Bandwidth Selection in Nonparametric Kernel Testing

Transcription

1 Te University of Adelaide Scool of Economics Researc Paper No January 2009 Bandwidt Selection in Nonparametric ernel Testing Jiti Gao and Irene Gijbels

2 Bandwidt Selection in Nonparametric ernel Testing By Jiti Gao and Irène Gijbels Te University of Adelaide and atolieke Universiteit Leuven Abstract. We propose a sound approac to bandwidt selection in nonparametric kernel testing. Te main idea is to find an Edgewort expansion of te asymptotic distribution of te test concerned. Due to te involvement of a kernel bandwidt in te leading term of te Edgewort expansion, we are able to establis closed form expressions to explicitly represent te leading terms of bot te size and power functions and ten determine ow te bandwidt sould be cosen according to certain requirements for bot te size and power functions. For example, wen a significance level is given, we can coose te bandwidt suc tat te power function is maximized wile te size function is controlled by te significance level. Bot asymptotic teory and metodology are establised. In addition, we develop an easy implementation procedure for te practical realization of te establised metodology and illustrate tis on two simulated examples and a real data example. eywords: Coice of bandwidt parameter, Edgewort expansion, nonparametric kernel testing, power function, size function. Jiti Gao is Professor of Economics, Scool of Economics, Te University of Adelaide, Adelaide SA 5005, Australia E mail: jiti.gao@adelaide.edu.au. Irene Gijbels is Professor of Statistics, Department of Matematics, University of Leuven, Celestijnenlaan 200B, B-300, Leuven, Belgium E mail: irene.gijbels@wis.kuleuven.be. Te first autor would like to tank Jiying Yin for is excellent computing assistance and te Australian Researc Council Discovery Grants under Grant Numbers: DP and DP for te financial support. Te second autor tanks te Scool of Matematics and Statistics at te University of Western Australia for its kind ospitality and support. Te second autor also gratefully acknowledges te Researc Fund.U.Leuven GOA/2007/4 and te Flemis Science Foundation, FWO Belgium Project G for financial support.

3 . Introduction Consider a nonparametric regression model of te form Y i = mx i + e i, i =, 2,..., n,. were {X i } is a sequence of strictly stationary time series variables, {e i } is a sequence of independent and identically distributed i.i.d. errors wit E[e ] = 0 and 0 < E[e 2 ] = σ 2 <, m is an unknown function defined over IR d for d, and n is te number of observations. We assume tat {X i } and {e j } are independent for all i j n. To avoid te so called curse of dimensionality problem, we mainly consider te case of d 3 in tis paper. For te case of d 4, various dimension reduction estimation and specification metods ave been discussed extensively in several monograps, suc as Fan and Gijbels 996, Hart 997, Fan and Yao 2003, Gao 2007, and Li and Racine Tere is a vast literature on testing a parametric regression model null ypotesis versus a nonparametric model, especially for te case of i.i.d. X i s random or fixed design case. Many goodness-of-fit testing procedures are based on evaluating a distance between a parametric estimate of te regression function m assuming te null ypotesis is true and a nonparametric estimate of tat function. Among te popular coices for a nonparametric kernel estimator for m are te Nadaraya-Watson estimator, te Gasser-Müller estimator and a local linear polynomial estimator. Earlier papers following tis approac of evaluating suc a distance include Härdle and Mammen 993, Weirater 993 and González-Manteiga and Cao 993, among oters. Härdle and Mammen 993 consider a weigted L 2 -distance between a parametric estimator and a nonparametric Nadaraya-Watson estimator of te regression function. Te asymptotic distribution of teir test statistic under te null ypotesis depends on te unknown error variance te conditional error variance function. Weirater 993 instead uses a Gasser-Müller nonparametric estimator 2

