Statistics and Probability Letters 79 (2009) 807–813
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A note on the consistency of a robust estimator for threshold autoregressive processes Li-Xin Zhang a , Wai-Sum Chan b,∗ , Siu-Hung Cheung c,d , King-Chi Hung c a
Department of Mathematics, Zhejiang University, Hangzhou 310028, PR China
b
Department of Finance, The Chinese University of Hong Kong, PR China
c
Department of Statistics, The Chinese University of Hong Kong, PR China
d
Department of Statistics, National Cheng Kung University, Tainan, Taiwan
article
a b s t r a c t
info
Article history: Received 23 June 2008 Received in revised form 28 October 2008 Accepted 30 October 2008 Available online 13 November 2008
The method of conditional least squares is commonly used for estimating threshold autoregressive parameters, and its consistency was derived by Chan [Chan, K.S., 1993. Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Annals of Statistics 21, 520–533]. In this note we consider a general class of robust estimators for threshold autoregressive models, and under some regularity conditions and a proper choice of the weight function, the consistency is demonstrated. © 2008 Elsevier B.V. All rights reserved.
1. Introduction The class of self-exciting threshold autoregressive (SETAR) models (Tong, 1978, 1983) has been widely applied to explain various empirical phenomena in many disciplines, ranging from describing the dynamics of a sheep population (Coulson et al., 2001) to pricing of financial options (Siu et al., 2006). If a time series Yt follows the SETAR(k; p, d) model for positive integers d ≤ p, then (j)
Yt = φ0 +
p X
φl(j) Yt −l + σj et ,
if rj−1 ≤ Yt −d < rj ,
(1.1)
l=1
where j = 1, 2, . . . , k, σj2 < ∞, and et is a white noise process with mean zero and unit variance. Parameters p and d denote the autoregressive (AR) order and the delay parameter, respectively. Each regime is divided by the threshold parameters, rj ’s, such that −∞ = r0 < r1 < · · · < rk = ∞. Among these k regimes, each of them possess different AR(p) models. (j)
(s)
If φl = φl for all l = 0, 1, . . . , p and j 6= s = 1, 2, . . . , k; then the model reduces to a linear AR(p) process. A more generalized version of the SETAR model allows different orders of the AR models inside different regimes. In this note we only consider a two-regime SETAR(2; p, d) model,
( φ0(1) + φ1(1) Yt −1 + · · · + φp(1) Yt −p + σ1 et , Yt = φ0(2) + φ1(2) Yt −1 + · · · + φp(2) Yt −p + σ2 et , (1)
where p is known, d ≤ p, and φl
if Yt −d ≤ r , if Yt −d > r ,
(1.2)
6= φl(2) for some l = 0, 1, . . . , p.
∗
Corresponding author. Tel.: +852 2609 7715; fax: +852 2603 6586. E-mail addresses:
[email protected] (L.-X. Zhang),
[email protected] (W.-S. Chan),
[email protected] (S.-H. Cheung),
[email protected] (K.-C. Hung). 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.10.036
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L.-X. Zhang et al. / Statistics and Probability Letters 79 (2009) 807–813
2. Robust estimation The SETAR model can exhibit jumps, limit cycle, chaos and harmonic distortion. On the other hand, the observed SETAR time series might be contaminated by outliers. See, for example, Tsay (1988), Hau and Tong (1989) and Franses and van Dijk (2000, p. 97). Analysts may face the dilemma of classifying aberrant observations as outliers or part of the genuine nonlinearity structure. Denby and Martin (1979) showed that the least squares (LS) estimator of linear AR processes not only lack robustness in terms of variability but also suffer from a severe bias problem when the observations are contaminated by additive outliers. In a simulation study, Chan and Cheung (1994) concluded that additive outliers have similar adverse effects on the least squares estimation of SETAR models, and proposed a generalized-M (GM) approach to outlier robust estimation of threshold autoregressive processes. Given observations {Yt : t = 1, . . . , N } generated from (1.2) and the threshold value r, the data can be segmented into two arranged autoregressions (Petruccelli and Davies, 1986). The GM algorithm under the Schweppe-type regression (Handschin et al., 1975) can be applied to estimating the autoregressive parameters in each regime robustly. However, in (j) (j) (j) practice, r is seldom known and needs to be estimated. Let Xt = (1, Yt −1 , . . . , Yt −p )T , Φj = (φ0 , . . . , φp )T , t = σj et , and (1)
(2)
j = 1, 2. Then (1.2) can be expressed in the matrix form, Yt = (Φ1T Xt + t )I{Yt −d ≤ r } + (Φ2T Xt + t )I{Yt −d > r }. We write Φ = (Φ1T , Φ2T )T , β = (βT1 , βT2 )T , θ 0 = (Φ T , r , d)T , and θ = (βT , z , q)T ; where θ is a general parameter and θ 0 is the true
