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.     

The Identification of Potential Longitudinal Biomarkers and Measurements of Effectiveness for Biomarkers as Surrogates in Multivariate Survival Data

A proposed method based on frailty models is used to identify longitudinal biomarkers or surrogates for a multivariate survival. This method is an extention of earlier models by Wulfsohn and Tsiatis (1997 Wulfsohn , M. S. , Tsiatis , A. A. ( 1997 ). A joint model for survival and longitudinal data measured with error . Biometrics 53 ( 1 ): 330 – 339 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Song et al. (2002 Song , X. , Davidian , M. , Tsiatis , A. A. ( 2002 ). A Semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data . Biometrics 58 ( 4 ): 742 – 753 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, similar to Henderson et al. (2002 Henderson , R. , Diggle , P. J. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), a joint likelihood function combines the likelihood functions of the longitudinal biomarkers and the multivariate survival times. We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudianl biomarkers influence the power to detect the association of a longitudinal biomarker and the multivariate survival time. The proposed method is illustrate by using the gastric cancer data.     

Quantile-Based Reliability Analysis

Quantile functions are equivalent alternatives to distribution functions in modeling and analysis of statistical data. The present article discusses the role of quantile functions in reliability studies. We present the hazard, mean residual, variance residual, and percentile residual quantile functions, their mutual relationships and expressions for the quantile functions in terms of these functions. Further, some theoretical results relating to the Hankin and Lee (2006 Hankin , R. K. S. , Lee , A. ( 2006 ). A new family of non-negative distributions . Austral. N.Z. J. Statist. 48 : 67 – 78 .[Crossref], [Web of Science ®] , [Google Scholar]) lambda distribution are discussed.     

Bayesian Robustness of the Positive False Discovery Rate

The positive false discovery rate was introduced by Storey (2003 Storey , J. D. (2003). The positive false discovery rate: a Bayesian interpretation and the q-value. Ann. Statist. 31:2013–2035.[Crossref], [Web of Science ®] , [Google Scholar]) as an alternative to the family wise error rate for the case in which we are simultaneously testing a large amount of hypotheses. The positive false discovery rate has a very nice Bayesian interpretation (as it was shown by Storey, 2003 Storey , J. D. (2003). The positive false discovery rate: a Bayesian interpretation and the q-value. Ann. Statist. 31:2013–2035.[Crossref], [Web of Science ®] , [Google Scholar]) and its robustness is analyzed. The emphasis is on the ε-contamination class (one of the most used classes of priors for Bayesian robustness) and it is shown that robustness is not obtained when the basic prior concentrates the probability on the null hypothesis.     

Tests for Multiple Outliers in an Exponential Sample

By applying the recursion of Huffer (1988 Huffer, F. 1988. Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability, 25: 346–354. [Crossref], [Web of Science ®] , [Google Scholar]) repeatedly, we propose an algorithm for evaluating the null joint distribution of Dixon-type test statistics for testing discordancy of k upper outliers in exponential samples. By using the critical values of Dixon-type test statistics determined from the proposed algorithm and those of Cochran-type test statistics presented earlier by Lin and Balakrishnan (2009 Lin, C. T. and Balakrishnan, N. 2009. Exact computation of the null distribution of a test for multiple outliers in an exponential sample. Computational Statistics & Data Analysis, 53: 3281–3290. [Crossref], [Web of Science ®] , [Google Scholar]), we carry out an extensive Monte Carlo study to investigate the powers and the error probabilities for the effects of masking and swamping when the number of outliers k = 2 and 3. Based on our empirical findings, we recommend Rosner’s (1975 Rosner, B. 1975. On the detection of many outliers. Technometrics, 17: 221–227. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) sequential test procedure based on Dixon-type test statistics for testing multiple outliers from an exponential distribution.     

Parameter Estimation in Credit Models Under Incomplete Information

We consider the filtering model of Frey and Schmidt (2012 Frey , R. , Schmidt , T. ( 2012 ). Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering . Fin. Stocha. 16 ( 1 ): 105 – 133 .[Crossref], [Web of Science ®] , [Google Scholar]) stated under the real probability measure and develop a method for estimating the parameters in this framework by using time-series data of CDS index spreads and classical maximum-likelihood algorithms. The estimation-approach incorporates the Kushner-Stratonovich SDE for the dynamics of the filtering probabilities. The convenient formula for the survival probability is a prerequisite for our estimation algorithm. We apply the developed maximum-likelihood algorithms on market data for historical CDS index spreads (iTraxx Europe Main Series) in order to estimate the parameters in the nonlinear filtering model for an exchangeable credit portfolio. Several such estimations are performed as well as accompanying statistical and numerical computations.     