4 in te fixed design regression case, divides by an estimator of te error variance and considers a discretized version of te L 2 -distance. González-Manteiga and Cao 993 also consider te fixed design regression case but rely on minimum distance estimation of te parametric model, seeking for minimizing a weigted L 2 -type distance between te parametric model and a pilot nonparametric estimator. Anoter approac to te same testing problem is introduced in Dette 999 wo focusses on te integrated conditional variance function, and uses as a test statistic te difference of a parametric estimator and a nonparametric Nadaraya-Watson based estimator of tis integrated variance. It is sown tat tis estimator asymptotically corresponds to test statistics based on a weigted L 2 -distance between a parametric and nonparametric estimator of te regression function, as in te above mentioned papers, using an appropriate weigt function in defining te L 2 -distance. Dette 999 studies te asymptotic distribution of te test statistic under fixed alternatives. Suc kind of alternatives are to be distinguised from te so-called sequences of local alternatives, were te difference between te regression function under te alternative and te one under te null ypotesis depends on te sample size n and decreases wit n. Te latter setup is te one considered in our study. Te above papers and several more recent goodness-of-fit tests see for example Zang and Dette 2004 and references terein ave in common tat tey rely on nonparametric kernel type regression estimators and tat te resulting test statistics are of a similar form at least in first-order asymptotics, and all depend on a bandwidt parameter. Te coice of te bandwidt parameter in suc goodness-of-fit testing procedures is te main concern in te present paper. Rougly speaking one can distinguis in te literature two approaces to deal wit tis bandwidt parameter coice in nonparametric and semiparametric kernel metods used for constructing model specification tests for te mean function of model.. A first approac is to use an estimation-based optimal bandwidt value, suc as a cross validation bandwidt. A second approac is to consider a set of suitable values for te bandwidt 3

5 and proceed furter from tere. Existing studies based on te first approac include Härdle and Mammen 993 for testing nonparametric regression wit i.i.d. designs and errors, Hjellvik and Tjøsteim 995, and Hjellvik, Yao and Tjøsteim 998 for testing linearity in dependent time series cases, Li 999 for specification testing in econometric time series cases, Cen, Härdle and Li 2003 for using empirical likeliood based tests, Jul and Xiao 2005 for testing structural cange in nonparametric time series regression, and oters. As pointed out in te literature, suc coices cannot be justified in bot teory and practice since estimation based optimal values may not be optimal for testing purposes. Nonparametric tests involving te second approac of coising eiter a set of suitable bandwidt values for te kernel case or a sequence of positive integers for te smooting spline case include Fan 996, Fan, Zang and Zang 200, and Horowitz and Spokoiny 200. Te practical implementation of coosing suc sets or sequences is owever problematic. Tis is probably wy Horowitz and Spokoiny 200 develop teir teoretical results based on a set of suitable bandwidts on te one and, but coose teir practical bandwidt values based on te assessment of te power function of teir test on te oter and. Apart from using suc test statistics based on nonparametric kernel, nonparametric series, spline smooting and wavelet metods, tere are test statistics constructed and studied based on empirical distributions. Suc studies ave recently been summarized in Zu To te best of our knowledge, te idea of coosing te appropriate smooting parameter suc tat te size of te test under consideration is preserved wile maximizing te power against a given alternative was only first explored analytically by ulasekera and Wang 997, in wic te autors propose using a nonparametric kernel test to ceck weter te mean functions of two data sets can be identical in a nonparametric fixed design setting. In some oter closely related studies, various discussions ave been given on te comparison of power values of te same test at 4

6 different bandwidts or different tests at te same bandwidt. Suc studies include Hart 997, Hjellvik, Yao and Tjøsteim 998, Hunsberger and Follmann 200, and Zang and Dette Te last paper compares tree main types of nonparametric kernel tests proposed in Härdle and Mammen 993, Zeng 996, and Fan, Zang and Zang 200. On te issue of size correction, tere ave recently been some studies. For example, Fan and Linton 2003 develop an Edgewort expansion for te size function of teir test and ten propose using corrected asymptotic critical values to improve te small medium sample size properties of te size of teir test. Some oter related studies include Nisiyama and Robinson 2000, Horowitz 2003, Nisiyama and Robinson 2005, wo develop some useful Edgewort expansions for bootstrap distributions of partial sum type of tests for improving te size performance. Te current paper is motivated by suc existing studies, especially by ulasekera and Wang 997, Fan and Linton 2003, Dette and Spreckelsen 2004, and Zang and Dette 2004, to develop a solid teory to support a power function based bandwidt selection procedure suc tat te power of te proposed test is maximized wile te size is under control wen using nonparametric kernel testing in parametric specification of a nonparametric regression model of te form. associated wit te ypotesis form of.2 below. To state te main results of tis paper, we introduce some notational details. Te main interest of tis paper is to test a parametric null ypotesis of te form H 0 : mx = m θ0 x versus a sequence of alternatives of te form H : mx = m θ x + n x for all x IR d,.2 were bot θ 0, θ Θ are unknown parameters and Θ is a parameter space of IR p, and n x is a sequence of nonparametrically unknown functions over IR d. Wit n x not being equal to zero, te function m θ x in H is in fact te projection of te true function on te null model. 5

