Edgeworth expansions in small noise asymptotics

Transcription

1 Edgeworth expansions in small noise asymptotics Lan Zhang Per A. Mykland and Yacine Aït-Sahalia First Draft: November Abstract he paper considers Edgeworth expansions for estimators of volatility. Unlike the usual exapsions we have found that in order to obtain meaningful terms one needs to let the size of the noise to go zero asymptotically. his is reflected in our expansions. he results have application to Cornish-Fisher inversion and bootstrapping. KEY WORDS: Bootstrapping; Edgeworth expasion; Measurement error; Subsampling; Market Microstructure; Martingale; Bias-correction; Realized volatility. Lan Zhang is Assistant Professor Department of Statistics Carnegie Mellon University Pittsburgh PA and Assistant Professor Department of Finance University of Illinois at Chicago Chicago IL Per A. Mykland is Professor Department of Statistics he University of Chicago Chicago IL Yacine Aït-Sahalia is Professor Department of Economics and Bendheim Center for Finance Princeton University and NBER Princeton NJ We gratefully acknowledge the support of the National Science Foundation under grants DMS (Zhang and Mykland and SBR (Aït-Sahalia.

2 Edgeworth expansions in small noise 2 1 Introduction In this paper we consider the Edgeworth expansion on the volatility estimator when the price process is noisy. Let {Y ti } 0 = t 0 t 1 t n = be the observed (log price of a security at time t i [0]. Suppose that these observed prices can be decomposed into an underlying (log price process X (the signal and a noise term. hat is at each observation time t i one can write Let the signal (latent process X follows an Itô process Y ti = X ti + ǫ ti. (1 dx t = µ t dt + σ t db t (2 where B t is a standard Brownian motion. ypically µ t the drift coefficient and σt 2 the instantaneous variance of the returns process X t will be (continuous stochastic processes. Let the noise ǫ ti in (1 satisfy the following assumption ǫ ti i.i.d. with Eǫ ti = 0 and V ar(ǫ ti = Eǫ 2. Also ǫ X process (3 where denotes independence between two random quantities. Note that our interest in the noise is only at the observation times t i s so model (1 does not require that ǫ t exists for every t. In Zhang Mykland and Aït-Sahalia (2003 our focus is to construct a statistically sound estimator for integrated volatility 0 σ2 t dt of the true process assuming model (1 and that Y ti s can be observed highly frequently. In search for a final estimator we have touched a sequence of RV estimators which are from the statistically least desiarble to the most desirable: the all estimator [YY ] (all the sparse estimator [YY ] (sparse the optimal sparse estimator [YY ] (sparseopt the averaging estimator [YY ] (avg the optimal averaging estimator [YY ] (avgopt and the final two scale estimator (SRV X X. While the SRV is consistent the first four estimators are biased typically in proportion to the sampling frequency. When one looks at the stochastic terms in all five estimators they should be asymptotically normal. However simulation results show that the distribution of the stochastic term in the sparse estimators and the averaging estimator is far from normality. We argue that the lack of normality is caused by the coexistence of small effective sample size and small noise. In the current paper we provide Edgeworth expansions to the sparse estimators and the averaging estimator. What makes the situation unusual is that the errors ǫ are very small and if they are taken to be of order O p (1 their impact on the Edgeworth expansion may be exaggerated. Consequently the coefficients in the expansion may not accurately reflect which terms are important. o deal with this we here find expansions under the hypothesis that the size of ǫ goes to zero as stated precisely at the beginning of Section 4.

