Abstract
The problem of testing skew-symmetry of a distribution is studied in a general model of skew distributions. Toward this end, an order statistic-based test is first introduced to test the null hypotheses of symmetry against the alternative of skew-symmetry of a distribution. Some properties of this test are also studied. Then, using the idea of ranked set sampling, some appropriate sampling schemes are used to test skew-symmetry of a given data set. The power of the proposed tests are compared numerically to determine the best ranked set sampling scheme in different situations. Further, a comparison with some of existing non-parametric tests has been done. A real data set is also used to illustrate the results of the paper. Finally, some conclusions are stated.
Similar content being viewed by others
References
Albatineh AN, Kibria BMG, Wicox ML, Zogheib B (2014) Confidence interval estimation for the population coefficient of variation using ranked set sampling: a simulation study. J Appl Stat 41:733–751
Arellano-Valle RB, Gomez HW, Quintana FA (2004) A new class of skew-normal distribution. Commun Stat Theory Methods 33:1465–1480
Arnold BC, Balakrishnan N, Nagaraja HN (2008) A first course in order statistics. SIAM, Philadelphia
Asgharzadeh A, Esmaeili L, Nadarajah S (2013) A generalized skew logistic distribution. REVSTAT 11:317–338
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178
Azzalini A, Capitano A (1999) Statistical applications of the multivariate skew-normal distribution. J R Stat Soc Ser B 61:579–602
Azzalini A, Dalla-Valle A (1998) The multivariate skew-normal distribution. Biometrika 83:715–726
Cook RD, Weisberg S (1994) An introduction to regression graphics. Wiley, Hoboken
Dalla Valle A (2007) A test for the hypothesis of skew-normality in a population. J Stat Comput Simul 77:63–77
David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, New York
Dell TR, Clutter JL (1972) Ranked set sampling with order statistics background. Biometrics 28:545–555
Gomez H, Venegas O, Bolfarine H (2007) Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics 18:395–407
Gupta AK, Chang FC, Huang WJ (2002) Some skew-symmetric models. Random Oper Stoch Equations 10:133–140
Gupta AK, Chen T (2001) Goodness of fit tests for the skew-normal distribution. Commun Stat Simul Comput 30:907–930
Gupta RD, Kundu D (2010) Generalized logistic distributions. J Appl Stat Sci 18:51–66
Hasanalipour P, Hassani Oskouei L, Jabbari Khamnei H, Khorshidian K (2017) The Beta skew-generalized normal distribution. Commun Stat Theory Methods 46:2270–2276
Hasanalipour P, Razmkhah M (2019) Inference on skew-normal distribution based on Fisher information in order statistics. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2019.1674869
Hasanalipour P, Sharafi M (2012) A new generalized Balakrishnan skew-normal distribution. Stat Pap 53:219–228
He X, Chen W, Qian W (2018) Maximum likelihood estimators of the parameters of the log-logistic distribution. Stat Pap. https://doi.org/10.1007/s00362-018-1011-3
Jamalizadeh A, Behboodian J, Balakrishnan N (2008) A two-parameter generalized skew-normal distribution. Stat Probab Lett 78:1722–1726
Kozubowski TJ, Nolan JP (2008) Infinite divisibility of skew Gaussian and Laplace laws. Stat Prob Lett 78:654–660
Liseo B, Loperfido N (2006) A note on reference priors for the scalar skew-normal distribution. J Stat Plan Interface 136:373–389
Ma Y, Hart JD (2007) Constrained local likelihood estimators for semiparametric skew-normal distributions. Biometrika 94:119–134
Mateu-Figueras G, Puing P, Pewsey A (2007) Goodness-of-fit tests for the skew-normal distribution when the parameters are estimated from the data. Commun Stat Theory Methods 36(9):1735–1755
Meintanis SG (2010) Testing skew-normality via the moment generating function. Math Methods Stat 19(1):64–72
McIntyre GA (1952) A method for unbiased selective sampling, using ranked sets. Crop Pasture Sci 3(4):385–390
McGill WJ (1962) Random fluctuations of response rate. Psychometrika 27:3–17
Nadarajah S (2009) The skew logistic distribution. Adv Stat Analy 93:187–203
Nadarajah S, Kotz S (2003) Skewed distributions generated by the normal kernel. Stat Prob Lett 65:269–277
Nekoukho V, Alamatsaz MH (2012) A family of skew-symmetric-Laplace distributions. Stat Pap 53:685–696
Pewsey A (2001) Problems of inference for Azzalini’s skew-normal distribution. J Appl Stat 27:859–870
Rahmani H, Razmkhah M (2017) Perfect ranking test in moving extreme ranked set sampling. Stat Pap 58:855–875
Rohatgi VK, Saleh AKME (2001) An introduction to probability and statistics. Wiley, New York
Salvan A (1986) Locally most powerful invariant tests of normality (in Italian), Atti della XXXIII Riunione Scientifica della Societa Italiana di. Statistica 2:173–179
Samawi HM (1996) Estimating the population mean using extreme ranked set sampling. Biometr J 38(5):577–586
Samawi HM, Al-Sagheer OAM (2001) On the estimation of the distribution function using extreme and median ranked set sampling. Biometr J 43(3):357–373
Sartori N (2006) Bias prevention of maximum likelihood estimates for scalar skew-normal and skew t distributions. J Stat Plan Inference 135:4259–4275
Sastry DVS, Bhati D (2016) A new skew logistic distribution: properties and applications. Braz J Probab Stat 30(2):248–271
Zhang L, Dong X, Xu X (2014) Sign tests using ranked set sampling with unequal set sizes. Stat Probab Lett 82:2213–2220
Acknowledgements
The authors would like to thank an anonymous referees and the associate editor for their useful comments and constructive criticisms on the original version of this manuscript which led to this considerably improved version. This research was supported by a grant from Ferdowsi University of Mashhad (No. 2/52143).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hasanalipour, P., Razmkhah, M. Testing skew-symmetry based on extreme ranked set sampling. Stat Papers 62, 2311–2332 (2021). https://doi.org/10.1007/s00362-020-01183-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-020-01183-3