A study of uniformity pattern for extended designs

The objective of this paper is to investigate the issue of projection discrepancy for extended U-type or nearly U-type extended designs along the line of Fang and Qin (2005 Fang, K. T., and H. Qin. 2005. Uniformity pattern and related criteria for two-level factorials. Science China Series A 48:1–11.[Crossref] , [Google Scholar]) based on the centered L₂-discrepancy proposed in Hickernell (1998 Hickernell, F. J. 1998. A generalized discrepancy and quadrature error bound. Mathematics of Computation 67:299–322.[Crossref], [Web of Science ®] , [Google Scholar]). Extended designs are obtained through augmenting optimally few runs (or points) to an optimal U-type design. Lower bounds to projection discrepancy with reference to the centered L₂-discrepancy of extended designs have been obtained. Some illustrative examples are also provided.     

Stochastic Comparison on General Inactivity Time and General Residual Life of k-Out-of-n Systems

This paper conducts stochastic comparison on general residual life and general inactivity time of (n ? k + 1)-out-of-n systems and investigates the stochastic behavior of the general inactivity time of a system with units having decreasing reversed hazard rate. These results strengthen some conclusions in both Khaledi and Shaked (2006 Khaledi , B. , Shaked , M. ( 2006 ). Ordering conditional residual lifetimes of coherent systems . Journal of Statistical Planning and Inference 137 : 1173 – 1184 .[Crossref], [Web of Science ®] , [Google Scholar]) and Hu et al. (2007 Hu , T. , Jin , W. , Khaledi , M. ( 2007 ). Ordering conditional distributions of generalized order statistics . Probability in the Engineering and Informational Sciences 21 : 401 – 417 .[Crossref], [Web of Science ®] , [Google Scholar]).     

D s -Optimal Designs for Quadratic Log Contrast Model for Experiments with Mixtures

The approximate D _s -optimal design for discriminating between linear and quadratic log contrast models for experiments with mixtures suggested by Aitchison and Bacon-Shone (1984 Aitchison , J. , Bacon-Shone , J. ( 1984 ). Log contrast models for experiments with mixtures . Biometrika 71 : 323 – 330 .[Crossref], [Web of Science ®] , [Google Scholar]) is investigated, where the experimental domain is restricted further as in Chan (1992 Chan , L. Y. ( 1992 ). D-Optimal design for a quadratic log contrast model for experiments with mixtures . Commun. Statist. Theor. Meth. 21 ( 10 ): 2909 – 2930 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). It is found that for a symmetric subspace of the finite dimensional simplex, there is a D _s -optimal design with the nice structure that puts a weight 1/2 ^k?1 on the centroid of this subspace and the remaining weight is uniformly distributed on the vertices of the experimental domain. Finally, the D _s -efficiency of the D-optimal design for quadratic model and the design given by Aitchison and Bacon-Shone (1984 Aitchison , J. , Bacon-Shone , J. ( 1984 ). Log contrast models for experiments with mixtures . Biometrika 71 : 323 – 330 .[Crossref], [Web of Science ®] , [Google Scholar]) are also discussed.     

On Testing Sample Selection Bias Under the Multicollinearity Problem

ABSTRACT

This paper reviews and extends the literature on the finite sample behavior of tests for sample selection bias. Monte Carlo results show that, when the “multicollinearity problem” identified by Nawata (1993 Nawata , K. ( 1993 ). A note on the estimation of models with sample-selection biases . Economics Letters 42 : 15 – 24 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) is severe, (i) the t-test based on the Heckman–Greene variance estimator can be unreliable, (ii) the Likelihood Ratio test remains powerful, and (iii) nonnormality can be interpreted as severe sample selection bias by Maximum Likelihood methods, leading to negative Wald statistics. We also confirm previous findings (Leung and Yu, 1996 Leung , S. F. , Yu , S. ( 1996 ). On the choice between sample selection and two-part models . Journal of Econometrics 72 : 197 – 229 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) that the standard regression-based t-test (Heckman, 1979 Heckman , J. J. ( 1979 ). Sample selection bias as a specification error . Econometrica 47 : 153 – 161 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) and the asymptotically efficient Lagrange Multiplier test (Melino, 1982 Melino , A. ( 1982 ). Testing for sample selection bias . Review of Economic Studies 49 : 151 – 153 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), are robust to nonnormality but have very little power.     

