Multivariate Statistics THEORY AND APPLICATIONS 8705hc_9789814449397_tp.indd 22/2/13 8:46 AM This page intentionally left blank Proceedings of IX Tartu Conference on Multivariate Statistics and XX International Workshop on Matrices and Statistics Multivariate Statistics THEORY AND APPLICATIONS Tartu, Estonia, 26 June – July 2011 Editor Tõnu Kollo University of Tartu, Estonia World Scientific NEW JERSEY • LONDON 8705hc_9789814449397_tp.indd • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 22/2/13 8:46 AM Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library MULTIVARIATE STATISTICS: THEORY AND APPLICATIONS Proceedings of IX Tartu Conference on Multivariate Statistics and XX International Workshop on Matrices and Statistics Copyright © 2013 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN 978-981-4449-39-7 Printed in Singapore HeYue - Multivariate Statistics.pmd 2/21/2013, 2:12 PM February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 v PREFACE This volume consists of selected papers presented at the IX Tartu Conference on Multivariate Statistics organized jointly with the XX International Workshop on Matrices and Statistics The conference was held in Tartu, Estonia from 26 June to July 2011 More than 100 participants from 30 countries presented in four days recent devolopments on various topics of multivariate statistics The papers cover wide range of problems in modern multivariate statistics including distribution theory and estimation, different models of multivariate analysis, design of experiments, new developments in highdimensional statistics, sample survey methods, graphical models and applications in different areas: medicine, transport, life and social sciences The Keynote Lecture by Professor N Balakrishnan was delivered as the Samuel Kotz Memorial Lecture Thorough treatment of multivariate exponential dispersion models is given by Professor B Jørgensen A new general approach to sampling plans is suggested by Professor Y K Belyaev Professor E Ahmed compares different strategies of estimating regression parameters C M Cuadras introduces a generalization of Farley-GumbelMorgenstern distributions As Editor I am thankful to the authors who have presented interesting and valuable results for publishing in the current issue The book will be useful for researchers and graduate students who work in multivariate statistics The same time numerous applications can give useful ideas to scientists in different areas of research My special thanks go to the anonymous Referees who have done great job and spent lot of time with reading the papers Due to their comments and suggestions the presentation of the material has been improved and the quality of the papers has risen I am extremely thankful to the technical secretary of the volume Dr Ants Kaasik who has efficiently organised correspondence with the authors and Referees T˜ onu Kollo Editor Tartu, Estonia October 2012 February 15, 2013 11:7 8705 - Multivariate Statistics This page intentionally left blank Tartu˙ws-procs9x6 February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 vii ORGANIZING COMMITTEES PROGRAMME COMMITTEE of The 9th Tartu Conference on Multivariate Statistics and The 20th International Workshop on Matrices & Statistics D von Rosen (Chairman) G P H Styan (Honorary Chairman of IWMS) T Kollo (Vice-Chairman) S E Ahmed J J Hunter S Puntanen G Trenkler H J Werner Swedish University of Agricultural Sciences, Linkăoping University, Sweden – McGill University, Canada – University of Tartu, Estonia – University of Windsor, Canada – Auckland University of Technology, New Zealand – University of Tampere, Finland – Technical University of Dortmund, Germany – University of Bonn, Germany ORGANIZING COMMITTEE of The 9th Tartu Conference on Multivariate Statistics and The 20th International Workshop on Matrices & Statistics K Păarna (Chairman) A Kaasik (Conference Secretary) – University of Tartu, Estonia – University of Tartu, Estonia February 15, 2013 11:7 8705 - Multivariate Statistics This page intentionally left blank Tartu˙ws-procs9x6 February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 ix CONTENTS Preface Organizing Committees v vii Variable Selection and Post-Estimation of Regression Parameters Using Quasi-Likelihood Approach S Fallahpour and S E Ahmed Maximum Likelihood Estimates for Markov-Additive Processes of Arrivals by Aggregated Data A M Andronov 17 A Simple and Efficient Method of Estimation of the Parameters of a Bivariate Birnbaum-Saunders Distribution Based on Type-II Censored Samples N Balakrishnan and X Zhu Analysis of Contingent Valuation Data with Self-Selected Rounded WTP-Intervals Collected by Two-Steps Sampling Plans Yu K Belyaev and B Kristră om 34 48 Optimal Classication of Multivariate GRF Observations K Duˇcinskas and L Dreiˇzien˙e 61 Multivariate Exponential Dispersion Models B Jørgensen and J R Mart´ınez 73 Statistical Inference with the Limited Expected Value Function M Kă aa ărik and H Kadarik 99 February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 160 Confidence regions The one-dimensional distribution of ρˆ depends only on unknown value of 1−ρ ρ, and the function 1+(p−1)ρ is monotone in the region of possible values Then, it is easy to provide − α confidence interval for ρ It is ) ( − c2 − c1 ; , (5) + (p − 1)c1 + (p − 1)c2 where c1 = ( − ρˆ α) Fn−r(X),(p−1)(n−r(X)) − + (p − 1)ˆ ρ and c2 = (α) − ρˆ Fn−r(X),(p−1)(n−r(X)) + (p − 1)ˆ ρ ˆ is that its distribution depends on both σ and The problem with σ ρ The most primitive (and conservative) method is to look how does the variance of the estimator depend on ρ and take the least favourable value A simple calculation yields that ( ) 2σ + (p − 1)ρ2 var σ ˆ = p (n − r(X)) Since it grows with ρ2 , we get the largest confidence interval for ρ = (and the smallest one for ρ = 0) Figures – displays 0.025 and 0.975 quantiles of this distribution as a function of ρ for σ = and n − r(X) = 20, 25, 50, 100 for selected p’s Fig p=3 Fig p=5 Fig p = 11 February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 161 Simple computation gives − α (conservative) confidence interval for σ to be ( ) σ2 (n − r(X))ˆ σ (n − r(X))ˆ ); ( ( ) (6) χ2n−r(X) − α2 χn−r(X) α2 −1 If it is reasonable to suppose that ρ < 0, we can better taking ρ = p−1 as the least favourable value This leads to the interval ) ( σ2 (p − 1)(n − r(X))ˆ σ (p − 1)(n − r(X))ˆ ); ( ( ) (7) χ2(p−1)(n−r(X)) − α2 χ(p−1)(n−r(X)) α2 Because the variance for ρ = does not depend on p, the difference between ρ = and ρ = increases with p Thus, especially for larger p it is better to look for simultaneous confidence region for pair (σ , ρ).