Estimation of cusp in nonregular nonlinear regression models

Journal of Multivariate Analysis 88 (2004) 243 251 Estimation of cusp in nonregular nonlinear regression models B.L.S. Prakasa Rao Indian Statistical Institute, 7, SJS Sansanwal Marg, New Delhi, 110016, India Received8 August 2002 Abstract The asymptotic properties of the least squares estimator of the cusp in some nonlinear nonregular regression models is investigated via the study of the weak convergence of the least squares process generalizing earlier results in Prakasa Rao (Statist. Probab. Lett. 3 (1985) 15). r 2003 Elsevier Inc. All rights reserved. AMS 2000 subject classifications: primary 62J02 Keywords: Cusp; Least squares process; Nonlinear regression; Nonregular problem; Fractional Brownian motion 1. Introduction Nonlinear regression models are widely used for modeling of stochastic phenomena. Several examples for such modeling are given by Bard [2]. The study of asymptotic properties of the least squares estimator (LSE) of the parameters occurring in nonlinear regression models has been the subject of investigation since it is in general difficult to obtain the exact distribution of the LSE for any fixed sample. Jennrich [9], Malinvaud [10], Bunke Schmidt [3] Wu [20], among others, have investigatedthe asymptotic properties of the LSE for nonlinear regression models. All the works cited above, on the asymptotic distribution theory for the LSE in nonlinear regression models, assume regularity conditions which include in particular the condition on the twice differentiability of the regression function with E-mail address: blsp@isid.ac.in. 0047-259X/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/s0047-259x(03)00102-7

244 B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 respect to the parameter in addition to other conditions (cf. [7]). We have given an alternate approach to the problem which circumvents the condition on the differentiability of the regression function in a series of papers in Prakasa Rao [15 17]. A comprehensive presentation of this approach is given in Prakasa Rao [18]. A review of these relatedresults is given in Prakasa Rao [19]. Detaileddiscussion of this approach for a special class of nonregular nonlinear regression models is given in Prakasa Rao [14]. Maximum likelihoodestimation for cusp-type nonregular problems was first startedin Prakasa Rao [11] later by Ibragimov Khasminskii [8]. Other works dealing with problems of estimation for nonregular models include Akahira Takeuchi [1], Dachian [4] for the estimation of the cusp of Poisson observations Dachian Kutoyants [5] on the cusp estimation for ergodic diffusion process among others. Consider the nonlinear regression model Y i ¼ Sðx i ; yþþe i ; ix1: ð1:1þ We now discuss the problem of estimation of the parameter y by the least squares approach when the parameter y is a cusp for the function Sðx; yþ ¼sðx yþ: This problem in general is not amenable to stard methods using the Taylor s expansion, for instance, the function Sðx; yþ ¼jx yj l ; 0olo 1 2 is not differentiable at y: We study the asymptotic properties of the LSE via the least squares process developed in Prakasa Rao [15] (cf. [18,19]). A special case of this problem was studied in Prakasa Rao [14]. 2. Main result We now have the following main result. Theorem 2.1. Consider the nonlinear regression model Y i ¼ Sðx i ; yþþe i ; ix1; ð2:1þ where Sðx; yþ ¼ajx yj l þ hðx yþ; xpy; Sðx; yþ ¼bjx yj l þ hðx yþ; xxy; ð2:2þ where aa0; ba0; 0olo 1 2 ; yay hð:þ satisfies the Holder condition of order b4l þ 1 2 : Suppose Y is compact. Let y 0 be the true parameter. Furthermore, suppose that fe i ; ix1g are independent identically distributed rom variables with mean zero variance one. Let fx i g be a real sequence satisfying X n fsðx i ; yþ Sðx i ; y 0 Þg 2 ¼ 2nCðlÞjy y 0 j 2lþ1 ð1 þ oð1þþ ð2:3þ

