Institute of Mathematical Statistics Lecture Notes - Monograph Series

On the Estimation of Symmetric Distributions under Peakedness Order Constraints

Javier Rojo and José Batún-Cutz

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Consider distribution functions F and G and suppose that F is more peaked about a than G is about b. The problem of estimating F or G, or both, when F and G are symmetric, arises quite naturally in applications. The empirical distribution functions Fn and Gm will not necessarily satisfy the order constraint imposed by the experimental conditions. Rojo and Batun-Cutz [Series in Biostatistics vol. 3, Advances in Statistical Modeling and Inference, (2007) 649–670] proposed some estimators that are strongly uniformly consistent when both m and n tend to infinity. However the estimators fail to be consistent when only either m or n tend to infinity. Here estimators are proposed that circumvent these problems and the asymptotic distribution of the estimators is delineated. A simulation study compares these estimators in terms of Mean Squared Error and Bias behavior with their competitors.

Chapter information

Javier Rojo, ed., Optimality: The Third Erich L. Lehmann Symposium (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009), 147-172

First available in Project Euclid: 3 August 2009

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Digital Object Identifier

Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions 62G20: Asymptotic properties
Secondary: 60F05: Central limit and other weak theorems

partial orders nonparametric statistics stochastic order empirical process weak convergence

Copyright © 2009, Institute of Mathematical Statistics


Rojo, Javier. On the Estimation of Symmetric Distributions under Peakedness Order Constraints. Optimality, 147--172, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-LNMS5710.

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  • [1] Arenaz, P., Bitticks L., Panell, K. H. and García, S. (1992). Genotoxic potential of crown ethers in mammalian cells; induction of sister chromatid exchanges. Mutation Research 280 109–115.
  • [2] Bartoszewicz, J. (1985). Moment inequalities for order statistics from ordered families of distributions. Metrika 32 383–389.
  • [3] Bartoszewicz, J. (1985). Dispersive ordering and monotone failure rate distributions. Adv. in Appl. Probab. 17 472–474.
  • [4] Bartoszewicz, J. (1986). Dispersive ordering and the total time on test transformation. Statist. Probab. Lett. 4 285–288.
  • [5] Bickel, P. J. and Lehmann, E. L. (1979). Descriptive statistics for nonparametric models. IV. Spread. In Contributions to Statistics, Jaroslav Hájek Memorial Volume (J. Jureckova, ed.) 33–40. Dordrecht, Riedel.
  • [6] Birnbaum, Z. W. (1948). On random variables with comparable peakedness. Ann. Math. Statist. 19 6–81.
  • [7] Brown, G. and Tukey, J. W. (1946). Some distributions of sample means. Ann. Math. Statist. 7 1–12.
  • [8] Doksum, K. A. (1969). Starshaped transformations and the power of rank tests. Ann. Math. Statist. 40 1167–1176.
  • [9] El Barmi, H. and Rojo, J. (1997). Likelihood ratio test for peakedness in multinomial populations. J. Nonparametr. Statist. 7 221–237.
  • [10] El Barmi, H. and Mukerjee, H. (2008). Peakedness and peakedness ordering in symmetric distributions. J. Multivariate Anal. To appear. doi:10.1016/j.jmva2008.06.011.
  • [11] Elston, R. C., Boxbaum, S. and Olson, M. (2000). Haseman and Elston revisited. Genetic Epidem. 19 1–17.
  • [12] Embury, S. H., Elias, L., Heller, P. H., Hood, C. E., Greenberg, P. L. and Schrier, S. L. (1977). Remission maintenance therapy in acute myelogenous lukemia. Western Journal of Medicine 126 267–272.
  • [13] Fraser, D. A. S. (1957). Nonparametric Methods in Statistics. Wiley, New York.
  • [14] Haseman, J. K. and Elston, R. C. (1972). The investigation of linkage between a quantitative trait and a marker locus. Behav. Genet. 2 2–19.
  • [15] Karlin, S. (1968). Total Positivity. Stanford Univ. Press, CA.
  • [16] Lehmann, E. L. (1955). Ordered families of distributions. Ann. Statist. 26 399–419.
  • [17] Lehmann, E. L. (1959). Testing Statistical Hypotheses. Wiley, New York.
  • [18] Lehmann, E. L. (1988). Comparing location experiments. Ann. Statist. 16 521–533.
  • [19] Lehmann, E. L. and Rojo, J. (1992). Invariant directional orderings. Ann. Statist. 20 2100–2110.
  • [18] Liu, B. H. (1988). Statistical Genomics, Linkage, Mappings, and QTL Analysis. CRC Press, New York.
  • [21] Lo, S. H. (1987). Estimation of distribution functions under order restrictions. Statist. Decisions 5 251–262.
  • [22] Marshall, A. W. and Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer, New York.
  • [23] Oh, M. (2004). Inference for peakedness ordering between two distributions. J. Kor. Statist. Soc. 33 303–312.
  • [24] Oja, H. (1981). On location, scale, skewness, and kurtosis of univariate distributions. Scand. J. Statist. 8 154–168.
  • [25] Proschan, F. (1965). Peakedness of distributions of convex combinations. Ann. Math. Statist. 36 1703–1706.
  • [26] Rojo, J. and He, G. Z. (1991). New properties and characterizations of the dispersive ordering. Statist. Probab. Lett. 11 365–372.
  • [27] Rojo, J. and Wang, J. (1994). Test based on L-statistics to test the equality in dispersion of two probability distributions. Statist. Probab. Lett. 21 107–113.
  • [28] Rojo, J. (1995). On the weak convergence of certain estimators of stochastically ordered survival functions. J. Nonparametr. Statist. 4 349–363.
  • [29] Rojo, J. (1995). Nonparametric quantile estimation under order constraints, J. Nonparametr. Statist. 5 185–200.
  • [30] Rojo, J. and Ma, Z. (1996). On the estimation of stochastically ordered survival functions. J. Statist. Comput. Simulation 55 1–21.
  • [31] Rojo, J. (1998). Estimation of the quantile function of an IFRA distribution. Scand. J. Statist. 25 293–310.
  • [32] Rojo, J. (1999). On the estimation of a survival function under a dispersive order constraint. J. Nonparametr. Statist. 11 107–135.
  • [33] Rojo, J. (2004). On the estimation of survival functions under a stochastic order constraint. In The First Erich L. Lehmann Symposium - Optimality (J. Rojo, ed.). IMS Lecture Notes Monogr. Ser. 44 37–61. Inst. Math. Statits., Beachwood, OH.
  • [34] Rojo, J. and Batun-Cutz, J. (2007). Estimation of symmetric distributions subjects to peakedness order. In Series in Biostatistics Vol 3, Advances in Statistical Modeling and Inference Chapter 13, 649–670.
  • [35] Rojo, J., Batun-Cutz, J. and Durazo, R. (2007). Inference under peakedness restrictions. Statist. Sinica 17, 1165–1189.
  • [36] Schuster, E. (1975). Estimating the distribution function of a symmetric distribution. Biometrika 62 631–635.
  • [37] Schweder, T. (1982). On the dispersion of mixtures. Scand. J. Statist. 9 165–169.
  • [38] Shaked, M. (1980). On mixtures from exponential families. J. Roy. Statist. Soc. Ser. B 42 192–198.
  • [37] Shaked, M. (1982). Dispersive orderings of distributions. J. Appl. Probab. 19 310–320.
  • [40] Shaked, M. and Shantikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • [41] Shibata, T. and Takeyama, T. (1977). Stochastic theory of pitting corrosion. Corrosion 33 243.