ˆ x, y) = |arctan(x) − arctan(y)|. The parameter space Θ parameter. Let R¯ = R ∪ {−∞, ∞} be equipped with the metric δ( is R 2p+2 × R¯ × {1, 2, . . . , p} equipped with the product metric. Define ρN (θ) =
N X
ρ(Yt − Eθ (Yt |Ft −1 )) = ρ1N (θ) + ρ2N (θ)
t =p
where
ρ1N (θ) =
N X
ρ(Yt − βT1 Xt )I{Yt −q ≤ z },
t =p
ρ2N (θ) =
N X
ρ(Yt − βT2 Xt )I{Yt −q > z },
t =p
Ft −1 is the information set up to time t − 1 (i.e., Ft −1 = σ (Xi : i ≤ t − 1)), and ρ is an objective function for M-estimation. The model parameter in (1.2) can be robustly estimated by
θˆ = θˆ N = arg min ρN (θ),
(2.1)
θ∈Θ
when r is unknown. 3. Consistency (j)
In the case when r is fixed, it is well known that robust estimators of φl ’s in each regime are consistent (e.g., see Bustos (1982)). In this section we demonstrate the consistency of the robust estimator in (2.1) when r is unknown. Condition 3.1. E[ρ(σj et + c )] > E[ρ(σj et )] if c 6= 0, j = 1, 2. Condition 3.2. E[ρ(σj et + uT Xt )] < ∞ for all u ∈ R p+1 , j = 1, 2, and pr(uT Xt 6= 0) > 0 for all u 6= 0. 0 < pr(Yt ≤ r ) < 1. Condition 3.3. ρ(·) is a convex non-negative function. It should be noted that Condition 3.2 implies that model (1.2) is identifiable and non-singular. Condition 3.3 ensures that (2.1) has a solution. Conditions 3.1–3.3 entail that the true value θ 0 = (Φ T , r , d)T is the unique parameter which minimizes E[ρN (θ)]. Theorem 3.1. Suppose {Xt } is stationary ergodic. Under Conditions 3.1–3.3,
θˆ N → θ 0 a.s.
(3.1)
The proof of Theorem 3.1 is given in the Appendix. 4. Remarks For the special case of squared loss ρ -function,Chan (1993) proved that rˆN , the conditional least squares estimator of r, is strongly consistent. Qian (1998) derived the similar results for the maximum likelihood estimators of the SETAR process under some regularity conditions on the error density, not necessarily Gaussian.
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Koul et al. (2003) proposed a class of M-estimators for the two-phase linear regression model, which is similar to the SETAR model (1.2), with p = 1, and obtained the strong consistency and asymptotic distribution of b θN under the condition that the parameter space of Φ is compact and some conditions on the derivative of the objective function ρ . It should be noted that Conditions 3.1–3.3 in this paper are weaker than those conditions (a.1)–(a.3) in Koul et al. (2003, p.127). Furthermore, we do not assume the compactness of the parameter space in this paper. Giordani (2006) cautioned that Chan and Cheung’s (1994) GM estimator of threshold models could lead to inconsistent estimates of the threshold value r, even when the model is correctly specified and the disturbances are normally distributed. It is known that the objective ρ -function can be chosen to provide the estimator desirable properties (in terms of bias and efficiency) when the data are truly from the assumed distribution or close to the assumed distribution. In this note, we require the objective function ρ be a convex non-negative function. Then, the consistency of the GM estimator is guaranteed by Theorem 3.1. For example, Huber ρ -function is a possible choice to replace the redescending weight function of Lucas et al. (1996) which was used in Giordani’s (2006) study. The GM estimator is robust against outliers since outlying observations are more likely to receive less weight and therefore to only marginally affect the estimates of Φj and fit of the remaining observations. However, if r is not fixed, the complexity in estimation will increase exponentially. In a simulation study, Kapetanios (2000) showed that despite the superconsistency of the least squares threshold parameter estimates, the estimator performs poorly in small sample size situations. This problem also applies to the robust estimator of the threshold parameter in this note. Therefore, it is still a challenge to come up with a threshold estimate with desirable finite-sample properties. Acknowledgements The authors thank the editor and two anonymous referees for their thoughtful and helpful comments on the earlier drafts of this paper. Appendix. Proof of Theorem 3.1 Let Rt (θ) =: ρ(Yt − Eθ (Yt |Ft −1 )). The proof will be completed if we have shown the following two lemmas. Only Lemma A.1 is needed if the parameter space of Φ is assumed to be compact and then the main idea of the proof is similar to that of Theorem 3.1 of Koul et al. (2003). Lemma A.1. For any M > δ > 0, lim inf inf ρN (θ)/N > E[Rt (θ 0 )] = E[ρ(σ1 et )I{Yt −d ≤ r }] + E[ρ(σ2 et )]I{Yt −d > r } a.s., N →∞ θ∈Θδ,M
(A.1)
where Θδ,M = {θ : kθ − θ 0 k ≥ δ, kβ − Φ k ≤ M }. Lemma A.2. For any M > kΦ2 − Φ1 k, lim inf
inf
N →∞ {θ:kβ−Φ k≥4M }
ρN (θ)/N > E[Rt (θ 0 )] a.s.