7 Note tat mx under H in.2 is semiparametric wen { n x} is unknown nonparametrically. Note also tat instead of requiring.2 for all x IR d, it may be assumed tat.2 olds wit probability one for x = X i. Some first order asymptotic properties for bot te size and power functions of a nonparametric kernel test for te case were n, corresponding to a class of fixed alternatives not depending on n, ave already been discussed in te literature, suc as Dette and Spreckelsen Tis paper focuses on studying iger order asymptotic properties of suc kernel tests for te case were { n } is a sequence of local alternatives in te sense tat lim n n x = 0 for all x IR d. Let be te probability kernel density function and be te bandwidt involved in te construction of a nonparametric kernel test statistic denoted by T n. To implement te kernel test in practice, we propose a new bootstrap simulation procedure to approximate te α quantile of te distribution of te kernel test by a bootstrap simulated critical value l α. Let α n = P T n > l α H 0 and β n = P T n > l α H be te respective size and power functions. In Teorem 2.2 we sow tat α n = Φl α s n κ n l α s n 2 φl α s n + o d,.3 β n = Φl α r n κ n l α r n 2 φl α r n + o d,.4 were s n = p d, r n = p 2 n δ 2 n d, κ n = p 3 d, and Φ and φ denote respectively te cumulative distribution and density function of te standard Normal random variable, in wic all p i s are positive constants and δ 2 n = 2 nxπ 2 xdx wit π being te marginal density function of {X i }. Our aim is to coose a bandwidt ew suc tat β n ew = max Hnα β n wit H n α = { : α c min < α n < α + c min } for some small 0 < c min < α. Our detailed study in Section 3 sows tat ew is proportional to n δ 2 n 3 2d. Suc establised relationsip between δ n and ew sows us tat te coice of an optimal rate of ew depends on tat of an order of δ n. 6

8 If δ n is cosen proportional to n d+2 6d+4 for a sequence of local alternatives under H, ten te optimal rate of ew is proportional to n d+4, wic is te order of a nonparametric cross validation estimation based bandwidt frequently used for testing purposes. Wen considering a sequence of local alternatives wit δ n = O n 2 loglogn being cosen as te optimal rate for testing in tis kind of kernel testing Horowitz and Spokoiny 200, te optimal rate of ew is proportional to loglogn 3 2d. Te rest of te paper is organised as follows. Section 2 points out tat existing nonparametric kernel tests can be decomposed wit quadratic forms of {e i } as leading terms in te decomposition. Tis motivates te discussion about establising Edgewort expansions for suc quadratic forms. In Section 3, we apply te Edgewort expansions to study bot te size and power functions of a representative kernel test. Section 4 presents several examples of implementation. Some concluding remarks are made in Section 5. Matematical assumptions and proofs are provided in te appendix. 2. Nonparametric kernel testing As mentioned in te introductory section, various autors ave discussed and studied nonparametric kernel test statistics based on a weigted L 2 distance function between a nonparametric kernel estimator and a parametric counterpart of te mean function. It can be sown tat te leading term of eac of tese nonparametric kernel test statistics is of a quadratic form see, for example, Cen, Härdle and Li 2003 P n = e i wx i L X i X j wx j e j, 2. i= j= were L = n L d, Lx = yx + ydy, and w is a suitable weigt function probably depending on eiter π, σ 2 or bot, in wic is a probability kernel function, is a bandwidt parameter and bot are involved in a nonparametric kernel estimation of m. 7

9 In tis paper, we concentrate on a second group of nonparametric kernel test statistics using a different distance function. Rewrite model. into a notational version of te form under H 0 Y = m θ0x + e, 2.2 were X is assumed to be random and θ 0 is te true value of θ under H 0. Obviously, E[e X] = 0 under H 0. Existing studies Zeng 996; Li and Wang 998; Li 999; Fan and Linton 2003; Dette and Spreckelsen 2004; Jul and Xiao 2005 propose using a distance function of te form E [ee e X πx] = E [ E 2 e X πx ], 2.3 were π is te marginal density function of X. Tis suggests using a normalized kernel based sample analogue of 2.3 of te form T n = n d σ n Xi X j e i e j, 2.4 i= j=, i were σn 2 = 2µ 2 2 ν 2 2 udu wit µ k = E[e k ] for k and ν l = E[π l X ] for l. It can easily be seen tat T n is te leading term of te following quadratic form Q n = n Xi X j e i e j. 2.5 d σ n i= j= In summary, bot equations 2. and 2.5 can be generally written as R n = e i φ n X i, X j e j, 2.6 i= j= were φ n, may depend on n, te bandwidt and te kernel function. Tus, it is of general interest to study asymptotic distributions and teir Edgewort expansions for quadratic forms of type 2.6. To present te main ideas of establising Edgewort expansions for suc quadratic forms, we focus on T n in te rest of tis paper. Tis is because te main tecnology for establising an Edgewort 8