as n-n where CðlÞa0 further suppose that there exists 0ok 1 ok 2 on such that nk 1 jy 1 y 2 j 2lþ1 p Xn B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 245 fsðx i ; y 1 Þ Sðx i y 2 Þg 2 pnk 2 jy 1 y 2 j 2lþ1 ð2:4þ for all y 1 y 2 in Y: Let # y n be an LSE of y based on the observations fy i ; 1pipng: Then there exists a constant C40 such that, for any t40 for any nx1; P y0 ½n r j # y n y 0 j4tšpct ð2lþ1þ ð2:5þ n r ð # y n y 0 Þ! L # f as n-n; ð2:6þ where r ¼ 1 2lþ1 # f is the location of the minimum of the gaussian process frðfþ; NofoNg with E½RðfÞŠ ¼ 2CðlÞjfj 2lþ1 ð2:7þ Cov½Rðf 1 Þ; Rðf 2 ÞŠ ¼ 4CðlÞ½jf 1 j 2lþ1 þjf 2 j 2lþ1 jf 1 f 2 j 2lþ1 Š: ð2:8þ The process frðfþ; NofoNg can be represented in the form p RðfÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 8CðlÞW H ðfþþ2cðlþjfj 2lþ1 ; ð2:9þ where W H is the fractional Brownian motion with mean zero the covariance function CovðW H ðf 1 Þ; W H ðf 2 ÞÞ ¼ 1 2 ½jf 1j 2lþ1 þjf 2 j 2lþ1 jf 1 f 2 j 2lþ1 Š ð2:10þ H ¼ l þ 1 2 is the Hurst parameter. Proof. Let y # n be an LSE of y obtainedby minimizing Q n ðyþ ¼ Xn ðy i Sðx i ; yþþ 2 : It is obvious that y # n minimizes Q n ðyþ Q n ðy 0 Þ ¼ Xn ðy i Sðx i ; yþþ 2 Xn ðy i Sðx i ; y 0 ÞÞ 2 ¼ 2 Xn e i ðsðx i ; yþ Sðx i ; y 0 ÞÞ þ Xn ðsðx i ; yþ Sðx i ; y 0 ÞÞ 2 : ð2:11þ ð2:12þ

246 B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 Observe that condition (2.3) implies that E y0 ½Q n ðyþ Q n ðy 0 ÞŠ ¼ Xn ðsðx i ; yþ Sðx i ; y 0 ÞÞ 2 Var y0 ½Q n ðyþ Q n ðy 0 ÞŠ ¼ 4 Xn ¼ 2nCðlÞjy y 0 j 2lþ1 ð1 þ oð1þþ ðsðx i ; yþ Sðx i ; y 0 ÞÞ 2 ¼ 8nCðlÞjy y 0 j 2lþ1 ð1 þ oð1þþ: ð2:13þ ð2:14þ In general, it follows from condition (2.4) there exists k 2 40 independent of n; y y 0 AY such that E y0 ½Q n ðyþ Q n ðy 0 ÞŠpnk 2 jy y 0 j 2lþ1 ð2:15þ Var y0 ½Q n ðyþ Q n ðy 0 ÞŠp4nk 2 jy y 0 j 2lþ1 : Furthermore, Cov y0 ½Q n ðy 1 Þ Q n ðy 0 Þ; Q n ðy 2 Þ Q n ðy 0 ÞŠ ¼ 4 Xn ðsðx i ; y 1 Þ Sðx i ; y 0 ÞÞðSðx i ; y 2 Þ Sðx i ; y 0 ÞÞ ¼ 4nCðlÞ½jy 1 y 0 j 2lþ1 þjy 2 y 0 j 2lþ1 jy 1 y 2 j 2lþ1 Šð1 þ oð1þþ from the relation jj f jj 2 þjjgjj 2 jjf gjj 2 ¼ 2/ f ; gs for any two vectors f g in R n : Let J n ðyþ ¼Q n ðyþ Q n ðy 0 Þ: ð2:16þ ð2:17þ In view of the above relations, it follows by arguments similar to those given in the Theorem 3.6 of Prakasa Rao [11] that there exists Z40 such that " # J n ðyþ lim lim sup P y0 inf t-n n-n jy y 0 j4tn r njy y 0 j 2lþ1pZ ¼ 0; ð2:18þ where r ¼ð2lþ1Þ 1 : In fact the same proof shows that, for any t40; P y0 ½n r j y # n y 0 j4tšpct ð2lþ1þ ; ð2:19þ where the constant C is independent of n t: In view of the above observation, the process fj n ðyþ; yayg has a minimum in the interval ½y 0 tn r ; y 0 þ tn r Š with probability approaching one for large t: For any such t40; let R n ðfþ ¼J n ðy 0 þ fn r Þ; fa½ t; tš ð2:20þ