A Measure of Multivariate Kurtosis with Principal Components

Srivastava (1984 Srivastava , M. S. ( 1984 ). A measure of skewness and kurtosis and a graphical method for assessing multivariate normality . Statist. Probab. Lett. 2 ( 5 ): 263 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) defined a measure of multivariate kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Some new results are obtained by generalizing Srivastava's theorem to an asymptotic expansion up to higher order. Finally, two numerical examples are presented.     

Change Point Detection in a General Class of Distributions

ABSTRACT

We consider the change point problem in a general class of distributions, and derive a test statistic T _n which reduces to the statistic obtained by Kander and Zacks (1966 Kander , Z. , Zacks , S. ( 1966 ). Test procedure for possible changes in parameter of statistical distributions occurring at unknown time points. Ann. Math. Statist. 37 : 1196 – 1210 . [CSA] [Crossref] , [Google Scholar]) for the exponential family. Properties of the test, including its asymptotic distribution, are discussed.     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

Use of Two Decks of Cards in Randomized Response Techniques for Complex Survey Designs

The randomized response (RR) technique with two decks of cards proposed by Odumade and Singh (2009 Odumade , O. , Singh , S. ( 2009 ). Efficient use of two deck of cards in randomized response sampling . Commun. Statist. Theor. Meth. 38 : 439 – 446 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can always be made more efficient than the RR techniques proposed by Warner (1965 Warner , S. L. ( 1965 ). Randomize response: A survey technique for eliminating evasive answer bias . J. Amer. Statist. Assoc. 60 : 63 – 69 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Mangat and Singh (1990 Mangat , N. S. , Singh , R. ( 1990 ). An alternative randomized response procedure . Biometrika 77 : 349 – 442 .[Crossref], [Web of Science ®] , [Google Scholar]), and Mangat (1994 Mangat , N. S. ( 1994 ). An improved randomized response strategy . J. Roy. Statist. Soc. B 56 : 93 – 95 . [Google Scholar]) by adjusting the proportion of cards in the decks. The proposed method of Odumade and Singh (2009 Odumade , O. , Singh , S. ( 2009 ). Efficient use of two deck of cards in randomized response sampling . Commun. Statist. Theor. Meth. 38 : 439 – 446 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is limited to simple random sampling with replacement (SRSWR) sampling only. In this article, generalization of Odumade and Singh strategy is provided for complex survey designs and a wider class of estimators. The results of Odumade and Singh (2009 Odumade , O. , Singh , S. ( 2009 ). Efficient use of two deck of cards in randomized response sampling . Commun. Statist. Theor. Meth. 38 : 439 – 446 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can be derived from the proposed method as a special case.     

Likelihood Ratio Order of m-Spacings in Multiple-Outlier Models

This article deals with the multiple-outlier exponential model. The likelihood ratio order between m-spacings of the combined sample is developed, some results extend the conclusions on simple spacings in Wen et al. (2007 Wen , S. , Lu , Q. , Hu , T. ( 2007 ). Likelihood ratio order of spacings of heterogeneous exponential random variables . J. Multivariate Anal. 98 : 743 – 756 .[Crossref], [Web of Science ®] , [Google Scholar]).     

