Penultimate gamma approximation in the CLT for skewed distributions
Boutsikas, Michael V.1
1 Department of Statistics and Insurance Science, University of Piraeus, Greece.
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 590-604.

Sharp upper bounds are offered for the total variation distance between the distribution of a sum of n independent random variables following a skewed distribution with an absolutely continuous part, and an appropriate shifted gamma distribution. These bounds vanish at a rate O(n -1 ) as n→∞ while the corresponding distance to the normal distribution vanishes at a rate O(n -1/2 ), implying that, for skewed summands, pre-asymptotic (penultimate) gamma approximation is much more accurate than the usual normal approximation. Two illustrative examples concerning lognormal and Pareto summands are presented along with numerical comparisons confirming the aforementioned ascertainment.

Reçu le :
DOI : 10.1051/ps/2015010
Classification : 60F05, 60G50, 62E17, 60E15
Mots-clés : Central limit theorem, gamma approximation, lognormal sums, rates of convergence, Pareto sums, total variation distance, quantile approximation
Boutsikas, Michael V. 1

1 Department of Statistics and Insurance Science, University of Piraeus, Greece. 
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Boutsikas, Michael V. Penultimate gamma approximation in the CLT for skewed distributions. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 590-604. doi : 10.1051/ps/2015010. http://www.numdam.org/articles/10.1051/ps/2015010/

M.V. Boutsikas, Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part. J. Stat. Plan. Inf. 141 (2011) 1250–1268. | MR | Zbl

M.V. Boutsikas and E. Vaggelatou, On the distance between convex-ordered random variables. Adv. Appl. Probab. 34 (2002) 349–374. | MR | Zbl

M.V. Boutsikas and E. Vaggelatou, A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables. Bernoulli 16 (2010) 301–330. | MR | Zbl

M. Denuit, C. Lefèvre and M. Shaked, The s-convex orders among real random variables, with applications. Math. Inequal. Appl. 1 (1998) 585–613. | MR | Zbl

D. Dufresne, The log-normal approximation in financial and other computations. Adv. Appl. Probab. 36 (2004) 747–773. | MR | Zbl

P. Hall, Chi squared approximations to the distribution of a sum of independent random variables. Ann. Probab. 11 (1983) 1028–1036. | MR | Zbl

S. Karlin and A. Novikoff, Generalized convex inequalities. Pac. J. Math. 13 (1963) 1251–1279. | MR | Zbl

N. Mehta, J. Wu, A. Molisch and J. Zhang, Approximating a Sum of Random Variables with a Lognormal. IEEE Trans. Wirel. Commun. 6 (2007) 2690–2699.

V.V. Petrov, Limit Theorems of Probability Theory. Clarendon Press, Oxford (1995). | MR | Zbl

Yu V. Prokhorov, On a local limit theorem for densities. Dokl. Akad. Nauk SSSR 83 (1952) 797–800. | MR | Zbl

S.T. Rachev, Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester, New York (1991). | MR | Zbl

S.T. Rachev and L. Ruschendorf, Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. Appl. Probab. 22 (1990) 350–374. | MR | Zbl

C.M. Ramsay, The Distribution of Sums of Certain I.I.D. Pareto Variates. Commun. Stat., Theory Methods 35 (2006) 395–405. | MR | Zbl

R. Reijnen, W. Albers and W.C.M. Kallenberg, Approximations for stop-loss reinsurance premiums. Insur. Math. Econ. 36 (2005) 237–250. | MR | Zbl

S. Kh. Sirazhdinov and M. Mamatov, On convergence in the mean for densities. Theory Probab. Appl. 7 (1962) 424–429. | MR | Zbl

I.V. Zaliapin, Y.Y. Kagan and F.P. Schoenberg, Approximating the Distribution of Pareto Sums. Pure Appl. Geophys. 162 (2005) 1187–1228.

Zolotarev V.M. Probability metrics. Theory Prob. Appl. 28 (1983) 278–302. | MR | Zbl

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