A Comparison of Procedures for Controlling the False Discovery Rate in the Presence of Small Variance Genes: A Simulation Study

The Significance Analysis of Microarrays (SAM; Tusher et al., 2001 Tusher , V. G. , Tibshirani , R. , Chu , G. ( 2001 ). Significance analysis of microarrys applied to the ionizing radiation response . Proceedings of the National Academy of Sciences 98 : 5116 – 5121 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) method is widely used in analyzing gene expression data while controlling the FDR by using resampling-based procedure in the microarray setting. One of the main components of the SAM procedure is the adjustment of the test statistic. The introduction of the fudge factor to the test statistic aims at deflating the large value of test statistics due to the small standard error of gene-expression. Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) pointed out that the fudge factor does not effectively improve the power and the control of the FDR as compared to the SAM procedure without the fudge factor in the presence of small variance genes. Motivated by the simulation results presented in Lin et al. (2008 Lin , D. , Shkedy , Z. , Burzykowski , T. , Göhlmann , H. W. H. , De Bondt , A. , Perera , T. , Geerts , T. , Bijnens , L. ( 2008 ). Significance analysis of microarray (SAM) for comparisons of several treatments with one control . Biometric Journal, MCP 50 ( 5 ): 801 – 823 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), in this article, we extend our study to compare several methods for choosing the fudge factor in the modified t-type test statistics and use simulation studies to investigate the power and the control of the FDR of the considered methods.     

Tests of Multivariate Independence for Ordinal Data

Population and sample versions of Kendall and Spearman measures of association suitable for multivariate ordinal data are defined. The latter generalize the indices of dependence of Ruymgaart and van Zuijlen (1978 Ruymgaart , F. H. , van Zuijlen , M. C. A. ( 1978 ). Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case . Ann. Statist. 6 : 588 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]), Joe (1990 Joe , H. ( 1990 ). Multivariate concordance . J. Multivariate Anal. 35 : 12 – 30 .[Crossref], [Web of Science ®] , [Google Scholar]), and Schmid and Schmidt (2007 Schmid , F. , Schmidt , R. ( 2007 ). Multivariate extensions of Spearman's rho and related statistics . Statist. Probab. Lett. 77 : 407 – 416 .[Crossref], [Web of Science ®] , [Google Scholar]) by allowing atoms in the underlying distribution. The representation of the proposed empirical measures as U-statistics enables to establish their asymptotic normality under general distributions. A special attention is given to tests of independence for multivariate ordinal data, where the power of the new methodologies are investigated under fixed and contiguous alternatives.     

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.     

Estimators for the Population Mean in the Case of Missing Data

We propose a class of estimators for the population mean when there are missing data in the data set. Obtaining the mean square error equations of the proposed estimators, we show the conditions where the proposed estimators are more efficient than the sample mean, ratio-type estimators, and the estimators in Singh and Horn (2000 Singh , S. , Horn , S. ( 2000 ). Compromised imputation in survey sampling . Metrika 51 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) and Singh and Deo (2003 Singh , S. , Deo , B. (2003). Imputation by power transformation. Statist. Pap. 44:555–579.[Crossref], [Web of Science ®] , [Google Scholar]) in the case of missing data. These conditions are also supported by a numerical example.     

Bayesian Estimation of Parameters and Comparison of Shared Gamma Frailty Models

In this article, we introduce shared gamma frailty models with three different baseline distributions namely, Weibull, generalized exponential and exponential power distributions. We develop Bayesian estimation procedure using Markov Chain Monte Carlo(MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these three models to a real life bivariate survival dataset of McGilchrist and Aisbett (1991 McGilchrist, C. A. and Aisbett, C. W. 1991. Regression with frailty in survival analysis. Biometrics, 47: 461–466. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to kidney infection data and a better model is suggested for the data.     