B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 247 let RðfÞ be the gaussian process on ½ t; tš with E½RðfÞŠ ¼ 2CðlÞjfj 2lþ1 ð2:21þ Cov½Rðf 1 Þ; Rðf 2 ÞŠ ¼ 4CðlÞ½jf 1 j 2lþ1 þjf 2 j 2lþ1 jf 1 f 2 j 2lþ1 Š: ð2:22þ Observe that the process f RðfÞ; NofoNg; where RðfÞ ¼ RðfÞ E½RðfÞŠ pffiffiffiffiffiffiffiffiffiffiffiffi 8CðlÞ ð2:23þ is a gaussian rom process with mean zero the covariance function Covð Rðf 1 Þ; Rðf 2 ÞÞ ¼ 1 2 ½jf 1j 2lþ1 þjf 2 j 2lþ1 jf 1 f 2 j 2lþ1 Š ð2:24þ which is the fractional Brownian motion with the Hurst parameter H ¼ l þ 1 2 : Let Z n ðfþ ¼ Xn e i ðsðx i ; y 0 þ fn r Þ Sðx i ; y 0 ÞÞ ¼ Xn a ni e i ð2:25þ T n ðfþ ¼ Xn ðsðx i ; y 0 þ fn r Þ Sðx i ; y 0 ÞÞ 2 : ð2:26þ Observe that T n ðfþ is continuous in f for any fixed nx1 T n ðfþ-2cðlþjfj 2lþ1 as n-n: ð2:27þ In addition, Z n ðfþ! L Nð0; 2CðlÞjfj 2lþ1 Þ as n-n ð2:28þ since fe i ; ix1g are independent identically distributed rom variables with mean zero finite positive variance fa nk ; 1pkpng satisfy the condition max 1pkpn a 2 nk P n a2 ni-0 as n-n: ð2:29þ

248 B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 This follows from a central limit theorem due to Eicker [6]. The above relation can be provedby the following arguments. Note that X n a 2 ni ¼ Xn ¼ T n ðfþ ðsðx i ; y 0 þ fn r Þ Sðx i ; y 0 ÞÞ 2 ð2:30þ which tends to 2CðlÞjfj 2lþ1 as n-n by relation (2.27) the latter in turn implies that max ðsðx i; y 0 þ fn r Þ Sðx i ; y 0 ÞÞ 2-0: 1pipn ð2:31þ This follows from the observation that if P 1pkpn a2 nk -c40 max 1pkpN a 2 nk -0 for every fixed NX1; then max 1pkpn a 2 nk-0asn-n: The above discussion proves (2.28). Similarly, it can be shown that all the finite-dimensional distributions of the process fz n ðfþ; tpfptg converge to the corresponding finite-dimensional distributions of the gaussian process fzðfþ; tpfptg with mean zero Cov½Zðf 1 Þ; Zðf 2 ÞŠ ¼ CðlÞ½jf 1 j 2lþ1 þjf 2 j 2lþ1 jf 1 f 2 j 2lþ1 Š: ð2:32þ In addition, observe that EjZ n ðf 1 Þ Z n ðf 2 Þj 2 ¼ Xn ðsðx i ; y 0 þ f 1 n r Þ Sðx i ; y 0 þ f 2 n r ÞÞ 2 p k 2 jf 1 f 2 j 2lþ1 ; ð2:33þ where k 2 is independent of n; f 1 f 2 : Hence, the family of measures fm n g generatedby the stochastic processes fz n ðfþ; tpfptg on the space C½ t; tš of continuous functions on the interval ½t; tš with supremum norm topology forms a tight family. This observation together with the fact that the finite-dimensional distributions of the process fz n ðfþ; tpfptg converge weakly to the corresponding finite-dimensional distributions of the process fzðfþ; tpfptg prove that the sequence of processes fz n ðfþ; tpfptg converge weakly to the gaussian process fzðfþ; tpfptg: Hence, the sequence of processes fr n ðfþ; tpfptg converge weakly to the gaussian process frðfþ; tpfptg with mean function covariance function given by (2.7) (2.8), respectively. Applying arguments similar to those given in Prakasa Rao [11], it follows that n 1 2lþ1 ð # yn y 0 Þ! L # f as n-n; ð2:34þ where # f has the distribution of the location of the minimum of the gaussian process frðfþ; NofoNg with E½RðfÞŠ ¼ 2CðlÞjfj 2lþ1 ð2:35þ