A Variance Estimator for Systematic Sampling from a Deliberately Ordered Population

ABSTRACT

The systematic sampling (SYS) design (Madow and Madow, 1944 Madow , L. H. , Madow , W. G. ( 1944 ). On the theory of systematic sampling . Ann. Math. Statist. 15 : 1 – 24 .[Crossref] , [Google Scholar]) is widely used by statistical offices due to its simplicity and efficiency (e.g., Iachan, 1982 Iachan , R. ( 1982 ). Systematic sampling a critical review . Int. Statist. Rev. 50 : 293 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]). But it suffers from a serious defect, namely, that it is impossible to unbiasedly estimate the sampling variance (Iachan, 1982 Iachan , R. ( 1982 ). Systematic sampling a critical review . Int. Statist. Rev. 50 : 293 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]) and usual variance estimators (Yates and Grundy, 1953 Yates , F. , Grundy , P. M. ( 1953 ). Selection without replacement from within strata with probability proportional to size . J. Roy. Statist. Soc. Series B 1 : 253 – 261 . [Google Scholar]) are inadequate and can overestimate the variance significantly (Särndal et al., 1992 Särndal , C. E. , Swenson , B. , Wretman , J. H. ( 1992 ). Model Assisted Survey Sampling . New York : Springer-Verlag , Ch. 3 .[Crossref] , [Google Scholar]). We propose a novel variance estimator which is less biased and that can be implemented with any given population order. We will justify this estimator theoretically and with a Monte Carlo simulation study.     

GMM in linear regression for longitudinal data with multiple covariates measured with error

Griliches and Hausman 5 Griliches, Z. and Hausman, J. A. 1986. Errors in variables in panel data. J. Econometrics, 32: 93–118. [Crossref], [Web of Science ®] [Google Scholar] and Wansbeek 11 Wansbeek, T. J. 2001. GMM estimation in panel data models with measurement error. J. Econometrics, 104: 259–268. [Crossref], [Web of Science ®] [Google Scholar] proposed using the generalized method of moments (GMM) to obtain consistent estimators in linear regression models for longitudinal data with measurement error in one covariate, without requiring additional validation or replicate data. For usefulness of this methodology, we must extend it to the more realistic situation where more than one covariate are measured with error. Such an extension is not straightforward, since measurement errors across different covariates may be correlated. By a careful construction of the measurement error correlation structure, we are able to extend Wansbeek's GMM and show that the extended Griliches and Hausman's GMM is equivalent to the extended Wansbeek's GMM. For illustration, we apply the extended GMM to data from two medical studies, and compare it with the naive method and the method assuming only one covariate having measurement error.     

An Extension of the Epsilon-Skew-Normal Distribution

This article is related with the probabilistic and statistical properties of an parametric extension of the so-called epsilon-skew-normal (ESN) distribution introduced by Mudholkar and Hutson (2000 Mudholkar , G. S. , Hutson , A. D. ( 2000 ). The epsilon-skew-normal distribution for analyzing near-normal data . J. Statist. Plann. Infer. 83 : 291 – 309 .[Crossref], [Web of Science ®] , [Google Scholar]), which considers an additional shape parameter in order to increase the flexibility of the ESN distribution. Also, this article concerns likelihood inference about the parameters of the new class. In particular, the information matrix of the maximum likelihood estimators is obtained, showing that it is non singular in the special normal case. Finally, the statistical methods are illustrated with two examples based on real datasets.     