3 Edgeworth expansions in small noise 3 We provide the coefficients both conditional and unconditional in the expansions for [YY ] (sparse and [YY ] (avg. In particular we shall see that not only does the latter have substantially less bias than the former but it is also much closer to a normal distribution cf. the end of Section 4. With the help of Cornish-Fisher expansions our Edgeworth expansions can be used for the purpose of setting intervals that are more accurate than the ones based on the normal distribution see for example Hall (1992. Since our expansions also hold in a triangular array setting they can also be used to analyse the behavior of bootstrapping distributions (for earlier theory on bootstrapping in this setting see Goncalves and Meddahi ( Estimators Our estimators have the following forms. First [YY ] (all uses all the observations [YY ] (all = t i G (Y ti+1 Y ti 2 (4 where G contains all the observation times t i s in [0] 0 = t 0 t 1... t n =. he sparse estimator uses a subsample of the data [YY ] (sparse = (Y tj+ Y tj 2 (5 t j t j+ H where H is a strict subset of G with sample size n sparse n sparse < n. And if t i H then t i+ denotes the following elements in H. he optimal estimator [YY ] (sparseopt has the same form as in (5 except replacing n sparse with n sparse where n sparse is determined by minimizing MSE of the estimator. he averaging estimator maintains a slow sampling scheme while using all the data [YY ] (avg = 1 K K (Y tj+ Y tj 2 (6 t j t j+ G } (k {{} [YY ] (k k=1 where G (k s are disjoint with union G. Let n k be the number of time points in G k n = k n k would then be the average sample size across different grids G k k = 1... K. One can also consider the optimal averaging estimator [YY ] (avgopt by substituting n by n where the latter is selected to balance the bias-variance trade-off in the error of averaging estimator.

4 Edgeworth expansions in small noise 4 A special case of (6 when the sampling points are regularly allocated is of the form [YY ] (avg = 1 K t j t j+k G (Y tj+k Y tj 2 where the sum-squared returns are computed only from subsampling every K-th observation times and then averaged with equal weights. he SRV has the form of XX = [YY ] (avg n [YY ](all (7 n that is the volatility estimator XX combines the sum squared estimators from two different from the returns on a slow time scale whereas [YY ] (all time scales [YY ] (avg a fast time scale. n in (7 is the average sample size across different grids. from the returns on From model (1 the distributions of various estimators can be studied by decomposing the sum-squared returns [YY ] [YY ] = [XX] + 2[Xǫ] + [ǫǫ]. (8 he above decomposion applies to all the estimators in this section with the samples suitably selected. 3 Why Do We Need the Edgeworth Expansions? 3.1 Asymptotical Normality in theory: sparse estimator and averaging estimator sparse estimator For the sparse estimator we have shown in Zhang Mykland and Aït-Sahalia (2003 that [YY ] (sparse L XX + 2n sparse Eǫ 2 + [V ar([ǫǫ] (sparse + 8[XX] (sparse Eǫ σt 4 dt }{{}}{{} n sparse 0 bias due to noise due to noise }{{} due to discretization }{{} total variance where V ar([ǫǫ] (sparse = 4n sparse Eǫ 4 2V ar(ǫ 2 and Z total is standard normal. ] 1/2 Z total If the sample size n sparse is large relative to the noise the variance due to noise in (9 would be dominated by V ar([ǫǫ] (sparse which is of order n sparse Eǫ 4. However at the co-presence of small (9