Block Design with Nested Rows and Columns for Research on Food Acceptability Limitation

Block designs with nested rows and columns were introduced in Srivastava (1978 Srivastava , J. N. ( 1978 ). Statistical design of agricultural experiments . J. Ind. Soc. Agric. Statist. 30 : 1 – 10 . [Google Scholar]) and more specifically defined in Singh and Dey (1979 Singh , M. , Dey , A. ( 1979 ). Block designs with nested rows and columns . Biometrika 66 ( 2 ): 321 – 326 .[Crossref], [Web of Science ®] , [Google Scholar]). We present properties of some partially balanced block designs with nested rows and columns. The designs are considered for near-factorial experiments, when there are a levels of experimental factor A and b levels of experimental factor B and there is one control treatment added. In this article we extend the theory of block designs with nested rows and columns described by \L acka and Koz?owska (2009 ?acka , A. , Koz?owska , M. ( 2009 ). Planning of factorial experiments in a block design with nested rows and columns for environmental research . Environmetrics 20 ( 6 ): 730 – 742 .[Crossref], [Web of Science ®] , [Google Scholar]) and Koz?owska et al. (2011 Koz?owska , M. , ?acka , A. , Krawczyk , R. , Koz?owski , R. J. ( 2011 ). Some block designs with nested rows and columns for research on pesticide dose limitation . Environmetrics 22 ( 6 ): 781 – 788 .[Crossref], [Web of Science ®] , [Google Scholar]). We carry out our consideration by the derived mixed linear model resulting from randomization of blocks, rows and columns. For this model of observations, some properties of estimation of treatment contrasts are discussed. We calculate the efficiency factors of estimation of treatment contrasts. We formulate theorems for some partially balanced block designs with nested rows and columns. Plant protection experiment on limitation of food acceptability for T. urticae is given to show how the obtained results can be applied.     

Testing for Unit Roots and the Impact of Quadratic Trends,with an Application to Relative Primary Commodity Prices

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modeled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated nonlinear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this article, we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modeled, we show that the quasi-differenced detrended Dickey–Fuller-type test of Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996 Elliott , G. , Rothenberg , T. J. , Stock , J. H. ( 1996 ). Efficient tests for an autoregressive unit root . Econometrica 64 : 813 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this article. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether the data admit a unit root.     

A Note on the Convergence Rate of Strong Law of Large Numbers for Positively Associated Sequences

By the inequalities established in this article, we obtain the convergence rate of strong law of large numbers for positively associated sequences. The results derived extend and improve the corresponding ones in Vronskii (1999 Vronskii , M. A. ( 1999 ). Rate of convergence in the slln for associated sequences and fields . Theory Probab. Appl. 43 ( 3 ): 449 – 462 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Bounds on the efficiency of unbalanced ranked-set sampling

Abstract

Takahasi and Wakimoto (1968 Takahasi, K., and K. Wakimoto. 1968. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20:1–31.[Crossref], [Web of Science ®] , [Google Scholar]) derived a sharp upper bound on the efficiency of the balanced ranked-set sampling (RSS) sample mean relative to the simple random sampling (SRS) sample mean under perfect rankings. The bound depends on the set size and is achieved for uniform distributions. Here we generalize the Takahasi and Wakimoto (1968 Takahasi, K., and K. Wakimoto. 1968. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20:1–31.[Crossref], [Web of Science ®] , [Google Scholar]) result by finding a sharp upper bound in the case of unbalanced RSS. The bound depends on the particular unbalanced design, and the distributions where the bound is achieved can be highly nonuniform. The bound under perfect rankings can be exceeded under imperfect rankings.     

Performance Evaluation of Different Tests for Location Parameters

In this article, our objective is to evaluate the performance of different tests which are used to compare the equality of more than two location parameters. We have considered six tests (including some commonly used) in this study, one of which is parametric and the others are nonparametric. These tests include the usual F test (Fisher and Mackenzie, 1923 Fisher , R. A. , Mackenzie , M. A. ( 1923 ). Studies in crop variation. II. The manurial response of different potato . Journal of Agricultural Science 13 : 311 – 320 .[Crossref], [Web of Science ®] , [Google Scholar]), Kruskal–Wallis test (Kruskall and Wallis, 1952 Kruskall , W. H. , Wallis , W. A. ( 1952 ). Use of ranks in one-criterion variance analysis . Journal of American Statistical Association 47 : 583 – 621 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Kolmogorov–Smirnov test (David, 1958 David , H. T. ( 1958 ). Three-sample Kolnogorov–Smirnov test . The Annals of Mathematical Statistics 29 : 842 – 851 .[Crossref] , [Google Scholar]), the g test (Stekler, 1987 Stekler , H. O. ( 1987 ). Who forecasters better? Journal of Business and Economic Statistics 5 : 155 – 158 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), f test (Batchelor, 1990 Batchelor , R. A. ( 1990 ). All forecasters are equal . Journal of Business and Economic Statistics 8 : 143 – 144 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Extension of Median test (as given in Daniel, 1990 Daniel , W. W. ( 1990 ). Applied Nonparametric Statistics. , 2nd ed. Duxbury Classic Series , Boston . [Google Scholar]). Performance of these tests are compared under different symmetric, skewed and contaminated probability distributions that include Normal, Cauchy, Uniform, Laplace, Lognormal, Exponential, Weibull, Gamma, t, Chi-square, Half Normal, Mixed Weibull, and Mixed Normal. Performances of these tests are measured in terms of power. We have suggested appropriate tests which may perform better under different situations. It is expected that researchers will find these results useful in decision making.     