( ) The smallest size confidence region must be based on α-cuts: if σ ˆ02 , ρˆ0 are the observed values of (estimators, − α confidence region of ) ( 2consists ) ˆ , ρˆ = cα , all pairs (σ , ρ) for which σ ˆ02 , ρˆ0 lies within contour f(σ2 ,ρ) σ where ∫∫ f(σ2 ,ρ) (x, y) dy dx = − α (8) f(σ2 ,ρ) (x,y)≥cα To find such a region is a computationally demanding ) To that, we ( task , ρ ˆ with respect to have to realize that p-value of the observed pair σ ˆ 0 ( ) true value σ , ρ is ∫∫ p=1− f(σ2 ,ρ) (x, y) dy dx f(σ2 ,ρ) (x,y)≥f(σ2 ,ρ) (σ ˆ02 ,ρˆ0 ) Then, the numerical algorithm can be the following one: ( ) ( ) (1) Start with σ , ρ = σ ˆ0 , ρˆ0 Choose step change ∆σ > (2) For for ρ, i.e such values ) ( current ) ( σ) , find left and right ( boundaries ρl σ , ρr σ that p-value of σ ˆ0 , ρˆ0 is equal to α If such values not exist, i.e all p-values are less then α, go to (4) (3) Exchange σ for σ + ∆σ , change starting point of ρ to the last (ρl + ρr )/2, and go to (2) (4) After the first run, set σ = σ ˆ02 , change the sign of ∆σ , and go to (3) After the second run, stop Now we have a set of discrete points, convex hull of which( (numerical solu) tion of (8)) is approximate optimal confidence region for σ , ρ February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 162 The previous algorithm takes days even with Mathematica on a good 64-bit machine That is why we have searched for something simpler, even if not optimal √ If we denote α∗ = − − α, then the following two inequalities hold with probability − α: ( ∗) σ [1 + (p − 1)ρ] χn−r(X) α2 ≤ Tr (V1 ) n − r(X) ( ) ∗ σ [1 + (p − 1)ρ] ≤ χn−r(X) − α2 n − r(X) ) ( ∗ σ (1 − ρ) χ(p−1)(n−r(X)) α2 ≤ Tr (V2 ) n − r(X) ( ) ∗ σ (1 − ρ) ≤ χ(p−1)(n−r(X)) − α2 n − r(X) ˆ [1 + (p − 1)ˆ ρ] and Tr (V2 ) = σ ˆ (p − 1) (1 − ρˆ), this can Since Tr (V1 ) = σ easily be turned into n − r(X) )σ ( ρ] ≤ σ [1 + (p − 1)ρ] ˆ [1 + (p − 1)ˆ ∗ − α2 χ2n−r(X) ≤ n − r(X) ( ∗ )σ ˆ [1 + (p − 1)ˆ ρ] χ2n−r(X) α (9) and (p − 1)(n − r(X)) ( )σ ˆ (1 − ρˆ) ≤ σ (1 − ρ) α∗ χ(p−1)(n−r(X)) − ≤ (p − 1)(n − r(X)) ( ∗ )σ ˆ (1 − ρˆ) (10) χ2(p−1)(n−r(X)) α2 If we take ρ as a function of σ , inequalities (9) defines two decreasing bounds for ρ for every σ : σ ˆ [1 + (p − 1)ˆ −1 ρ] n − r(X) )· ( + ≤ρ ∗ (p − 1) α p − χ2 σ n−r(X) − ≤ ρ] n − r(X) σ ˆ [1 + (p − 1)ˆ −1 ( ∗) · + , (11) α p − χ2 σ (p − 1) n−r(X) February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 163 and inequalities (10) two increasing bounds: 1− ˆ (1 − ρˆ) (p − 1)(n − r(X)) σ ( ∗) · ≤ρ σ2 χ2(p−1)(n−r(X)) α2 ≤1− σ ˆ (1 − ρˆ) (p − 1)(n − r(X)) ) ( (12) · ∗ σ2 χ2(p−1)(n−r(X)) − α2 The region marked out by these bounds will be called pseudo-rectangular confidence region It can also produce another conservative confidence interval for σ , which is given by the intersections of upper increasing with lower decreasing and lower increasing with upper decreasing boundaries: σl2 = ˆ2 + (p − 1)ˆ ρ (n − r(X)) σ ( ) · ∗ p χ2n−r(X) − α2 + σu2 = ˆ2 (p − 1)(n − r(X)) σ (p − 1) (1 − ρˆ) ), ( · ∗ p χ2(p−1)(n−r(X)) − α2 (13) + (p − 1)ˆ ρ (n − r(X)) σ ˆ2 ( ∗) · p χ2n−r(X) α2 + ˆ2 (p − 1) (1 − ρˆ) (p − 1)(n − r(X)) σ ( ∗) · p χ2(p−1)(n−r(X)) α2 (14) It is worth noting, that the bounds are convex combinations of (6) and (7) using estimated value ρˆ and α replaced with α∗ Change from α to α∗ does not enlarge the interval too much, as shown in the following figure, usually only by several percent Figure shows prolongation of the interval as a function of degrees of freedom for α = 0.05, 0.01, 0.001 (from top to bottom) It follows that if ρ is substantially less than 1, the interval can be shorter than (6), especially for large p The other two intersection points of boundaries are given by the same linear combination of mutually interchanged terms Comparison of the two types of confidence regions (exact – dotted line and pseudo-rectangular – dashed line) for