5 Edgeworth expansions in small noise 5 n sparse and small noise (say Eǫ 2 8[XX] sparse Eǫ 2 is not necessarily smaller than V ar([ǫǫ] (sparse. One then needs to add 8[XX] sparse Eǫ 2 into the approximation. We call this correction smallsample small-error adjustment. his type of adjustment is often useful since the magnitude of the microstructure noise is typically smallish as documented in the empirical literature (cite. Of course n sparse is selected either arbitrarily or in some ad hoc manner. In contrast the sampling frequency in the optimal-sparse estimator [YY ] (sparseopt can be determined by minimizing the MSE of the estimator analytically. Distribution-wise the optimal-sparse estimator has the same form as in (9 but one replaces n sparse by the optimal sampling frequency n sparse where for equidistant observations n sparse = ( ( Eǫ 2 2/3 1/3 σt dt 4. ( is optimal in the sense of minimizing the mean square error of the sparse estimator. n sparse No matter whether n sparse is selected optimally or not one can see from (9 that the de-biased sparse estimator would be asymptotically normal averaging estimator he optimal-sparse estimator only uses a fraction n sparse /n of the data and it faces the arbitrarity of picking the beginning point of the sample. he averaging estimator overcomes both shortcomings. Based on the decomposition (8 analysis in Zhang Mykland and Aït-Sahalia (2003 leads to [YY ] (avg L XX + } 2 neǫ {{ 2 } + [V ar([ǫǫ] (avg + 8 K [XX](avg Eǫ σt 4 bias due to noise }{{} 3 n dt 0 }{{} due to noise due to discretization }{{} where and Z total is a standard normal term. total variance V ar([ǫǫ] (avg = 4 n K Eǫ4 2 K V ar(ǫ2 ] 1/2 (11 Z total For the optimal-averaging estimator [YY ] (avgopt its distribution has the same form as in (11 but substituting n with the optimal sub-sampling size n. o find n one determines K from the bias-variance trade-off in (11 and then set K n/ n. In the equidistantly sampled case ( 1/3 n = 6(Eǫ 2 2 σt 4 dt. (12 0 or in [YY ] (avgopt the next term would follow asymptot- If one removes the bias in [YY ] (avg ically normal.

6 Edgeworth expansions in small noise 6 However the distributions of the de-biased sparse estimators and the de-biased averaging estimators are not exactly normal from the simulation results. 4 Edgeworth Expansions for the Distribution of the Estimators An Edgeworth expansion up to second order can be found separately for each of the components in (8 by first considering expansions for n 1/2 ([ǫǫ] (all 2nEǫ 2 and n 1/2 K([ǫǫ] (avg 2 neǫ 2. Each of these can be represented exactly as a triangular array of martingales. Results deriving such an expansion can be found in Mykland ( ba. It is easily seen that in the current case the expansion takes on the usual Edgeworth form see for example Section 5.3 of McCullagh (1987. Note that with the exception of the term of type n 1/2 K([ǫǫ] (avg 2 neǫ 2 the expansion can also be found from Bickel Götze and van Zwet (1986. We assume that the size of the law of ǫ goes to zero formally that E ǫ p 0 for all p (08]. In particlular say O p (E ǫ 5 = o p (E ǫ Conditional Cumulants We start with the conditional cumulants for [YY ] and [YY ] (avg given the latent process X. All the expressions about [YY ] hold for both [YY ] (all and [YY ] (sparse in the former case n remains to be the total sample size in G while in the latter n is replaced by n sparse. Similar notations apply for [ǫǫ] and for [XX] third-order conditional cumulants Denote where [ǫǫ] = (ǫ t i+1 ǫ ti 2. c 3 (n = cum 3 ([ǫǫ] 2nEǫ 2 (13 From Lemma 1 in the Appendix [ c 3 (n = 8 (n 3 4 cum 3(ǫ 2 7(n 6 7 cum 3(ǫ 2 + 6(n 1 ] 2 var(ǫvar(ǫ2 (14 = O p (ne[ǫ 6 ]

7 Edgeworth expansions in small noise 7 and also because the ǫ s from the different grids are independent cum 3 (K([ǫǫ] (avg 2 neǫ 2 = For the conditional third cumulant of [YY ]: K cum 3 ([ǫǫ] (k 2n k Eǫ 2 = Kc 3 ( n. (15 k=1 cum 3 ([YY ] X = cum 3 ([ǫǫ] + 2[Xǫ] X = cum 3 ([ǫǫ] + 6cum([ǫǫ] [ǫǫ] [Xǫ] X + 12cum([ǫǫ] [Xǫ] [Xǫ] X + 8cum 3 ([Xǫ] X o proceed define and Note that [Xǫ] = n b iǫ ti. a i = { 1 if 1 i n if i = 0n X ti 1 X ti if 1 i n 1 b i = X t if i = n X t0 if i = 0 (16 (17 hen it follows that cum([ǫǫ] [ǫǫ] [Xǫ] X = n b i cum([ǫǫ] [ǫǫ] ǫ ti = (b 0 +b n [2Eǫ 2 Eǫ 3 3Eǫ 5 ] = O p (n 1/2 E[ ǫ 5 ] because cum([ǫǫ] [ǫǫ] ǫ ti = cum([ǫǫ] [ǫǫ] ǫ t1 for i = 1 n 1. Also cum([ǫǫ] [Xǫ] [Xǫ] X n n n = cum(2 b j ǫ tj b k ǫ tk X cum(2 ǫ ti ǫ ti+1 = 2 n a i b 2 i V ar(ǫ 2 4 b i b i+1 (V ar(ǫ 2 = 4[XX] Eǫ 4 + O p (n 1/2 E[ǫ 4 ] n b j ǫ tj n b k ǫ tk X Finally cum 3 ([Xǫ] X = n b 3 i cum 3(ǫ = E(ǫ 3 [ 3 ( X ti 1 2 ( X ti + 3 ( X ti 1 ( X ti 2 ] i=1 = O p (n 1/2 E[ ǫ 3 ] i=1