The Alpha-power Tobit Model

The main object of this article is to propose an extension of the tobit model for which the error distribution follows the power-normal distribution (Gupta and Gupta, 2008 Gupta , D. , Gupta , R. C. ( 2008 ). Analyzing skewed data by power normal model . Test 17 : 197 – 210 .[Crossref], [Web of Science ®] , [Google Scholar]). Inference is dealt with by using the likelihood approach. Simulation studies and application to a real data set are used to demonstrate the usefulness of the extension.     

Regression M-estimators with One Modified Form of Doubly Censored Data

In this article, we consider the M-estimators for the linear regression model when both response and covariate variables are subject to double censoring. The proposed estimators are constructed as some functional of three types of estimators for a bivariate survival distribution. The first two estimators are the generalizations of the Campbell and Földes (1982 Campbell, G. and Földes, A. 1982. “Large sample properties of nonparametric statistical inference”. In Nonparametric Statistical Inference., Edited by: Gnredenko, B. V., Puri, M. L. and Vineze, I. 103–122. Amsterdam: North-Holland.  [Google Scholar]) and Dabrowska (1988 Dabrowska, D. M. 1988. Kaplan-Meier estimate on the plane. Annals of Statistics, 18: 1475–1489. [Crossref], [Web of Science ®] , [Google Scholar]) estimators proposed by Shen (2009 Shen, P. S. 2009. Nonparametric estimation of the bivariate survival function one modified form of doubly censored data. Computational Statistics, 25: 203–313. [Crossref], [Web of Science ®] , [Google Scholar]). The third estimator is the generalization of the Prentice and Cai (1992 Prentice, R. L. and Cai, J. 1992. Covariance and survivor function estimation using censored multivariate failure time data. Biometrika, 79: 495–512. [Crossref], [Web of Science ®] , [Google Scholar]) estimator. The consistency of the proposed M-estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators. Furthermore, the simple bootstrap methods are used to estimate standard deviations and construct interval estimators.     

An Improved Solution for Multiple Comparisons with a Control in Response Surface Methodology with Two Predictor Variables

Sa and Edwards (1993 Sa , P. , Edwards , D. ( 1993 ). Multiple comparisons with a control in response surface methodology . Technometrics 35 ( 4 ): 436 – 445 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) first proposed the Multiple Comparisons with a Control problem in Response Surface Methodology. They provided an exact solution for one predictor variable and a conservative solution when number of predictor variables is more than one. Merchant et al. (1998 Merchant , A. , McCann , M. , Edwards , D. ( 1998 ). Improved multiple comparisons with a control in response surface analysis . Technometrics 40 ( 4 ): 291 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]) improved the solution for the latter case. This article improves Merchant et al.'s solution for the case of rotatable designs in two predictor variables.     

Shortest Expected Length Confidence Interval for the Power of the t-Test

Tarasińska (2005 Tarasińska , J. ( 2005 ). Confidence intervals for the power of student's t-test . Statistics & Probability Letters 73 ( 2 ): 125 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]) considered a method to construct the shortest length confidence interval on the power of the t-test using a confidence interval for the population standard deviation in the non centrality parameter. Gilliland and Li (2008 Gilliland , D. , Li , M. ( 2008 ). A note on confidence intervals for the power of t-test . Statistics & Probability Letters 78 ( 5 ): 488 – 489 .[Crossref], [Web of Science ®] , [Google Scholar]) used simulations to show that this confidence interval has less than the nominal coverage, particularly in small samples. We propose to find the shortest expected length confidence interval for the power of the t-test by accounting for the variation in the sample standard deviation, and provide the necessary constants for its implementation for some selected sample and shift sizes. It is seen that the proposed interval is reasonably robust to the specification of the population standard deviation and maintains the nominal coverage.