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Statistics
A Journal of Theoretical and Applied Statistics
Volume 23, 1992 - Issue 3
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Original Articles

Order Statistics from Discrete Distributions

Pages 189-216 | Received 27 Nov 1990, Published online: 27 Jun 2007

References

  • Abdel-Aty , S. H. 1954 . Ordered variables in discontinuous distributions . Statist. Neerlandica , 8 : 61 – 82 .
  • Anderson , C. W. 1970 . Extreme value theory for a class of discrete distributions with applications to some stochastic processes . J. Appl. Probab. , 7 : 99 – 113 .
  • Anderson , C. W. 1980 . Local limit theorems for the maxima of discrete random variables . Math. Proc. Camb. Phil. Soc. , 88 : 161 – 165 .
  • Arnold , B. C. 1980 . Two characterizations of the geometric distribution . J. Appl. Probab. , 17 : 570 – 573 .
  • Arnold , B. C. 1990 . Private Communication
  • Arnold , B. C. and Balakrishnan , N. 1989 . Relations, Bounds and Approximations for Order Statistics , New York : Springer-Verlag . Lecture Notes in Statistics No. 53
  • Arnold , B. C. , Becker , A. , Gather , U. and Zahedi , H. 1984 . On the Markov property of order statistics . J. Statist. Plann. Inference , 9 : 147 – 154 .
  • Azlarov , T. A. and Volodin , N. A. 1986 . “ Characterization Problems Associated with the Exponential Distribution ” . In , Original Russian Edition, 1982 , New York : Springer-Verlag .
  • Balakrishnan , N. 1986 . Order statistics from discrete distributions . Commun. Statist.-Theor. Meth. , 15 : 657 – 675 .
  • Burr , I. W. 1955 . Calculation of exact sampling distribution of ranges from a discrete population . Ann. Math. Statist , 26 : 530 – 532 . Correction 38, 280
  • Cohen , A. C. 1954 . Estimation of the Poisson parameter from truncated samples and from censored samples . J. Amer. Statist. Assoc. , 49 : 158 – 168 .
  • Connor , R. J. 1969 . The sampling distribution of the range from discrete uniform finite populations and a range test from homogeneity . J. Amer. Statist. Assoc. , 64 : 1223 – 1235 .
  • Craig , C. C. 1962 . On the mean and variance of the smaller of two drawings from a binomial population . Biometrika , 49 : 566 – 569 .
  • David , H. A. 1981 . Order Statistics , Second Edition , New York : John Wiley .
  • David , H. A. and Mishriky , R. A. 1968 . Order statistics for discrete populations and for grouped samples . J. Amer. Statist. Assoc. , 63 : 1390 – 1398 .
  • Deheuvels , P. 1984 . The characterization of distributions by order statistics and record values—A unified approach . J. Appl. Probab , 21 : 326 – 334 . Correcton 22, 997
  • Doss , S. A. D. C. 1963 . On the efficiency of best asymptotically normal estimates of the Poisson parameter based on singly and doubly truncated or censored samples . Biometrics , 19 : 588 – 594 .
  • El-Neweihi , E. and Govindarajulu , Z. 1979 . Characterizations of geometric distribution and discrete IFR(DFR) distributions using order statistics . J. Statist. Plann. Inference , 3 : 85 – 90 .
  • Ferguson , T. S. 1965 . A characterization of the geometric distribution . Amer. Math. Monthly , 72 : 256 – 260 .
  • Galambos , J. 1975a . “ Characterizations of probability distributions by properties of order statistics I ” . In Statistical Distributions in Scientific Work , Edited by: Patil , G. P. 71 – 88 . Dordrecht : D. Reidel .
  • Galambos , J. 1975b . “ Characterizations of probability distributions by properties of order statistics II ” . In Statistical Distributions in Scientific Work , Edited by: Patil , G. P. 89 – 101 . Dordrecht : D. Reidel .
  • Galambos , J. 1987 . The Asymptotic Theory of Extreme order Statistics , Second Edition , Florida : Kreiger . Malabar
  • Galambos , J. and Kotz , S. 1978 . Characterizations of Probability Distributions , Berlin : Springer-Verlag . Lecture Notes in Mathematics No. 675
  • Gnedenko , B. 1943 . Sur la distribution limité du terme maximum d'une série aléatoire . Ann. Math. , 44 : 423 – 453 .
  • Govindarajulu , Z. 1963 . On moments of order statistics and quasi-ranges from normal populations . Ann. Math. Statist. , 34 : 633 – 651 .
  • Govindarajulu , Z. 1980 . Characterization of the geometric distribution using properties of order statistics . J. Statist. Plann. Inference , 4 : 237 – 247 .