(A.2)
In fact, notice that ρN (θ 0 )/N → E[Rt (θ 0 )] a.s. as N → ∞ by the ergodicity. Combining Lemmas A.1 and A.2 yields that for any δ > 0, lim inf
inf
N →∞ kθ−θ 0 k≥δ
(ρN (θ)/N − ρN (θ 0 )/N ) > 0 a.s.
which implies that with probability one kθˆ N − θ 0 k < δ if N is large enough by the definition of θˆ . This completes the proof. For proving Lemmas A.1 and A.2, we need two more lemmas. We first list several relevant properties of convex functions which will be used in the proofs. (P1) If f (u), u ∈ R , is a convex function, then sup f (u0 + u) ≤ max{f (u0 + δ), f (u0 + δ)}. |u|≤δ
(P2) If f (u), u ∈ R p , is a convex function with infkuk=δ (f (u) − f (0)) ≥ 0, then inf (f (u) − f (0)) ≥ inf (f (u) − f (0)) .
kuk≥δ
kuk=δ
(P3) If a sequence {fn (u); n ≥ 1} of convex functions on R p converges to a function f (u), then it will converge to f (u) uniformly on every bounded convex set. In particular, sup |fn (u) − f (u)| → 0. kuk≤δ
Property (P3) is a direct corollary of Theorem 10.8 of Rockafellar (1970, p.90). Property (P1) follows from the fact that for any |u| ≤ δ ,
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δ+u δ−u (u0 + δ) + (u0 − δ) 2δ 2δ δ+u δ−u ≤ f (u0 + δ) + f (u0 − δ) 2δ 2δ due to the convexity of f . For kuk ≥ δ , we let uδ = kuδ k u. Property (P3) follows from the facts that kuδ k = δ and δ δ 0 ≤ f (uδ ) − f (0) = f u+ 1− 0 − f (0) kuk ku k δ δ ≤ f (u) + 1 − f (0) − f (0) ku k kuk δ = (f (u) − f (0)) ≤ f (u) − f (0), ku k f (u0 + u) = f
where the second inequality is due to the convexity of f . Now, write Rt (θ) = R1t (θ) + R2t (θ) + R3t (θ) + R4t (θ), where R1t (θ) = ρ(σ1 et − (β1 − Φ1 )T Xt )I{Yt −q ≤ z , Yt −d ≤ r }, R2t (θ) = ρ(σ2 et − (β1 − Φ2 )T Xt )I{Yt −q ≤ z , Yt −d > r }, R3t (θ) = ρ(σ1 et − (β2 − Φ1 )T Xt )I{Yt −q > z , Yt −d ≤ r }, R4t (θ) = ρ(σ2 et − (β2 − Φ2 )T Xt )I{Yt −q > z , Yt −d > r }. Lemma A.3. Let U = U (θ, δ) denote an open neighborhood of θ . Then for each θ , ∀i = 1, 2, 3, 4,
E
sup |Rit (θ ∗ ) − Rit (θ)|
θ ∗ ∈U
→ 0 as U shrinks towards θ
(A.3)
and consequently,
E
sup |Rt (θ ) − Rt (θ)| ∗
→ 0 as U shrinks towards θ .