On Optimal Estimating Functions in the Presence of Nuisance Parameters

When there is only one interesting parameter θ₁ and one nuisance parameter θ₂, Godambe and Thompson (1974 Godambe , V. P. , Thompson , M. E. ( 1974 ). Estimating equations in the presence of a nuisance parameter . Ann. Statist. 2 : 568 – 571 .[Crossref], [Web of Science ®] , [Google Scholar]) showed that the optimal estimating function for θ₁ essentially is a linear function of the θ₁-score, the square of the θ₂-score, and the derivative of θ₂-score with respect to θ₂. Mukhopadhyay (2000b) generalized this result to m nuisance parameters. Mukhopadhyay (2000 Mukhopadhyay , P. ( 2000 ). On some lower bounds for the variance of an estimating function . Int. J. Math. Statist. Sci. 9 ( 2 ). [Google Scholar] 2002a Mukhopadhyay , P. ( 2002a ). On estimating functions in the presence of a nuisance parameter . Commun. Statist. Theor. Mem. 31 ( 1 ): 31 – 36 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] b Mukhopadhyay , P. ( 2002b ): Some lower bounds on variance of estimating functions . J. Statist. Res. 36 ( 2 ): 189 – 197 . [Google Scholar]) obtained lower bounds to the variance of regular estimating functions in the presence of nuisance parameters. Taking cues from these results we propose a method of finding optimal estimating function for θ₁ by taking the multiple regression equation on θ₁ score and Bhattacharyya's (1946 Bhattacharyya , A. ( 1946 ). On some analogues of the amount of information and their use in statistical estimation . Sankhya A 8 : 1 – 14 . [Google Scholar]) scores with respect to θ₂. The result is extended to the case of m nuisance parameters.     

Revisiting Beal's Confidence Intervals for the Difference of Two Binomial Proportions

Confidence interval construction for the difference of two independent binomial proportions is a well-known problem with a full panoply of proposed solutions. In this paper, we focus largely on the family of intervals proposed by Beal (1987 Beal , S. ( 1987 ). Asymptotic confidence intervals for the difference between two binomial parameters for use with small samples . Biometrics 43 : 941 – 950 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This family, which includes the Haldane and Jeffreys–Perks intervals as special cases, assumes a symmetric prior distribution for the population proportions p ₁ and p ₂. We propose new methods that allow the currently observed data to set the prior distribution by taking a parametric empirical-Bayes approach; in addition, we also provide an investigation of the new interval' behaviors in small-sample situations. Unlike other solutions, our intervals can be used adaptively for experiments conducted in multiple stages over time. We illustrate this notion using data from an Argentinean study involving the Mal Rio Cuarto virus and its transmission to susceptible maize crops.     

A Study of Expansions of Posterior Distributions

Johnson (1970 Johnson , R. ( 1970 ). Asymptotic expansions associated with posterior distributions . Ann. Math. Statist. 41 : 851 – 864 .[Crossref] , [Google Scholar]) obtained expansions for marginal posterior distributions through Taylor expansions. Here, the posterior expansion is expressed in terms of the likelihood and the prior together with their derivatives. Recently, Weng (2010 Weng , R. C. ( 2010 ). A Bayesian Edgeworth expansion by Stein's Identity . Bayesian Anal. 5 ( 4 ): 741 – 764 .[Crossref], [Web of Science ®] , [Google Scholar]) used a version of Stein's identity to derive a Bayesian Edgeworth expansion, expressed by posterior moments. Since the pivots used in these two articles are the same, it is of interest to compare these two expansions.

We found that our O(t ^?1/2) term agrees with Johnson's arithmetically, but the O(t ^?1) term does not. The simulations confirmed this finding and revealed that our O(t ^?1) term gives better performance than Johnson's.     

A Generalization of the Skew-Normal Distribution: The Beta Skew-Normal

We consider a new generalization of the skew-normal distribution introduced by Azzalini (1985 Azzalini , A. ( 1985 ). A class of distributions which includes the normal ones . Scand. J. Statis. 12 ( 2 ): 171 – 178 .[Web of Science ®] , [Google Scholar]). We denote this distribution Beta skew-normal (BSN) since it is a special case of the Beta generated distribution (Jones, 2004 Jones , M. C. ( 2004 ). Families of distributions of order statistics . Test 13 ( 1 ): 1 – 43 .[Crossref], [Web of Science ®] , [Google Scholar]). Some properties of the BSN are studied. We pay attention to some generalizations of the skew-normal distribution (Bahrami et al., 2009 Bahrami , W. , Agahi , H. , Rangin , H. ( 2009 ). A two-parameter Balakrishnan skew-normal distribution . J. Statist. Res. Iran 6 : 231 – 242 . [Google Scholar]; Sharafi and Behboodian, 2008 Sharafi , M. , Behboodian , J. ( 2008 ). The Balakrishnan skew-normal density . Statist. Pap. 49 : 769 – 778 .[Crossref], [Web of Science ®] , [Google Scholar]; Yadegari et al., 2008 Yadegari , I. , Gerami , A. , Khaledi , M. J. ( 2008 ). A generalization of the Balakrishnan skew-normal distribution . Statist. Probab. Lett. 78 : 1165 – 1167 .[Crossref], [Web of Science ®] , [Google Scholar]) and to their relations with the BSN.     

