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Concomitants of Order Statistics Shie-Shien Yang Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1976 Concomitants of order statistics Shie-Shien Yang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Yang, Shie-Shien, "Concomitants of order statistics " (1976). Retrospective Theses and Dissertations. 5720. https://lib.dr.iastate.edu/rtd/5720 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This material was produced from a microfilm copy of the original document. 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Xerox Liiilversiiy microfiims 300 North Zeeb Road Ann Arbor, Michigan 48106 77-1495 YANG, Shle-Shien, 1946* CONCOMITANTS OF ORDER STATISTICS. Iowa State University, Ph.D., 1976 Statistics Xerox University Microfilms, Ann Arbor, Michigan 48106 Concomitants of order statistics by Shie-Shien Yang A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF EHILOSOHÎY Major: Statistics Approved: Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For the Graduate College Iowa State University Ames,- Iowa 1976 ii TABLE OF CONTEKTS REVIEW OF LITERATURE AND SCOPE OF THE PRESENT STUDY 1 A. Introduction 1 B. Theory and Applications of the Distributional Properties of the Concomitants 3 1. A brief review 3 2. Scope and content of the present investigation k C. Concomitants in Parameter Estimation 8 THEORY OF CONCOMITANTS OF ORDER STATISTICS 11 A. Introduction 11 B. Bivariate Normal Case 11 1. Review of distribution theory 11 a. Basic assumptions 11 b. Small-sample theory 12 c. Large-sample theory 15 2. Further development of the theory 17 C. General Theory of Concomitants of Order Statistics 21 1- Assumptions and notation 21 2. Exact distribution of the concomitants 22 3» Exact moments of the concomitants 2h h. Exact distribution of the ranks of concomitants 25 5» Exact moments of the ranks of concomitants 28 6. Asymptotic distribution of the concomitants 32 7» Asymptotic distribution of the ranks of concomitants 33 D. Equicorrelated Multinormal Case 39 FLUCTUATIONS H TIE RAMS OF q. CU^SJLATIVS SUΫÎS OF RANDOM VARIABLES 1^3 A. Introduction i+3 B. General Theory ^5 iiî Page C. Prediction Problems in a Round Robin Tournament 50 1. Preliminary formulation 50 a. General model and assumptions 50 b. Joint distribution of the scores 51 c. The variances and covariances of the scores 52 2. Round Robin tournament among q teams of similar caliber 5^ 3- Round Robin tournament among q teeims in the presence of a superior team 55 i)-. Round Robin tournament among q teams of similar caliber when q is large 68 IV. THE MULTIVARIATE GENERALIZATION OF THE THEORY OF CONCOMITANTS OF ORDER STATISTICS 70 A. Introduction 70 B. Asymptotic Theory of the Concomitants 72 1. Asymptotic distribution of the concomitants 72 2. Asymptotic distribution of the ranks of concomitants 7^ C. Exact Distribution of the Ranks of Concomitants 82 V. CONCOMITANTS IN PARAMETER ESTIMATION 87 A. Introduction 87 "D T 4 tf* "C m "t" 4 Mm U' «5 Avs 4 ^ /-* w O 1 ^ #> Multivariate Normal Population 88 1. Introduction 88 2. Univariate case 89 3- Bivariate case 92 a. Type A or B sample 93 b- Type C sample 98 4. Multivariate case 98 C. Linear Estimation in a Doubla Sampling Situation 99 1. Introduction 99 2. Bivariate normal case 99 3. Multivariate normal case 10^- iv Page VI. BIBLIOGRAHIÏ IO9 A. References Cited IO9 B. Related References 110 VII. ACKNCWIEDGEMENTS 111 1 I. REVIEW OF LITERATURE MD SCOPE OF THE PRESENT STUDY A. Introduction Let (X^, Y^) (i = 1, 2, n) be n independent random vari­ ables from some bivariate distribution. If we arrange the X-variates in ascending order as ''im 2 5 ^ then the Y-variate corresponding to these order statistics are denoted by ^[l:n]' ^[2:n]' ' ^[n:n] ' and termed the concomitants of the order statistics- In particular, for r = 1, 2,.n , we call Y|.^the concomitant of the r-th order statistic. Note that &oes not necessarily have rank r among the Y's . These concomitants have recently been studied extensively and have been put to a variety of uses. Examples include Gross (1973), David and Galambos (197^), O'Connell (197^), and O'Connell and David (1976). The distribution theory of concomitants and of the ranks of concomitants is of interest in selection problems when selection is based on the ranks of the X's ' For instance, when m (< n) individuals having the high= est X-scores are selected, we may wish to know something about the 2 behavior of the corresponding Y-scores. In particular, the following question can be answered with the help of the distribution of the rank of \r-n]' ^ individual has rank r among n candidates scoring Xg; ..., X^ in a certain test, what is the probability that will have rank s in some future test with scores Y^, Y^ ? Gross (1973) describes a procedure based on a least squares regression model which allows a college counselor to predict from a sample of cur­ rently enrolled students, the performance of individuals applying for future admission. Here the X-variate is the score on some entrance entrance examination of a prospective student, and the Y-variate his score based on actual performance in college. On the other hand, the X's may refer to a characteristic in a parent and the Y's to the same characteristic in an offspring. See e.g. David and Galambos (197^) where it is also pointed out that in an "errors in variable" problem, one may wish to select on the basis of true values T , but actually must select on the basis of observed values V , where V = T + E and T, E are independent variates. Let Y = T , X = V • Then the correla­ tion of X and Y is p = 0^/(0^ + , where and are respec­ tively the variances of T and E • Here the ranks of the true values are of interest, while the actual ranking is done on the observed values. Concomitants of order statistics also enter naturally problems of estima­ tion in a multivariate population when the sample is ordered with respect to one variate. See e.g. Watterson (1959) and O'Connell and David (1976)- 3 B. Theory and Application of the Distributional Properties of the Concomitants 1. A brief review Let (X^; Y^) (i = 1, 2, n) be n independent random vari­ ables having a common bivariate distribution. Suppose further that may be expressed in the form (Xi - My) , X / V Yi = My + pf^Y 0 ^ (i = 1, 2, n) , (l.l) where the Z. are lid variates; p is the correlation coefficient of and Y^ ; 1-^ are the means and a® the variances of X and Y , respectively. The and X^ form a set of 2n mutually independent variates. Since the X^^ and are independent, ordering the X. will not affect the distributional properties of the . Hence from (l.l), Y[i.-n] (? = 1; 2, ..., n) may be expressed in the form 7[r:n] = + P°Y —^" %[r] ' where Z^ -, is that Z. associated with X^, . The Zr -, are inde- [rj 1 r:n Lrj pendent and identically distributed according to the distribution of Z^ and are independent of the X^.^ . For example, a random sample from a bivariate normal population has the above properties. Under the linear model assumption (l.l), the small-sample theory of concomitants has been studied extensively by O'Connell (197^)= He has obtained expressions for the exact distribution of the concomitants k and of the ranks of concomitants, as well as for the moments of the ranks of concomitants.
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