# Asymptotic theory for Rank inversion estimators from iid to weakly dependence: the case of Generalized Lehmann's alternative models.

#### Séminaire Probabilités & Statistique

31/05/2012 - 14:00
Ryozo Miura *(Hitotsubashi University, Japan)*
Salle 1 - Tour IRMA

I would like to present an asymptotic theory of statistical inferences based on rank statistics, for a simple regression model. A specific feature of our model is that the residual terms are modeled with Generalized Lehmann’s Alternative model which can be regarded as a semi-nonparametric version of skew-symmetric distributions. Also, we will take advantage of rank inversion estimator of regression coefficient β, that is, we can estimateβwithout estimating the intercept parameter α. αwill be estimated afterward. Here we note that α is different from usual intercepts in Least square estiomation frameworks. In Miura and Tsukahara (1985,1993) and Dabrowska, Doksum and Miura (1989), Generalized Lehmann’s Alternative models were introduced and implemented in One and Two Sample Problems. The model represents a distribution that is a transformation of an original distribution (See also Lehmann (1954)). This transformation makes an original symmetric distribution skewed in any plausible and semi-nonparametric manner. We note that this model can be regarded as a nonparametric version of skew symmetric distribution in Azzalini(1985). In our work, the estimation theories, based on rank sstatistics, for skew or transformation parameters, and for location parameters in our Lehmann’s Alternative Models are developed. These rank inversion estimators are the estimates derived from rank statistics for statistical testing. The asymptotic results have been already given in Miura and Tsukahara (1985,1993) for the case where the observations are independent and identically distributed (e.g. iid cases). The mathematical tools used there were a convergence of empirical distribution functions for the iid observations. In recent years, the study of a convergence of empirical distributions have developed enough to cover the case of non-iid cases such as weakly dependent cases ( Shao & Yu (1996), Louhichi (2000) ). We will take advantage of their results in order to extend our old results of iid cases, to those of the weakly dependent cases. Our work is almost complete in mathematical details, except that we need to determine the final analytical conditions for our statements. But we will show outline of the proof. We will also show further attempts of ours to construct estimators for regression parameters in a simple linear regression model with skew-symmetric residual terms described by our Lehmann’s Alternative Models. It should be remarked that this estimators work for the iid case as well. So we first talk about iid cases where we utilize the results by Jureckova(1969,1971) and Jaeckel (1972) for estimating β in iid cases. Then, using a sequence of residuals with estimated β, we go onto estimation of transformation parameters and intercepts. Next we will talk about the weakly dependent cases where we need to establish estimation for β. We are currently working on mathematical proof for this part. Once this can be done, it will be implemented into the estimation with residuals in the same way as above. A possible application of our model may be found in a field of quantitative finance. As discussed in Miura, Yokouchi and Aoki (2009), Individual Hedge Fund return data in many cases are not iid, but behaves like an order-changing auto-regressive process. It means that they are realization of weakly dependent processes. Also, their distributions are not symmetric, but rather skewed. The Generalized Lehmann’s Alternative Models can describe such statistical structure ( order changing autoregressive natures and skewed distributions ) of hedge fund return data with a transformation parameter that may represent a degree of skewness. The estimators defined here are all asymptotically normally distributed. The asymptotic variances of the estimator in our weakly dependent cases are different from that in the iid case. The difference comes from the covariances of the observations. In an application we need to estimate these asymptotic variances. Tsukahara (2011) worked on estimation for Distortion Risk Measures, where he introduced general L-statistics as estimators and proved their asymptotic normality in a weakly dependent case. Then, he proceeded to define estimators of their asymptotic variance and proved their consistency using the results by Shao and Yu (1996). The asymptotic variance is different in this case from the iid case only in that it adds covariance terms. Other formal figures are very similar to each other. Our work here takes advantage of these new results including estimators of asymptotic variance since the formulation of the problem is very similar. Louhichi’s result is on a same line of Shao and Yu and it improves a convergence condition of the sum of covariances. References [1] Azzalini, A.(1985) .“A Class of Distributions which includes the Normal Ones.” Scandinavian Journal of Statistics. Vol.12. 171-178. [2] Dabrowska D.M., Doksum K.A. and Miura R.(1989). “Rank Estimates in a Class of Semiparametric Two-Sample Models.” Annals of Institute of Statistical Mathematics, 41 (1989), 63-79. [3] Jaeckel, L.A. (1972) “Estimating regression coefficients by minimizing the dispersion of the residuals.” Vol.43. No,5. 1449-1458. [4] Jureckova, J. (1971) “Nonparametric estimate of regression coefficients.” Ann. Math. Statist.Vol.42. 1328-1338. [5] Lehmann, E.L.(1953) . “The power of rank tests.” Annals of Mathematical Statistics. Vol.24. 23-43. [6] Miura R. (1985). “Hodges–Lehmann type estimators and Generalized Lehmann’s Alternatives.” A special lecture (in Japanes) at Annual Meeting of Japan Mathematical Associations. April 1985. [7]. Miura R. , D. Yokouchi and Y. Aoki (2009). “A Note on Statistical Models for Individual Hedge Fund Returns.” Math.Meth. Oper. Res. 69. 553-577. [8].Tsukahara, H. & Miura, R.(1993). “One sample estimation for generalized Lehmann’s alternative models.” Statistica Sinica. Vol.3. 83-101. [9] Louhichi S.(2000) “Weak convergence for empirical processes of associated sequences. “Ann.Inst.Henri Poincare. Probabilites et Statistiques. 36 (2000), 5, 547-567. [10] Shao Q.M. and Yu H. (1996) “Weak Convergences for Weighted Empirical Processes of Dependent Sequences.” Annals of Probability. 24. 2098-2127 [11] Tsukahara H. (2011). “Estimation of Distortion Risk Measures.” Working Paper. Department of Economics at Seijo University