  • Greenwood , R. E. and Glasgow , M. O. 1950 . Distribution of maximum and minimum frequencies in a sample drawn from a multinomial distribution . Ann. Math. Statist. , 21 : 416 – 424 .
  • Gupta , P. L. and Gupta , R. C. 1981 . Probability of ties and Markov property in discrete order statistics . J. Statist. Plann. Inference , 5 : 273 – 279 .
  • Gupta , R. C. 1974 . Characterization of distributions by property of discrete order statistics . Commun. Statist. Theo. Meth. , 3 : 287 – 289 .
  • Gupta , S. S. 1960 . Order statistics from the gamma distribution . Technometrics , 2 : 243 – 262 .
  • Gupta , S. S. 1966 . “ Selection and ranking procedures and order statistics for the binomial distribution ” . In Classical and Contagious Discrete Distributions , Edited by: Patil , G. P. 219 – 230 . New York : Pergamon Press .
  • Gupta , S. S. and Nagel , K. 1967 . On selection and ranking procedures and order statistics from the multinomial distribution . Sankhyā B , 29 : 1 – 34 .
  • Gupta , S. S. and Panchapakesan , S. 1974 . On moments of order statistics from independent binomial populations . Ann. Inst. Statist. Math. Supplement , 8 : 95 – 113 .
  • Haan , L. 1970 . On Regular Variation and Its Application to the Weak Convergence of Sample Extremes , Amsterdam : Mathematics Centre . Mathematical Centre Tract 32
  • Herzel , A. 1963 . On order statistics and on the mean deviation about the median value in samples taken with or without replacement from discrete and finite populations . Bibliotecca Metron Serie C , 2 : 163 – 198 . In Italian
  • Hong , C.-J. 1985 . Some characterizations of the geometric distribution by the independence of minimum and difference (Abstract) . Inst. Math. Staist. Bulletin , 14 : 215
  • Huang , J. S. 1989 . Moment problem of order statistics: A review . International Statist. Review , 57 : 59 – 66 .
  • Jha , V. D. 1974 . A note on order statistics for discrete heterogeneous populations . J. Indian Statist. Assoc. , 12 : 21 – 30 .
  • Kabe , D. G. 1969 . Some distribution problems of order statistics from discrete populations . Ann. Inst. Statist. Math. , 21 : 551 – 556 .
  • Karlin , S. and Rinott , Y. 1980 . Classes of orderings of measures and related correlation inequalities . J. Multivariate Anal. , 10 : 467 – 498 .
  • Kemperman , J. H. B. 1971 . Product measures for which duration and minimum are independent . Sankhyā A , 33 : 271 – 288 .
  • Khatri , C. G. 1962 . Distributions of order statistics for discrete case . Ann. Inst. Statist. Math. , 14 : 167 – 171 .
  • Kirmani , S. N. U. A. and Alam , S. N. 1980 . Characterization of the geometric distribution by the form of a predictor . Commun. Statist.-Theory Meth. , 9 : 541 – 547 .
  • Kozelka , R. M. 1956 . Approximate upper percentage points for extreme values in multinomial sampling . Ann. Math. Statist. , 27 : 507 – 512 .
  • Margolin , B. H. and Winokur , H. S. Jr. 1967 . Exact moments of the order statistics of the geometric distribution and their relation to inverse sampling and reliability of redundant systems . J. Amer. Statist. Assoc. , 62 : 915 – 925 .
  • Melnick , E. L. 1964 . “ Moments of ranked Poisson variates ” . Virginia Polytechnic Institute . M.S. Thesis
  • Mises , R. 1936 . La distribution de la plus grande de n valeurs . Rev. Math. Union Interbalcanique , 1 : 141 – 160 . Reproduced in Selected Papers of Richard von Mises, Amer. Math. Soc. II, (1964), 271–294
  • Nagaraja , H. N. 1982 . On the non-Markovian structure of discrete order statistics . J. Statist. Plann. Inference , 7 : 29 – 33 .
  • Nagaraja , H. N. 1986a . A note on conditional Markov property of discrete order statistics . J. Statist. Plann. Inference , 13 : 37 – 43 .
  • Nagaraja , H. N. 1986b . Structure of discrete order statistics . J. Statist. Plann. Inference , 13 : 165 – 177 .
  • Nagaraja , H. N. 1988 . Some characterizations of discrete distributions based on linear regressions of adjacent order statistics . J. Statist. Plann. Inference , 20 : 65 – 75 .
  • Nagaraja , H. N. and Srivastava , R. C. 1987 . Some characterizations of geometric type distribution based on order statistics . J. Statist. Plann. Inference , 17 : 181 – 191 .