some choices of parameters and design is in Figures – for α = 0.05 We see that the approximate confidence region performs satisfactorily Figures also contain (solid line) and α∗ -cut (dash-dotted line) of the ( α-cut ) joint density f σ ˆ , ρˆ for true values of parameters equal to the observed estimates We can see that while α-cut produces regions with coverage February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 164 Fig Fig Relative prolongation of the CI σ ˆ = 1, ρˆ = 0.9 Fig Fig σ ˆ = 0.8, ρˆ = −0.4 σ ˆ = 0.8, ρˆ = 0.3 February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 165 less than − α, α∗ -cut produces coverage greater than − α, at least in all situations considered It was confirmed by a limited number of other simulations for different α’s As a rule of thumb we can say that α∗ -cut of the estimated density can be used as a conservative (1 − α)-confidence region Conclusion Depending on our needs, we can construct confidence regions for ρ and σ independently, or simultaneously Because of rather skew joint distribution of the estimators, we recommend to use the simultaneous confidence region approach Exact (optimal) confidence region can be numerically computed Pseudo-rectangular and conservative α∗ -cut confidence regions, which are much easier to construct, can also be recommended for use Acknowledgement ˇ SR 1/0131/09, VEGA The research was supported by grants VEGA MS ˇ ˇ MS SR 1/0325/10, VEGA MS SR 1/0410/11 and by the Agency of the Slovak Ministry of Education for the Structural Funds of the EU, under project ITMS:26220120007 References ˇ zula, Statistics 24, 321 (1993) I Zeˇ J C Lee, J Amer Statist Assoc 83, 432 (1988) T Kanda, Ann Inst Statist Math 4, 519 (1992) T Kanda, Hiroshima Math J 24, 135 (1994) N R Chaganty, J Statist Plann Inference 117, 123 (2003) ˇ zula, J Multivariate Anal 97, 606 (2006) I Zeˇ Q.-G Wu, J Statist Plann Inference 69, 101 (1998) Q.-G Wu, J Statist Plann Inference 88, 285 (2000) ˇ zula, On uniform correlation structure, in Proc MathematD Klein and I Zeˇ ical Methods In Economics And Industry (MMEI 2007), (Herl’any, Slovakia, 2007) 10 R.-D Ye and S.-G Wang, J Statist Plann Inference 139, 2746 (2009) ˇ zula and D Klein, Acta Comment Univ Tartu Math 14, 35 (2010) 11 I Zeˇ February 15, 2013 11:7 8705 - Multivariate Statistics This page intentionally left blank Tartu˙ws-procs9x6 February 15, 2013 11:7 8705 - Multivariate Statistics Tartu˙ws-procs9x6 167 AUTHOR INDEX Ahmed, S E., Andronov, A M., 17 Klein, D., 157 Kristră om, B., 48 Balakrishnan, N., 34 Belyaev, Yu K., 48 Liski, A., 112 Liski, E., 112 Dreiˇzien˙e, L., 61 Duˇcinskas, K., 61 Mă ols, M., 130 Martnez, J R., 73 Fallahpour, S., Pă arna, K., 130 Hă akkinen, U., 112 Tamm, A E., 141 Tamm, A O., 141 Traat, I., 141 Jứrgensen, B., 73 Kă aă arik, M., 99 Kaasik, A., 130 Kadarik, H., 99 Kangro, R., 130 von Rosen, T., 141 ˇ zula, I., 157 Zeˇ Zhu, X., 34 ... the British Library MULTIVARIATE STATISTICS: THEORY AND APPLICATIONS Proceedings of IX Tartu Conference on Multivariate Statistics and XX International Workshop on Matrices and Statistics Copyright.. .Multivariate Statistics THEORY AND APPLICATIONS 8705hc_9789814449397_tp.indd 22/2/13 8:46 AM This page intentionally left blank Proceedings of IX Tartu Conference on Multivariate Statistics and. .. Conference on Multivariate Statistics and XX International Workshop on Matrices and Statistics Multivariate Statistics THEORY AND APPLICATIONS Tartu, Estonia, 26 June – July 2011 Editor Tõnu Kollo University