8 Edgeworth expansions in small noise 8 Summing up cum 3 ([YY ] X = cum 3 ([ǫǫ] + 48[XX]Eǫ 4 + O p (n 1/2 E[ ǫ 3 ] (18 where cum 3 ([ǫǫ] is given in (14. For [YY ] (avg it is now obvious that cum 3 (K[YY ] (avg X = cum 3 (K[ǫǫ] (avg + 48K[XX] (avg Eǫ 4 + O p (K n 1/2 E[ ǫ 3 ] ( fourth-order conditional cumulants For the fourth-order cumulant denote c 4 (n = cum 4 ([ǫǫ] (all 2nEǫ 2. (20 It follows from Lemma 2 in the Appendix that Also here c 4 (n = 16{(n 7 8 cum 4(ǫ 2 + n(eǫ 4 2 3n(Eǫ (n 1var(ǫ 2 Eǫ 4 32(n Eǫ3 cov(ǫ 2 ǫ (n 7 4 Eǫ2 (Eǫ (n 3 4 cum 3(ǫ 2 Eǫ 2 } cum 4 (K([ǫǫ] (avg 2 neǫ 2 = K cum 4 ([ǫǫ] (k 2n k Eǫ 2 = Kc 4 ( n. (21 k=1 For the conditional fourth-order cumulant we know that cum 4 ([YY ] X = cum 4 ([ǫǫ] + 24cum([ǫǫ] [ǫǫ] [Xǫ] [Xǫ] X + 8cum([ǫǫ] [ǫǫ] [ǫǫ] [Xǫ] X + 32cum([ǫǫ] [Xǫ] [Xǫ] [Xǫ] X + 16cum 4 ([Xǫ] X. (22 Similar arguement as in deriving the third cumulant shows that the latter three terms in the right hand side of (22 are of order O p (n 1/2 E[ ǫ 5 ]. For the second term in equation (22 cum([ǫǫ] [ǫǫ] [Xǫ] [Xǫ] X (23 = ij = i b i b j cum([ǫǫ] [ǫǫ] ǫ ti ǫ tj b 2 i cum([ǫǫ] [ǫǫ] ǫ ti ǫ ti + 2 b i b i+1 cum([ǫǫ] [ǫǫ] ǫ ti ǫ ti+1 (24

