Some asymptotic expansions and distribution approximations outside a CLT context 1

Transcription

1 6th St.Petersburg Workshop on Simulation (2009) Some asymptotic expansions and distribution approximations outside a CLT context 1 Manuel L. Esquível 2, João Tiago Mexia 3, João Lita da Silva 4, Luís P. C. Ramos 5 Abstract Some asymptotic expansions non necessarily related to the central limit theorem are discussed. After observing that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation, two instances of this observation are presented. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(x + µ n)) n IN, where g is some smooth function, X is a random variable having a moment and a bounded density and (µ n ) n IN is a sequence going to infinity; the multivariate case as well as the proofs and a complete set of references will be published elsewhere. We next present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas, namely, a generic Laplace s type integral, by the sequence (µ n X) n IN, X being a Gamma distributed random variable. Finally, a simulation study of this last example is presented in order to stress the quality of asymptotic approximations proposed. 1. Asymptotics for random variables The set-up for the generic question studied in this paper is given by [A], the following set of conditions. A1 There is a real parameter sequence (µ n ) n IN such that µ + := lim µ n = +. A2 We are given three sequences of random variables, depending on the parameter sequence 1, (X n) n IN, (Y n 0) n IN and (Z n) n IN, and verifying: X n = Y n + Z n or X n = Y n 1 + Zn Z n and lim = 0. (1) Y n n + Y n 1 This work was partially supported by Financiamento Base 2008 ISFL from FCT/MCTES/PT. 2 Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (FCT/UNL), Portugal, mle@fct.unl.pt 3 FCT/UNL, Portugal, jtm@fct.unl.pt 4 FCT/UNL, Portugal, jfls@fct.unl.pt 5 FCT/UNL, Portugal, lpcr@fct.unl.pt

2 where the limit is taken is some prescribed sense: almost surely, in probability, in the mean, etc. A3 Possibly, the values X := X(µ + ), Y := Y (µ + ), Z := Z(µ + ) are not defined. The generic question we want to consider is the following. Given (1), can something be said about the asymptotic distribution of X n knowing the distribution of Y n? It is to be expected that under a set of hypothesis including [A], we may approximate, for large values of the parameter µ n, the distribution of X n, which may be hard to compute, by the distribution of Y n, that is, X n d Y n as µ n 1, in a sense to be determined, preferably, in the Kolmogorov distance sense. Remark 1. The first equality in 1 is a sort of asymptotic equality between the random variables X n and Y n. To the best of our knowledge, a comprehensive theory of asymptotic relations for random variables is not yet available although some authors tackled this topic in particular points. (see [1, p. 443]). Remark 2. The answer for the question above is not, in general, a central limit theorem issue due in particular to the fact that, possibly, the limiting values X, Y and Z are not defined. Our approach is based in the following asymptotic results. Lemma 1 (Esseen s type estimate). Let (X n ) n IN, (Y n ) n IN and (Z n ) n IN be sequences of random variables satisfying the hypotheses [A] above and formula 1. Suppose, furthermore that Y n admits a density and that for some δ ]0, 1] we have IE[ Z n δ ] < +. Then, for C δ = (1/π)2 2 δ 1+δ 24 1+δ δ (1 + (1/δ)): [ sup F Xn (x) F Yn (x) C δ IE Z n δ] δ 1 sup x x F Y n (x) δ 1+δ. (2) Theorem 1. Let (X n ) n IN, (Y n ) n IN and (Z n ) n IN be sequences of random variables satisfying the conditions [A] above and formula 1. Suppose, furthermore, that for each n 1 the random variable Y n admit a density F Y n and that for some δ ]0, 1] we have IE[ Z n δ ] < +. Then, if we have that: [ lim IE Z n δ] δ 1 sup x F Y n (x) = 0, lim sup F Xn (x) F Yn (x) = 0, (3) x that is, we have the uniform approximation, for large values of the parameter µ n, of the distribution function of X n by the distribution function of Y n. 445

3 2. A linear transform approximation result We now present some approximation results, consequences of theorem 1, where the first term in the asymptotic expansion is an affine transform of the initial random variable. The driving tool, both in the unidimensional and multidimensional cases, is to consider asymptotic Taylor type expansions. A first idea to deal with the problem studied in this question would be to apply some form of the delta method (see [2]). As pointed out in remark 2, the delta method relies on the central limit theorem which, in general, is not applicable to the situation under scrutiny. The proof is a consequence of Taylor s theorem and of theorem 1. Theorem 2. Suppose a non-negative absolutely continuous r.v. X such that IE X p < + for some p > 0, with d.f. F X verifying sup F X(x) C. Consider also a sequence x R (X n) n N of r.v. s defined by X n := g(x + µ n) where µ n is a non-random real sequence verifying lim µ n = + and g : R R is a C 2 (R) map such that: (a) E g (X) < +. (b) g is convex on R. (c) g (t) g (t) is decreasing on ]t0, + [ and tends to zero as t +. Then, with Y n := g(µ n ) + g (µ n )X, the law of X n is uniformly approximate by the law of Y n for large values of n, that is, lim sup F Xn (x) F Yn (x) = 0. x R Remark 3. This theorem can be extended to the cases where the support of X is ]a, + [ for some a R. 3. Non linear approximations In this instance of an application of theorem 1 we study an integral transform of a rescaled random variable. We will write X G(u, δ, λ) if the density of X is given by: x IR f (u,δ,λ) (x) = 1 δ u Γ(u) (x λ)(u 1) e x λ δ 1I [λ,+ [ (x), 1I [λ,+ [ being the indicator function of the interval [λ, + [. Recall that for such a random variable we have IE[X] = uδ + λ, IV[X] = uδ 2 and sup x IR f (u,δ,λ) (x) = sup f (u,δ,0) (x) = 1 x IR δγ(u) (u 1)(u 1) e (u 1). Proposition 1. Let X G(u, δ, λ) and (µ n ) n 1 lim µ n = +. Consider also a non-random sequence satisfying X n := a 0 e µ nxt dt (a > 0). 1 + t2 Then, we have lim sup x IR F Xn (x) F 1/(µn X)(x) = 0, that is, the law of X n is approximate by the law of 1/(µ n X) for n sufficiently large. 446