θ ∗ ∈U
Proof of Lemma A.3. We only consider R1t . Write θ ∗ = (β∗T , z ∗ , q∗ )T . Suppose δ > 0 is small enough. If z ∈ R , θ ∗ ∈ U (θ, δ), then q∗ = q and
|R1t (θ ∗ ) − R1t (θ)| ≤ |ρ(σ1 et − (β∗1 − Φ1 )T Xt ) − ρ(σ1 et − (β1 − Φ1 )T Xt )| + ρ(σ1 et − (β1 − Φ1 )T Xt )|I{Yt −q ≤ z , Yt −d ≤ r } − I{Yt −q ≤ z ∗ , Yt −d ≤ r }| ≤ sup |ρ(σ1 et − (β1 − Φ1 )T Xt + bT Xt ) − ρ(σ1 et − (β1 − Φ1 )T Xt )| kbk≤δ
+ ρ(σ1 et − (β1 − Φ1 )T Xt )|I{|Yt −q − z | ≤ |z ∗ − z |}| ≤ |ρ(σ1 et − (β1 − Φ1 )T Xt ± δ|Xt |) − ρ(σ1 et − (β1 − Φ1 )T Xt )| + ρ(σ1 et − (β1 − Φ1 )T Xt )|I{|Yt −q − z | ≤ |z ∗ − z |}| by Property (P1) of the convex function. It follows that
E
sup |R1t (θ ) − R1t (θ)| ∗
θ ∗ ∈U
→ 0 as U shrinks towards θ
by the continuity of ρ and Condition 3.2. When z = ±∞, the proof is similar.
Lemma A.4. Let f ≥ 0 be a bounded function and pr(f (Xt ) > 0) > 0. For any given δ > 0, let γi = infkuk=δ E[ρ(σi et + uT Xt )f (Xt )], i = 1, 2. Then for i = 1, 2, γi > E[ρ(σi et )f (Xt )] and lim inf N →∞
lim inf N →∞
1
inf
N X
N kuk≥δ t =p 1 N
inf u
N X
ρ(σi et + uT Xt )f (Xt ) ≥ γi a.s.,
ρ(σi et + uT Xt )f (Xt ) ≥ E[ρ(σi et )f (Xt )] a.s.
t =p
Further, (A.4) and (A.5) holds obviously when f (Xt ) = 0 a.s.
(A.4)
(A.5)
L.-X. Zhang et al. / Statistics and Probability Letters 79 (2009) 807–813
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Proof of Lemma A.4. Without loss of generality, assume that σ1 = σ2 = 1 and write γ = γ1 = γ2 . By Conditions 3.1 and 3.2, it is easily seen that
E[ρ(et + uT Xt )f (Xt )] > E[ρ(et )f (Xt )]. By the continuity of E[ρ(et + uT Xt )f (Xt )] and the compactness of the set {u : kuk = δ}, we conclude that γ = infkuk=δ E[ρ(et + uT Xt )f (Xt )] > E[ρ(et )f (Xt )]. Denote η = γ − E[ρ(et )f (Xt )]. Notice that for every u, N 1 X
[ρ(et + uT Xt ) − ρ(et )]f (Xt ) → E[(ρ(et + uT Xt ) − ρ(et ))f (Xt )]
N t =p
a.s.,
by the ergodicity. Noticing the convexity, we have
N 1 X T T sup [ρ(et + u Xt ) − ρ(et )]f (Xt ) − E[(ρ(et + u Xt ) − ρ(et ))f (Xt )] → 0 a.s. kuk≤δ N t =p by Property (P3) of the convex function. So, for any 0 < < η, with probability one there is a N large enough such that 1
N X [ρ(et + uT Xt ) − ρ(et )]f (Xt ) > η − ,
inf
N kuk=δ t =p
which, together with Property (P2) of the convex function, implies that 1
N X [ρ(et + uT Xt ) − ρ(et )]f (Xt ) > η − .
inf
N kuk≥δ t =p It follows that lim inf N →∞
1
inf
N X
N kuk≥δ t =p
ρ(et + uT Xt )f (Xt ) = lim inf N →∞
1
inf
N N X 1 X [ρ(et + uT Xt ) − ρ(et )]f (Xt ) + lim ρ(et )f (Xt )
N kuk≥δ t =p
N →∞
N t =p
≥ η − + E[ρ(et )f (Xt )] = γ − a.s. and then (A.4) is proved. On the other hand, lim inf N →∞
1
inf
N X
N kuk≤δ t =p
ρ(et + uT Xt )f (Xt ) ≥ lim inf N →∞
=E which, together with (A.4), implies (A.5).