9 Edgeworth expansions in small noise 9 Note that cum([ǫǫ] [ǫǫ] ǫ ti ǫ ti and cum([ǫǫ] [ǫǫ] ǫ ti ǫ ti+1 are independent of i except close to the edges. One can take α and β to be α = n 1 i β = n 1 i cum([ǫǫ] [ǫǫ] ǫ ti ǫ ti cum([ǫǫ] [ǫǫ] ǫ ti ǫ ti+1. Now following the two identities: cum([ǫǫ] [ǫǫ] ǫ i ǫ i = cum 3 ([ǫǫ] [ǫǫ] ǫ 2 i 2(Cov([ǫǫ] ǫ i 2 V ar([ǫǫ] Eǫ 2 cum([ǫǫ] [ǫǫ] ǫ i ǫ i+1 = cum 3 ([ǫǫ] [ǫǫ] ǫ i ǫ i+1 2Cov([ǫǫ] ǫ i Cov([ǫǫ] ǫ i+1 also observing that that Cov([ǫǫ] ǫ i = Cov([ǫǫ] ǫ i+1 except at the edges 2(α β = n 1 cum 3 ([ǫǫ] 2V ar([ǫǫ] Eǫ 2 + O p (n 1/2 E[ ǫ 6 ] Hence (24 becomes cum 4 ([ǫǫ] [ǫǫ] [Xǫ] [Xǫ] X n n = b 2 iα + 2 b i b i+1 β + O p (n 1/2 E[ ǫ 6 ] = n 1 [XX] cum 3 ([ǫǫ] 2[XX] V ar([ǫǫ] Eǫ 2 + O p (n 1/2 E[ ǫ 6 ] where the last line is because n b 2 i = 2[XX] + O p (n 1/2 herefore in the final analysis n b i b i+1 = [XX] + O p (n 1/2. cum 4 ([YY ] X = cum 4 ([ǫǫ] + 24[XX] n 1 cum 3 ([ǫǫ] 48[XX] V ar([ǫǫ] Eǫ 2 For the average estimator + O p (n 1/2 E[ ǫ 5 ] (25 cum 4 (K[YY ] (avg X = cum 4 (K[ǫǫ] (avg + 24K[XX] (avg c 3 ( n n 48Eǫ2 k [XX] (k V ar([ǫǫ](k + O p(k n 1/2 E[ ǫ 5 ] 4.2 Unconditional Cumulants o pass to the unconditional third cumulant we use the general formulas (Brillinger (1969 Speed (1983 see also Chapter 2 in McCullagh (1987:

10 Edgeworth expansions in small noise 10 cum 3 (A = E[cum 3 (A F] + 3Cov[V ar(a FE(A F] + cum 3 [E(A F] cum 4 (A = E[cum 4 (A F] + 4Cov[cum 3 (A FE(A F] + 3V ar[v ar(a F] + 6cum 3 (V ar(a FE(A FE(A F + cum 4 (E(A F unconditional cumulants for sparse estimators Recall that in Zhang Mykland and Aït-Sahalia (2003 we have derived that also E([YY ] X process = [XX] + 2nEǫ 2 (26 V ar([yy ] X = 4nEǫ 4 2V ar(ǫ 2 +8[XX] }{{} Eǫ 2 + O p (E ǫ 2 n 1/2 (27 V ar([ǫǫ] Put together the third and fourth conditional cumulants with the results from (26-(27 and we obtain the unconditional cumulants as in the following: and cum 3 ([YY ] XX = c 3 (n + 48E(ǫ 4 E[XX] + 24V ar(ǫcov([xx] [XX] XX (28 + cum 3 ([XX] XX + O(n 1/2 E[ ǫ 3 ] cum 4 ([YY ] XX = c 4 (n n c 3(nE[XX] 48E[XX] Eǫ 2 V ar([ǫǫ] + 192Eǫ 4 Cov([XX] [XX] XX + 192(V ar(ǫ 2 V ar([xx] + 48V ar(ǫcum 3 ([XX] [XX] XX [XX] XX (29 + cum 4 ([XX] XX + O(n 1/2 E[ ǫ 5 ] Example 1 (constant σ and equidistant case. Suppose t i = t = /n (the equidistant case and when σ t = σ is a constant XX is a constant which does not contribute to either of the above cumulants. Also [XX] has distribution σ 2 tχ 2 n (χ2 with n degrees of freedom so that cum p ([XX] = σ 2p ( t p n cum p (χ 2 1 = n (p 1 (σ 2 p cum p (χ 2 1 ; recall that p = 1 p = 2 p = 3 p = 4 cum p (χ

