Abstract
General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.
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References
Ahsanullah M (1981) On characterizations of the exponential distribution by spacings. Statist Hefte 22:316–320
Arnold BC (1988) Bounds on the expected maximum. Comm Statist A 17:2135–2150
Arnold BC, Balakrishnan N (1989) Relations, Bounds and Approximations for Order Statistics. Lecture Notes in Statistics. Springer, New York Berlin
Balakrishnan N (1986) Order statistics from discrete distributions. Comm Statist A 15:657–675
David HA (1981) Order Statistics, 2nd ed. Wiley, New York
David HA (1988) General bounds and inequalities for order statistics. Comm Statist A 17:2119–2134
Dziubdziela W, Kopociński B (1976) Limiting properties of thek-th record value. Appl Math 15:187–190
Gajek L (1985) Limiting properties of difference between the successivek-th record values. Probability & Math Statist 5:221–224
Gajek L, Gather U (1989) Characterizations of the exponential distribution by failure rate and moment properties. In: Hülser J, Reiss R-D (eds) Springer Lecture Notes 51, pp 114–124
Galambos J, Kotz S (1978) Characterizations of Probability Distributions. Lecture Notes in Mathematics 675. Springer, Berlin New York
Grudzién Z, Szynal D (1983) On the expected values of thek-th record values and associated characterizations of distributions. Prob Stat Decision Theory Vol A, Proc 4th Pannonian Symp on Math Stat Austria 1983. Konecny et al (eds)
Gumbel EJ (1954) The maxima of the mean largest value and of the range. Ann Math Statist 25:76–84
Hartley HO, David HA (1954) Universal bounds for mean range and extreme observation. Ann Math Statist 25:85–99
Huang JS (1989) Moment problem of order statistics: a review. Int Stat Rev 57:59–66
Khan AH, Ali MM (1987) Characterization of probability distributions through higher order gap. Comm Statist A 16:1281–1287
Lin GD (1984) A note on equal distributions. Ann Inst Statist Math 36:451–453
Lin GD (1988) Characterizations of uniform distributions and of exponential distributions. Sankhya, Ser A 50:64–69
Mitrinović DS (1970) Analytic Inequalities. Springer, Berlin
Moriguti S (1951) Extremal property of extreme value distributions. Ann Math Statist 22:523–536
Nagaraja HN (1978) On the expected values of record values. Austral J Statist 20:176–182
Nassar MM, Mahmoud MR (1985) On characterizations of a mixture of exponential distributions. IEEE Trans Reliability 34:484–488
Plackett RL (1947) Limits of the ratio of mean range to standard deviation. Biometrika 34:120–122
Resnick SI (1973) Limit laws for record values. J Stoch Proc Appl 1:67–82
Srivastava RC (1978) Some characterizations of the exponential distribution based on record values. Abstract. Bull Inst Math Stat 7:283
Srivastava RC (1979) Two characterizations of the geometric distribution by record values. Sankhya, Ser B 40:276–278
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Gajek, L., Gather, U. Moment inequalities for order statistics with applications to characterizations of distributions. Metrika 38, 357–367 (1991). https://doi.org/10.1007/BF02613633
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DOI: https://doi.org/10.1007/BF02613633