. It is assumed that after-warranty data are reported with probability p₁(<1), while within-warranty data are reported with probability 1. Methods of obtaining maximum likelihood estimators are outlined, their asymptotic properties are studied, and specific formulas for Weibull distribution are obtained. An estimation procedure using the expectation and maximization algorithm is also proposed when the reporting probability is unknown. Simulation studies are performed to investigate the properties of the estimates.     

Dating multiple change points in the correlation matrix

A nonparametric procedure for detecting and dating multiple change points in the correlation matrix of sequences of random variables is proposed. The procedure is based on a recently proposed test for changes in correlation matrices at an unknown point in time. Although the procedure requires constant expectations and variances, only mild assumptions on the serial dependence structure are assumed. The convergence rate of the change point estimators is derived and the asymptotic validity of the procedure is proved. Moreover, the performance of the proposed algorithm in finite samples is illustrated by means of a simulation study and the analysis of a real data example with financial returns. These examples show that the algorithm has large power in finite samples.     

Maximum likelihood estimator for jitter noise models [HF samplingscopes]

High-frequency sampling scopes suffer from both additive noise and time jitter. The classical techniques for identifying the additive noise and time jitter noise are based on linear least squares (LS) estimators. This work derives a maximum likelihood (ML) estimator and compares its performance with the LS estimator. Simulation results show the gain in efficiency of the proposed method