11 Edgeworth expansions in small noise 11 It follows that in this case cum 3 ([YY ] XX = c 3 (n + 48E(ǫ 4 (σ V ar(ǫn 1 (σ 2 2 (30 + 8n 2 (σ O(n 1/2 E[ ǫ 3 ] Similarly for the fourth cumulant cum 4 ([YY ] XX = c 4 (n n c 3(n(σ 2 48(σ 2 Eǫ 2 V ar([ǫǫ] + 384(Eǫ 4 + V ar(ǫ 2 n 1 (σ V ar(ǫn 2 (σ 2 3 ( n 3 (σ O(n 1/2 E[ ǫ 5 ] It is obvious that one needs ǫ n = o p (n 1/2 to keep all the terms in (30 and (31 non-neglegible. In the case of optimal-sparse estimator (10 lends to ǫ = O p (n 3/4 in particular ǫ = o p (n 1/2. Hence the expression works in this case and also for many suboptimal choices of n. For the special case of constant σ and equidistant sampling times the optimal sampling size is ( n sparse = σ 2 2/3 2 Eǫ 2. (32 Plug (32 into (30 and (31 for the choice of n then and respectively. cum 3 ([YY ] (sparseopt XX cum 4 ([YY ] (sparseopt XX Note that under the optimal sampling V ar([yy ] (sparseopt = E = 48(σ (Eǫ (σ (2Eǫ O(E ǫ 11 3 (33 1 = 192(σ Eǫ 4 ( Eǫ (Eǫ 4 + V ar(ǫ 2 (σ 2 4 3(2Eǫ (σ (Eǫ (σ 2 2 (2Eǫ O(E ǫ 17 3 (34 ( V ar([yy ] (sparseopt X = 8 < XX > Eǫ n (σ2 2 + o p (Eǫ 2 = 2(σ (2Eǫ O p (Eǫ 2 ( + V ar E([YY ] (sparseopt X

12 Edgeworth expansions in small noise 12 hence ( cum 3 (Eǫ 2 1/3 ([YY ] (sparseopt XX ( cum 4 (Eǫ 2 1/3 ([YY ] (sparseopt XX = O((E ǫ 2/3 = O((E ǫ 4/3 in other words the third-order and the fourth-order cumulants indeed vanish as n and Eǫ unconditional cumulants for averaging estimator Similarly for the averaging estimators E([YY ] (avg X process = [XX] (avg + 2 neǫ 2 (35 V ar([yy ] (avg X = V ar([ǫǫ] (avg + 8 K [XX](avg Eǫ 2 + O p (E[ ǫ 2 (nk 1/2 ] (36 with V ar([ǫǫ] (avg = 4 n K Eǫ4 2 K V ar(ǫ2. Invoking the relations between the conditional and the unconditional cumulants one gets the unconditional cumulants for the average estimator: and cum 3 ([YY ] (avg XX = 1 K 2c 3( n K 2E(ǫ4 E[XX] (avg K V ar(ǫcov([xx](avg [XX] (avg XX (37 + cum 3 ([XX] (avg XX + O(K 2 n 1/2 E[ ǫ 3 ] cum 4 ([YY ] XX = 1 K 3c 4( n c 3 ( n K 3 n E[XX](avg K 2 Eǫ4 Cov([XX] (avg [XX] (avg XX K 2 (V ar(ǫ2 V ar([xx] (avg K 4E[XX](avg Eǫ 2 V ar([ǫǫ] (all K V ar(ǫcum 3([XX] (avg [XX] (avg XX [XX] (avg XX + cum 4 ([XX] (avg XX + O(K 3 n 1/2 E[ ǫ 5 ] (38 In the special case where σ t is constant and t i = [XX] (avg [XX] (all namely σ 2 tχ 2 n. has the same distribution as