B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 249 Cov½Rðf 1 Þ; Rðf 2 ÞŠ ¼ 4CðlÞ½jf 1 j 2lþ1 þjf 2 j 2lþ1 jf 1 f 2 j 2lþ1 Š: ð2:36þ 3. Remarks (i) Observe that r4 1 2 if 0olo1 2 the asymptotic variance is of the order Oðn 2r Þ which is small comparedto the case when the regression function Sðx; yþ is smooth the asymptotic variance is of the order Oðn 1 Þ (cf. [13]). Furthermore, for any t40; P y0 ½n r j y # n y 0 j4tšpct ð2lþ1þ ð3:1þ where the constant C is independent of n t: (ii) Let ( a if xo0 dðxþ ¼ b if x40: Conditions (2.3) (2.4) on the sequence fx i g are plausible conditions that can be assumed. This can be justified by the following arguments. Suppose the sequence fx i ; ix1g is the realization of a sequence of independent identically distributed rom variables fx i ; ix1g with the probability density function ( f ðx; y 0 Þ¼ hðx y 0Þexpfajx y 0 j l g if xpy 0 hðx y 0 Þexpfbjx y 0 j l g if xxy 0 : Then the strong law of large numbers implies that Xn 1 n fsðx i ; y 1 Þ SðX i ; y 0 ÞgfSðX i ; y 2 Þ SðX i ; y 0 Þg converges almost surely to E½fSðX 1 ; y 1 Þ SðX 1 ; y 0 ÞgfSðX 1 ; y 2 Þ SðX 1 ; y 0 ÞgŠ which is equal to CðlÞ½jy 1 j 2lþ1 þjy 2 j 2lþ1 jy 1 y 2 j 2lþ1 Š: This can be seen from Lemma 4 in Dachian Kutoyants [5]. In fact, Z N N ½dðx y 1 Þjx y 1 j l dðx y 0 Þjx y 0 j l Š½dðx y 2 Þjx y 2 j l dðx y 0 Þjx y 0 j l Š dx ¼ CðlÞ½jy 1 j 2lþ1 þjy 2 j 2lþ1 jy 1 y 2 j 2lþ1 Š; ð3:2þ

250 B.L.S. Prakasa Rao / Journal of Multivariate Analysis 88 (2004) 243 251 where CðlÞ ¼ Gð2l þ 1ÞGð1 2 lþ pffiffiffi ½a 2 þ b 2 2ab cosðplþš: 2 2lþ1 p ð2l þ 1Þ A special case of this result, when a ¼ b; is provedin Prakasa Rao [12]. For the general case, see Ibragimov Khasminskii [8]. (iii) If l ¼ 1 2 in model (2.1), the conditions statedin (2.3) (2.4) hold, then it can be checkedby similar arguments as before that there exists a constant C40 such that P y0 ðn 1=2 j # y n y 0 j4tþpct 2 ð3:3þ n 1=2 ð # y n y 0 Þ! L # f as n-n; ð3:4þ where f # is the location of the minimum of the gaussian process frðfþ; NofoNg with mean function EðRðfÞÞ ¼ 2C f 2 CovðRðf 1 Þ; Rðf 2 ÞÞ ¼ 8C f 1 f 2 for some constant C 40: It is easy to see that the process RðfÞ can be representedin the form RðfÞ ¼2C f 2 þ Lfc; ð3:5þ p where L ¼ ffiffiffiffiffiffiffiffi 8C c is a stard normal rom variable. Hence, #f ¼ L 4C c: ð3:6þ Combining the above remarks, we obtain that #y n! p y 0 as n-n; ð3:7þ there exists a constant C40 independent of n t such that P y0 ðn 1=2 j # y n y 0 j4tþpct 2 ð3:8þ n 1=2 ð # y n y 0 Þ! L Nð0; ð2c Þ 1 Þ as n-n: ð3:9þ Note that the limiting distribution of the LSE y # n is normal if l ¼ 1 2 in the model. This case illustrates the situation when the stard regularity conditions do not holdyet the estimator is strongly consistent asymptotically normal. Acknowledgments The author thanks the referee for his comments suggestions.

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