  • Pinsker , I. S. , Kipnis , V. and Grechanovsky , E. 1986 . A recursive formula for the probability of occurrence of at least m out of N events . Amer. Statistician , 40 : 275 – 276 .
  • Puri , P. S. and Rubin , H. 1970 . A characterization based on the absolute difference of two i.i.d. random variables . Ann. Math. Statist. , 41 : 2113 – 2122 .
  • Rao , C. R. and Shanbhag , D. N. 1986 . Recent results on characterization of probability distributions: a unified approach through extensions of Deny's theorem . Adv. Appl. Probab. , 18 : 660 – 678 .
  • Rider , P. R. 1951 . The distribution of the range in samples from a discrete rectangular distribution . J. Amer. Statist. Assoc. , 46 : 375 – 378 .
  • Reiss , R. D. 1989 . Approximate Distributions of Order Statistics , Berlin : Springer-Verlag .
  • Resnick , S. I. 1987 . Extreme Values, Regular Variation, and Point Processes , New York : Springer-Verlag .
  • Rüschendorf , L. 1985 . Two remarks on order statistics . J. Statist. Plann. Inference , 11 : 71 – 74 .
  • Scheffé , H. and Tukey , J. W. 1945 . Non-parametric estimation—I. Validation of order statistics . Ann. Math. Statist. , 16 : 187 – 192 .
  • Sedransk , J. and Meyer , J. 1978 . Confidence intervals for the quantiles of a finite population: simple random and stratified simple random sampling . J. R. Statist. Soc. B , 40 : 239 – 252 .
  • Sedransk , J. and Smith , P. J. 1988 . “ Inference for finite population quantiles ” . In Handbook of Statistics 6: Sampling , Edited by: Krishnaiah , P. R. and Rao , C. R. 267 – 289 . Amsterdam : North Holland .
  • Shah , B. K. 1966 . A note on Craig's paper on the minimum of binomial variates . Biometrika , 53 : 614 – 615 .
  • Shanbhag , D. N. 1977 . An extension of the Rao-Rubin characterization of the Poisson distribution . J. Appl. Probab. , 14 : 640 – 646 .
  • Shorack , G. R. and Wellner , J. A. 1986 . Empirical Processes With Applications to Statistics , New York : John Wiley .
  • Sillitto , G. P. 1951 . Interrelations between certain linear systematic statistics of samples from any continuous population . Biometrika , 38 : 377 – 382 .
  • Siotani , M. 1956 . Order statistics for discrete case with a numerical application to the binomial distribution . Ann. Inst. Statist. Math. , 8 : 95 – 104 .
  • Siotani , M. and Ozawa , M. 1958 . Tables for testing the homogeneity of k independent binomial experiments on a certain event based on the range . Ann. Inst. Statist. Math. , 10 : 47 – 53 .
  • Smirnov , N. V. 1949 . Limit distributions for the term of a variational series . Trudy Mat. Inst. Steklov , 25 : 1 – 60 . (In Russian). English translation in Amer. Math. Soc. Transl. 67, (1952), 1–64
  • Smith , P. J. and Sedransk , J. 1983 . Lower bounds for confidence coefficients for confidence intervals for finite populations quantiles . Commun. Statist.-Theory Meth. , 12 : 1329 – 1344 .
  • Sreehari , M. 1983 . A characterization of the geometric distribution . J. Appl. Probab. , 20 : 209 – 212 .
  • Srivastava , R. C. 1974 . Two characterizations of the geometric distribution . J. Amer. Statist. Assoc. , 69 : 267 – 269 .
  • Srivastava , R. C. 1981 . “ On some characterizations of the geometric distribution ” . In Statistical Distributions in Scientific Work , Edited by: Taillie , C. Vol. 4 , 349 – 355 . Dordrecht : D. Reidel .
  • Srivastava , R. C. 1986 . On characterizations of the geometric distribution using properties of order statistics . J. Appl. Probab. , 23 : 227 – 232 .
  • Wilks , S. S. 1962 . Mathematical Statistics , New York : John Wiley .
  • Young , D. H. 1970 . The order statistics of the negative binomial distributions . Biometrika , 57 : 181 – 186 .
  • Young , D. H. 1971 . Moment relations for order statistics of the standardized gamma distribution and the inverse multinomial distribution . Biometrika , 58 : 637 – 640 .
  • Young , D. H. 1973 . Some results for the order statistics of the negative binomial distribution under slippage configurations . Biometrika , 60 : 206 – 209 .
  • Zijlstra , M. 1983 . Characterizations of the geometric distribution by distributional properties . J. Appl. Probab. , 20 : 843 – 850 .

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