N 1 X
inf ρ(et + uT Xt )f (Xt ) N t =p kuk≤δ
inf ρ(et + uT Xt )f (Xt ) → E[ρ(et )f (Xt )]
kuk≤δ
a.s. as δ → 0,
Now we tend to the proof of Lemmas A.1 and A.2. Proof of Lemma A.1. Without loss of generality, we can assume that σ1 = σ2 = 1. It follows from Condition 3.1 that
E[R1t (θ)] ≥ E[ρ(et )I{Yt −q ≤ z , Yt −d ≤ r }] and the equality holds only when β1 = Φ1 or pr(Yt −q ≤ z , Yt −d ≤ r ) = 0. For R2t , R3t and R4t we have similar estimates. It follows that
E[Rt (θ)] > E[ρ(et )] if θ 6= θ 0 . According to Lemma A.3, E[Rt (θ)] is a continuous function of θ . Notice that Θδ,M is a compact set. We conclude that
η =: inf E[Rt (θ)] − E[ρ(et )] > 0. θ∈Θδ,M
By Lemma A.3, for every θ ∈ Θδ,M , there is an open neighborhood U (θ) of θ such that
E
inf Rt (θ ∗ ) ≥ E[Rt (θ)] − η/2 ≥ E[ρ(et )] + η/2.
θ ∗ ∈U (θ)
Hence, there are open sets Ui , i = 1, 2, . . . , m, such that m [ i =1
Ui ⊃ Θδ,M
and E
inf Rt (θ) ≥ E[ρ(et )] + η/2.
θ∈Ui
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L.-X. Zhang et al. / Statistics and Probability Letters 79 (2009) 807–813
It follows that
( lim inf inf ρN (θ)/N ≥ min N →∞ θ∈Θδ,M
i
)
N 1 X
lim
N →∞
inf Rt (θ) N t =p θ∈Ui
≥ min E inf Rt (θ) ≥ E[ρ(et )] + η/2 a.s., i
θ∈Ui
by the ergodicity. The proof is now completed.
Proof of Lemma A.2. We will show that lim inf
inf
N →∞ {θ:kβ1 −Φ1 k≥2M }
ρN (θ)/N > E[Rt (θ 0 )] a.s.
(A.6)
For each (q∗ , z ∗ ), we denote U ∗ (δ) be its δ neighborhood. Notice the fact that R¯ × {1, 2, . . . , p} is compact. Expression (A.6) is true if we had proved that for each (q∗ , z ∗ ), there is a constant γ ∗ > 0 and an δ ∗ > 0 small enough such that lim inf
inf
inf
N →∞ (q,z )∈U ∗ (δ ∗ ) kβ1 −Φ1 k≥2M
ρN (θ)/N > E[Rt (θ 0 )] + γ ∗ a.s.
(A.7)
Without loss of generality, we assume σ1 = σ2 = 1. Write ϕ = kΦ1 − Φ2 k/2, and A1 = inf E[ρ(et + uT Xt )I{Yt −q∗ ≤ z ∗ , Yt −d ≤ r }], kuk=ϕ
B1 = E[ρ(et )I{Yt −q∗ ≤ z ∗ , Yt −d ≤ r }], and define Ai (and Bi ), i = 2, 3, 4, similarly by replacing the term I{Yt −q∗ ≤ z ∗ , Yt −d ≤ r } by I{Yt −q∗ ≤ z ∗ , Yt −d > r }, I{Yt −q∗ > z ∗ , Yt −d ≤ r } and I{Yt −q∗ > z ∗ , Yt −d > r }, respectively. It is obvious that Ai ≥ Bi by Condition 3.1, i = 1, 2, 3, 4. Noticing that pr(Yt −q∗ ≤ z ∗ , Yt −d ≤ r ) and pr(Yt −q∗ > z ∗ , Yt −d ≤ r ) will not be zero simultaneously, we have A1 + A3 > B1 + B3 by Lemma A.4. And similarly, A2 + A4 > B2 + B4 . It follows that C = b min {A1 + A2 + A3 + B4 , A1 + A2 + B3 + A4 }
> B1 + B2 + B3 + B4 = E[Rt (θ 0 )].