13 Edgeworth expansions in small noise 13 hus cum 3 ([YY ] (avg XX = 24 1 K V ar(ǫ2( nk 1 (σ 2 2 (39 + 8( nk 2 (σ O(K 2 n 1/2 E[ ǫ 3 ] and cum 4 ([YY ] (avg XX = 48 1 K 4(σ2 Eǫ 2 (4Eǫ 4 nk K 2(Eǫ4 + V ar(ǫ 2 2( nk 1 (σ K V ar(ǫ8( nk 2 (σ 2 3 ( ( nk 3 (σ O(K 3 n 1/2 E[ ǫ 5 ] Also the optimal average subsampling size for the constant σ is ( σ n 4 2 1/3 = 6(Eǫ 2 2. Also the unconditional variance of the averaging estimator under the optimal sampling V ar([yy ] (avgopt = 8 K Eǫ2 (σ 2 + o(eǫ 2 K 1 + (σ 2 2 2( n K 1 }{{}}{{} E V ar([yy ] (avgopt X V ar = 2 K 61 3 (Eǫ (σ O(Eǫ 2 K 1 E([YY ] (avgopt X hence ( cum 3 (Eǫ 2 1/3 K 1/2 ([YY ] (avgopt XX ( cum 4 (Eǫ 2 1/3 K 1/2 ([YY ] (avgopt XX = O((E ǫ 2/3 K 1/2 0 = O((E ǫ 4/3 K 1 0 as n and Eǫ 2 0. By comparing to the expression for the sparse case it is clear that the average volatility is substantially closer to normal that the sparsely sampled volatility. 5 Appendix: Proofs of c 3 (n and c 4 (n Lemma 1. c 3 (n = 8 [ (n 3 4 cum 3(ǫ 2 7(n 6 7 cum 3(ǫ 2 + 6(n 1 ] 2 var(ǫvar(ǫ2

14 Edgeworth expansions in small noise 14 where Proof of Lemma 1: Let a i be defined as in (16. One can then write cum( c 3 (n = cum 3 (2 n = 8[cum 3 ( n n n +3cum( a i (ǫ 2 t i Eǫ 2 2 ǫ ti ǫ ti+1 cum 3 ( ǫ ti ǫ ti+1 3cum( n n n a j ǫ 2 t j ǫ tk ǫ tk+1 ǫ tj ǫ tj+1 ǫ tk ǫ tk+1 ] (41 a j ǫ 2 t j ǫ tk ǫ tk+1 = 2 a k a k+1 cum(ǫ 2 t k ǫ 2 t k+1 ǫ tk ǫ tk+1 = 2(n 1(Eǫ 3 2 (42 since a ka k+1 = n 1 and the summation is non-zero only when (i = kj = k + 1 or (i = k + 1j = k. cum( Also n ǫ tj ǫ tj+1 ǫ tk ǫ tk+1 = 2 a j cum(ǫ 2 t j ǫ tj ǫ tj+1 ǫ tj ǫ tj+1 = 2(n 1 2 (Eǫ2 V ar(ǫ 2 since a j = n 1 2 and the summation is non-zero only when j = k = (i or i 1. And finally cum( cum( ǫ ti ǫ ti+1 n with n a3 i = n 3 4. n a j ǫ 2 t j ǫ tj ǫ tj+1 n a k ǫ 2 t k = Inserting (42-(45 in (41 yields (14. Lemma 2. (43 ǫ tk ǫ tk+1 = cum 3 (ǫ ti ǫ ti+1 = n(eǫ 3 2 (44 n a 3 icum 3 (ǫ 2 t i = (n 3 4 cum 3(ǫ 2 (45 c 4 (n = 16{(n 7 8 cum 4(ǫ 2 + n(eǫ 4 2 3n(Eǫ (n 1var(ǫ 2 Eǫ 4 32(n Eǫ3 cov(ǫ 2 ǫ (n 7 4 Eǫ2 (Eǫ (n 3 4 cum 3(ǫ 2 Eǫ 2 }