4 Our second example has its source in the general theme of Laplace integrals. Often, the asymptotic behavior of models in the applied sciences is studied using this kind of integrals. The example presented here amounts to a randomization of these models. Proposition 2. Consider p, q > 0, a ]0, + ], f C([0, a[) verifying f(0) 0 and X G(u, δ, λ). Suppose also a non-random sequence (µ n ) n 1 satisfying lim µ n = + and let a f(0)γ p X n := t p 1 f(t)e µ nxt q q dt and Y n := (µ nx) p q. q 0 We then have that lim sup x IR F Xn (x) F Yn (x) = 0, that is, the law of X n is approximate by the law of Y n for n sufficiently large. 4. A simulation study as a general practical approximation methodology Follows a simulation study having as an ultimate purpose to sketch a methodology that allows to use the approximations proposed above. Let us suppose that we are faced with the problem of using a distribution given, for instance, by the one treated in propositions 1 or 2. A natural question is to determine the size of the parameter µ n necessary to use the asymptotic approximation instead of the true distribution. We propose the following protocol for a simulation study in order to determine the order n for which it is safe to use the distribution of Y n instead of the distribution of X n. We present next a simulation study for the asymptotic approximation studied in proposition 2 in the case where X G(u, δ, 0): X n := + 0 t p 1 cos(t)e µnxtq dt and Y n := Γ( p q ) q(µ nx) p q. The simulation protocol implemented in Wolfram s Mathematica was the following. 1. For specific values of p, q, simulate samples of dimension 1000 of a Gamma distributed random variable with parameters u, δ and the correspondent values of random variables X n and Y n. 2. Compare the samples of the random variables X n and Y n by means, for instance, of the Kolmogorov-Smirnov test determining the p-value. 3. Repeat the previous steps for different values of parameters u, p, q studying the p-value as a function of the shape parameter u. In figure 1 we show the variation of the p-value as a function of of µ n for three values of the shape parameter u of the random variable X. It is clear that as the value of u increases the variation becomes steeper. In table 1 we present the size of µ n necessary to achieve a p-value equal to one, with an error less than 10 10, for different choices of the parameters p, q. For such a p-value the distributions of the samples of X n and Y n are impossible to differentiate by means of the Kolmogorov-Smirnov test. In table 2 we present the size of µ n necessary to achieve a p-value equal to 0.01 for different choices of the parameters p, q. For such a p-value the distributions of the samples of X n and Y n are impossible to differentiate, by the Kolmogorov-Smirnov test up to a confidence level of 99%. 447

5 p Value u 2 u 5 u Μ n Figure 1: p-values as function of µ n for 3 values of u (q = 3, p = 1, δ = 2) q = 2 p = p = q = 3 p = p = Table 1: First value of µ n for which the p-value is equal to 1 for u = 2,..., q = 2 p = p = q = 3 p = p = Table 2: First value of µ n for which the p-value is equal to 0.01 for u = 2,..., 10 The simulations above confirm what was to be expected from the theoretical results presented. The method for practical use of the approximations may be so summarized. Given an asymptotic approximation simulate samples of the two distributions varying the parameter µ n and performing, for instance, a Kolmogorov-Smirnov test until the p-value reaches a prescribed value. For values of µ n greater the the one determined it would be safe, up to the confidence level associated with the p-value chosen, to use the asymptotic approximation instead of the original distribution. 5. Conclusion and final remarks In this work we have shown that under mild technical hypothesis it is possible to prove the validity of asymptotic approximations, in the Kolmogorov s distance sense, to distributions of random variables tied by some non trivial almost sure asymptotic relation. 448

6 Other simulation results, not presented in this work, make believe that the validity of these asymptotic approximations is much more comprehensive, thus indicating the need of further studies. References [1] Hoffmann-Jørgensen, J. (1994), Probability with a View Towards Statistics, volume II, Chapman & Hall. [2] Oehlert, G. W. (1992), A Note on the Delta Method, The American Statistician, 46, no. 1,