Write U ∗ = U ∗ (δ ∗ ). Let f1 (Xt ) = f2 (Xt ) = f3 (Xt ) = f4 (Xt ) =
inf
I{Yt −q ≤ z , Yt −d ≤ r },
inf
I{Yt −q ≤ z , Yt −d > r },
inf
I{Yt −q > z , Yt −d ≤ r },
inf
I{Yt −q > z , Yt −d > r }.
(q,z )∈U ∗ (q,z )∈U ∗ (q,z )∈U ∗ (q,z )∈U ∗
Then for (q, z ) ∈ U ∗ , we have
ρN (θ) ≥
N X
ρ(et − (β1 − Φ1 )T Xt )f1 (Xt ) +
t =p
+
N X
ρ(et − (β1 − Φ2 )T Xt )f2 (Xt )
t =p
N X
ρ(et − (β2 − Φ1 )T Xt )f3 (Xt ) +
t =p
N X
ρ(et − (β2 − Φ2 )T Xt )f4 (Xt ).
t =p
Suppose M > kΦ1 − Φ2 k = 2ϕ and kβ1 − Φ1 k ≥ 2M. Then kβ1 − Φ1 k ≥ ϕ and kβ1 − Φ2 k ≥ kβ1 − Φ1 k − kΦ1 − Φ2 k ≥ ϕ . Also, at least one of the inequalities kβ2 − Φ1 k ≥ ϕ and kβ2 − Φ2 k ≥ ϕ will hold. It follows that for (q, z ) ∈ U ∗ and kβ1 − Φ1 k ≥ 2M, N X
ρ(et − (β1 − Φ1 )T Xt )f1 (Xt ) +
t =p
N X
ρ(et − (β1 − Φ2 )T Xt )f2 (Xt )
t =p
≥ min
kuk≥ϕ
N X t =p
ρ(et + uT Xt )f1 (Xt ) + min
kuk≥ϕ
N X t =p
ρ(et + uT Xt )f2 (Xt )
L.-X. Zhang et al. / Statistics and Probability Letters 79 (2009) 807–813
813
and N X
ρ(et − (β2 − Φ1 )T Xt )f3 (Xt ) +
t =p
N X
ρ(et − (β2 − Φ2 )T Xt )f4 (Xt )
t =p
( ≥ min
min
kuk≥ϕ
min kuk
N X
ρ(et + uT Xt )f3 (Xt ) + min kuk
t =p
N X
ρ(et + u Xt )f3 (Xt ) + min T
kuk≥ϕ
t =p
N X
ρ(et + uT Xt )f4 (Xt ),
t =p N X
) ρ(et + u Xt )f4 (Xt ) . T
t =p
We conclude that for (q, z ) ∈ U ∗ and kβ1 − Φ1 k ≥ 2M,
ρN (θ) ≥ min
kuk≥ϕ
N X
ρ(et + uT Xt )f1 (Xt ) + min
kuk≥ϕ
t =p
( + min
min
N X
kuk≥ϕ
min kuk
N X t =p
ρ(et + uT Xt )f3 (Xt ) + min kuk
t =p
N X
ρ(et + uT Xt )f2 (Xt )
ρ(et + u Xt )f3 (Xt ) + min T
kuk≥ϕ
t =p
N X
ρ(et + uT Xt )f4 (Xt ),
t =p N X
) ρ(et + u Xt )f4 (Xt ) . T
t =p
It follows from Lemma A.4 that lim inf N →∞
1
inf inf ρN (θ)/N ≥ inf E[ρ(et + uT Xt )f1 (Xt )] + inf E[ρ(et + uT Xt )f2 (Xt )] kuk=ϕ kuk=ϕ N (q,z )∈U ∗ kβ1 −Φ1 k≥2M
+ min
inf E[ρ(et + uT Xt )f3 (Xt )] + E[ρ(et )f4 (Xt )] ,
kuk=ϕ
E[ρ(et )f3 (Xt )] + inf E[ρ(et + u Xt )f4 (Xt )] T
kuk=ϕ ∗ ∗
a.s.
→ C as U ∗ shrinks towards (q , z ). Expression (A.7) is proved by letting γ ∗ = (C − E[Rt (θ)])/2 and choosing δ ∗ small enough. Therefore, expression (A.6) is proved, and similarly lim inf
inf
N →∞ {θ:kβ2 −Φ2 k≥2M }
ρN (θ)/N > E[Rt (θ 0 )] a.s.
Combining the above inequality with (A.6) yields (A.2). The proof of the lemma is now completed.
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