Sains Malaysiana 43(11)(2014): 1801–1809

An Empirical Assessment of the Closeness of Hidden Truncation and Additive Component based Skewed Distributions

(Penilaian Empirik Keakraban Pemangkasan Tersembunyi dan Komponen

Tambahan berasaskan Taburan Terpencong)

PARTHA JYOTI HAZARIKA¹ & SUBRATA CHAKRABORTY²*

¹Department of Statistics, North-Eastern Hill University, Shillong 793022, Meghalaya, India

²Department of Statistics, Dibrugarh University, Dibrugarh 786004, Assam, India

Received: 26 June 2013/Accepted: 31 March 2014

ABSTRACT

Hidden truncation (HT) and additive component (AC) are two well-known paradigms of generating skewed distributions from known symmetric distribution. In case of normal distribution it has been known that both the above paradigms lead to Azzalini’s (1985) skew normal distribution. While the HT directly gives the Azzalini’s (1985) skew normal distribution, the one generated by AC also leads to the same distribution under a re-parameterization proposed by Arnold and Gomez (2009). But no such re-parameterization which leads to exactly the same distribution by these two paradigms has so far been suggested for the skewed distributions generated from symmetric logistic and Laplace distributions. In this article, an attempt has been made to investigate numerically as well as statistically the closeness of skew distributions generated by HT and AC methods under the same re-parameterization of Arnold and Gomez (2009) in the case of logistic and Laplace distributions.

Keywords: KS test; Kullback–Leibler (KL) distance; Monte Carlo integration; simulation; skew Laplace distribution; skew logistic distribution

ABSTRAK

Pemangkasan tersembunyi (HT) dan komponen tambahan (AC) adalah dua paradigma yang terkenal dalam menghasilkan taburan terpencong daripada taburan simetri. Dalam taburan normal ia telah diketahui bahawa kedua-dua paradigma di atas membawa terus kepada taburan pencongan normal (Azzalini 1985). Manakala HT terus memberikan taburan pencongan normal (Azzalini 1985), yang dijana oleh AC juga membawa kepada taburan yang sama di bawah pemparameteran semula yang dicadangkan oleh Arnold dan Gomez (2009). Tetapi tiada pemparameteran semula yang membawa kepada taburan yang sama oleh kedua-dua paradigma ini disarankan untuk taburan pencongan yang dihasilkan daripada simetri logistik dan taburan Laplace. Dalam artikel ini, usaha telah dibuat untuk mengkaji secara berangka dan statistik keakraban taburan pencongan yang dijana oleh kaedah HT dan AC di bawah pemparameteran semula Arnold dan Gomez (2009) bagi kes logistik dan taburan Laplace.

Kata kunci: Integrasi Monte Carlo; jarak Kullback-Leibler (KL); simulasi; taburan terpencong Laplace; taburan terpencong logistik; ujian KS

REFERENCES

Arnold, B.C. & Beaver, R.J. 2002. Skewed multivariate models related to hidden truncation and/or selective reporting (with discussion). Test 11: 7-54.

Arnold, B.C. & Beaver, R.J. 2000a. The skew Cauchy distribution. Statist. Prob. Lett. 49: 285-290.

Arnold, B.C. & Beaver, R.J. 2000b. Hidden truncation models. Sankhya A62: 22-35.

Arnold, B.C., Beaver, R.J., Groeneveld, R.A. & Meeker, W.Q. 1993. The non-truncated marginal of a truncated bivariate normal distribution. Psychometrika 58: 471-478.

Arnold, B.C. & Gomez, H.W. 2009. Hidden truncation and additive components: Two alternative skewing paradigms. Calcutta Statistical Association Bulletin 61: 241-244.

Azzalini, A. 1986. Further results on a class of distributions which includes the normal ones. Statistica 46: 199-208.

Azzalini, A. 1985. A class of distributions which includes the normal ones. Scand. J. Stat. 12: 171-178.

Burnham, K.P. & Anderson, D.R. 2002. Model Selection and Multimodel Inference: A Practical Information Theoretic Approach. 2nd ed. New York: Springer.

Chakraborty, S. & Hazarika, P.J. 2011. A survey of the theoretical developments in univariate skew normal distributions. Assam. Statist. Rev. 25(1): 41-63.

Chakraborty, S., Hazarika, P.J. & Ali, M. Masoom. 2014a. A multimodal skew laplace distribution. Pak. J. Statist. 30(2): 253-264.

Chakraborty, S., Hazarika, P.J. & Ali, M. Masoom. 2014b. A multimodal skewed extension of normal distribution: Its properties and applications. Statistics: A Journal of Theoretical and Applied Statistics DOI: 10.1080/02331888.2014.908880.

Chakraborty, S., Hazarika, P.J. & Ali, M. Masoom. 2012. A new skew logistic distribution and its properties. Pak. J. Statist. 28(4): 513-524.

Gupta, R.D. & Kundu, D. 2004. Discriminating between gamma and generalized exponential distributions. J. Statist. Comput. Simul. 74(2): 107-121.

Gupta, R.D. & Kundu, D. 2003a. Closeness of gamma and generalized exponential distribution. Communications in Statistics-Theory and Methods 32(4): 705-721.

Gupta, R.D. & Kundu, D. 2003b. Discriminating between Weibull and generalized exponential distributions. Computational Statistics & Data analysis 43(2): 179-196.

Huang, W.J. & Chen, Y.H. 2007. Generalized skew Cauchy distribution. Statist. Probab. Lett. 77: 1137-1147.

Kotz, S., Kozubowski, T.J. & Podgorski, K. 2001. The Laplace Distribution and Generalizations. Berlin: Birk Hauser.

Nadarajah, S. 2009. The skew logistic distribution. Adv. Stat. Anal. 93: 187-203.

Nekoukhou, V. & Alamatsaz, M.H. 2012. A family of skew-symmetric-Laplace distributions. Statistical Papers 53(3): 685-696.

Pakyari, R. 2011. Discriminating between generalized exponential, geometric extreme exponential and Weibull distributions. J. Statist. Comput. Simul. 80(12): 1403-1412.

Robert, C.P. & Casella, G. 2004. Monte Carlo Statistical Methods. 2nd ed. New York: Springer.

Wahed, A.S. & Ali, M. Masoom. 2001. The skew logistic distribution. J. Stat. Res. 35: 71-80.

*Corresponding author; email: subrata_arya@yahoo.co.in