Linear Combinations,Products and Ratios of Simplicial or Spherical Variates

The density level sets of the two types of measures under consideration are l _{2, p} -circles with p = 1 and p = 2, respectively. The intersection-percentage function (ipf) of such a measure reflects the percentages which the level set corresponding to the p-radius r shares for each r > 0 with a set to be measured. The geometric measure representation formulae in Richter (2009 Richter , W.-D. (2009). Continuous l _{n, p} -symmetric distributions. Lithuanian Mathemat. J. 49:93–108.[Crossref], [Web of Science ®] , [Google Scholar]) is based upon these ipf's and will be used here for evaluating exact cdf's and pdf's for the linear combination, the product, and the ratio of the components of two-dimensional simplicial or spherically distributed random vectors.     

A New Nonparametric Test for the Umbrella Alternative Problem

In this article, we propose a new nonparametric test for detecting umbrella alternatives. It is designed to improve the power of Pan (1996 Pan , G. ( 1996 ). Distribution-free tests for umbrella alternatives . Communications in Statistics—Theory and Methods 25 ( 12 ): 3185 – 3194 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar])'s test, which is based on the breakdown of umbrella alternatives into a union of simple-ordered alternatives. Distribution-free tests are proposed in the case where the peak of the umbrella is unknown. Some actual data examples and the results of a Monte Carlo power study are also discussed.     

Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

We consider non-parametric estimation of a continuous cdf of a random vector (X ₁, X ₂). With bivariate RC data, it is stated in van der Laan (1996 Van der Laan , M. J. ( 1996 ) Efficient estimation in the bivariate censoring model and repairing NPMLE . Ann. Statist. 24 : 596 – 627 .[Crossref], [Web of Science ®] , [Google Scholar], p. 598¹⁰, Ann. Statist.), Quale et al. (2006 Quale , C. M. , van der Laan , M. J. , Robins , J. R. ( 2006 ). Locally efficient estimation with bivariate right-censored data . JASA. 101 : 1076 – 1084 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar])).” The claim is based on a result in Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], p.1352, Ann. Statist.) that if X ₁ is right censored but not X ₂, then common ways for defining one NPMLE lead to inconsistency. If X ₁ is right censored and X ₂ is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], Ann. Statist.).     

D-Optimal Chemical Balance Weighing Designs with N ≡ 0 (mod 4) and 3 Objects

In this article, the problem of estimation of the individual weights of three objects using a chemical balance weighing design is considered. We use the criterion of D-optimality. We assume that the covariance matrix of errors is the matrix of first-order autoregressive process. Such problems were discussed in Li and Yang (2005 Li , C. H. , Yang , S. Y. ( 2005 ). On a conjecture in D-optimal designs with n ≡ 0 (mod 4) . Lin. Alg. Applic. 400 : 279 – 290 .[Crossref], [Web of Science ®] , [Google Scholar]) and also in Yeh and Lo Huang (2005 Yeh , H. G. , Lo Huang , M. N. ( 2005 ). On exact D-optimal designs with 2 two-level factors and n autocorrelated observations . Metrika 61 : 261 – 275 .[Crossref], [Web of Science ®] , [Google Scholar]). We present some results of D-optimal designs in certain class of designs with the design matrix X  ∈ M _n×3(±1) such that each column of matrix X has at least one 1 and one ?1.