15 Edgeworth expansions in small noise 15 Proof of Lemma 2: = cum( n n j= ǫ tj ǫ tj+1 a i [1 {l=j} 1 {i=jor j+1} + ǫ tk ǫ tk+1 l=0 ǫ tl ǫ tl+1 ( 3 (1 2 {l=j+1i=j+2} + 1 {l=i=j 1} ]cum(ǫ 2 t i ǫ tj ǫ tj+1 ǫ tk ǫ tk+1 ǫ tl ǫ tl+1 = 2(n 1 2 Eǫ3 cov(ǫ 2 ǫ 3 + 6(n 3 2 (Eǫ3 2 Eǫ 2 (46 = cum( n n n l=0 +1 {k=l(ij=(kk+1[2]} a j ǫ 2 t j ǫ tk ǫ tk+1 ǫ tl ǫ tl+1 l=0 n a i a j [1 {i=jk=li=(k+1 or k} + (1 {l=k 1(ij=(k+1k 1[2]} + 1 {l=k+1(ij=(kk+2[2]} = 2(n 3 4 cum 3(ǫ 2 Eǫ 2 + 4(n 2(Eǫ 3 2 Eǫ 2 + 2(n 1(V ar(ǫ 2 2 (47 where the notation (ij = (k + 1k 1[2] means that (i = k + 1j = k 1 or (j = k + 1i = k 1. he last equation above holds because n i=1 a2 i = n 3 4 i=1 a i 1a i+1 = n 2 and a ia i+1 = n 1. = cum( n n l=0 n a j ǫ 2 t j n a k ǫ 2 t k ǫ tl ǫ tl+1 l=0 n ( 3 a i a j a k [1 2 {i=j=lk=l+1} + 1 {i=j=l+1k=l} ]cum(ǫ 2 t i ǫ 2 t j ǫ 2 t k ǫ tl ǫ tl+1 = 6 a 2 i a i+1cum(ǫ 2 ǫ 2 ǫeǫ 3 = 6(n 5 4 cum(ǫ2 ǫ 2 ǫeǫ 3 (48 since a2 i a i+1 = n 5 4.

16 Edgeworth expansions in small noise 16 = cum 4 ( ǫ ti ǫ ti+1 [1 {i=j=k=l} + l=0 ( {i=jk=li=(k+1k 1} ]cum(ǫ ti ǫ ti+1 ǫ tj ǫ tj+1 ǫ tk ǫ tk+1 ǫ tl ǫ tl+1 = n((eǫ 4 2 3(Eǫ (n 1(Eǫ 2 2 V ar(ǫ 2 (49 n cum 4 ( = n a 4 i cum 4(ǫ 2 = (n 7 8 cum 4(ǫ 2 (50 Putting together (46-(50: c 4 (n = cum 4 (2 = 16[cum 4 ( n 2 ǫ ti ǫ ti+1 n ( 4 n cum( 3 + cum 4 ( ( 4 n + cum( a i ǫ 2 t 2 i n ǫ ti ǫ ti+1 ǫ tj ǫ tj+1 ( 4 cum( 1 ǫ tk ǫ tk+1 l=0 n ǫ tl ǫ tl+1 a j ǫ 2 t j ǫ tk ǫ tk+1 ǫ tl ǫ tl+1 ] n a j ǫ 2 t j (46 (50 = 16{(n 7 8 cum 4(ǫ 2 + n(eǫ 4 2 3n(Eǫ (n 1var(ǫ 2 Eǫ 4 l=0 n a k ǫ 2 t k ǫ tl ǫ tl+1 32(n Eǫ3 cov(ǫ 2 ǫ (n 7 4 Eǫ2 (Eǫ (n 3 4 cum 3(ǫ 2 Eǫ 2 } (51 since cov(ǫ 2 ǫ 3 = Eǫ 5 Eǫ 2 Eǫ 3 and cum(ǫ 2 ǫ 2 ǫ = Eǫ 5 2Eǫ 2 Eǫ 3. l=0 REFERENCES Bickel P. J. Götze F. and van Zwet W. R. (1986 he Edgeworth Expansion for U-statistics of Degree wo he Annals of Statistics Brillinger D. R. (1969 he Calculation of Cumulants via Conditioning Annals of the Institute of Statistical Mathematics

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