10 expansion for te asymptotic distribution of eac of suc tests is te same as tat for T n. Since T n involves some unknown quantities, we estimate it by a stocastically normalized version of te form T n = ni= nj=, i ê i X i X j ê j n, 2.7 d σ n were ê i = Y i m θx i and σ n 2 = 2 µ 2 2 ν 2 2 udu wit µ 2 = ni= ê 2 n i ν 2 = ni= π 2 X n i, in wic θ is a n consistent estimator of θ 0 under H 0 and πx = ni= x X i is te conventional nonparametric kernel density estimator wit b cv being a bandwidt parameter cosen by cross validation see for n bd cv example Silverman 986. Similarly to existing results Li 999, it may be sown tat for eac given and T n = T n + o P d. 2.8 Tus, we may use te distribution of T n to approximate tat of T n. Let l e α 0 < α < be te α quantile of te exact finite sample distribution of T n. Because l e α may not be evaluated in practice, we terefore suggest coosing eiter a non random approximate α level critical value, l α, or a stocastic approximate α level critical value, l α by using te following simulation procedure: We generate Y i = m θx i + µ 2 e i for i n, were {e i } is a sequence of i.i.d. random samples drawn from a pre-specified distribution, suc as N0,. Use te data set {X i, Yi : i =, 2,..., n} to estimate θ by θ and compute T n. Let l α be te α quantile of te distribution of ni= nj=, i ê T n i X i X j ê j =, 2.9 were ê i = Y i m θ X i and σ 2 n n d σ n = 2 µ 2 2 ν 2 2 udu wit µ 2 = n ni= ê 2 i. In te simulation process, te original sample X n = X,, X n acts in te resampling as a fixed design even wen {X i } is a sequence of random regressors. 9

11 Repeat te above step M times and produce M versions of T n denoted by T n,m for m =, 2,..., M. Use te M values of T n,m to construct teir empirical distribution function. Te bootstrap distribution of T n given W n = {X i, Y i : i n} is defined by P T n x = P T n x W n. Let lα 0 < α < satisfy P T n lα = α and ten estimate lα by lα. Note tat bot l α = l α and lα = lα depend on. It sould be pointed out tat te coice of a pre specified distribution does not ave muc impact on bot te teoretical and practical results. In addition, we may also use a wild bootstrap procedure to generate a sequence of resamples for {e i }. Note also tat te above simulation is based on te so called regression bootstrap simulation procedure discussed in te literature, suc as Li and Wang 998, Franke, reiss and Mammen 2002, and Li and Racine Wen X i = Y i, we may also use a recursive simulation procedure, commonly-used in te literature. See for example, Hjellvik and Tjøsteim 995, and Franke, reiss and Mammen Since te coice of a simulation procedure does not affect te establisment of our teory, our main results are establised based on te proposed simulation procedure. We now ave te following results in Teorems 2. and 2.2; teir proofs are provided in te appendix. Teorem 2.. Suppose tat Assumptions A. and A.2 listed in te appendix old. Ten under H 0 sup P T n x P T n x = O d x R olds in probability wit respect to te joint distribution of W n, and 2.0 P T n > lα = α + O d. 2. For an equivalent test, Li and Wang 998 establis some results weaker tan 2.0. Fan and Linton 2003 consider some iger order approximations to te size function of te test discussed in Li and Wang

12 For eac we define te following size and power functions α n = P T n > l α H 0 and βn = P T n > l α H. 2.2 Correspondingly, we define α n, β n wit l α replaced by l α. Before we discuss ow to coose an optimal bandwidt in Section 3, we give Edgewort expansions of bot te size and power functions in Teorem 2.2 below. In order to express te Edgewort expansions, we need to introduce te following notation. Let were ν l wit itself. d µ µ3 n κ n = d 2 ν , 2.3 σ 3 n = E[π l X ] = π l+ xdx, and 3 is te tree time convolution of Teorem 2.2. i Suppose tat Assumptions A. and A.2 listed in te appendix old. Ten α n = Φl α s n κ n l α s n 2 φl α s n + o d, 2.4 α n = Φl α s n κ n l α s n 2 φl α s n + o d 2.5 old in probability wit respect to te joint distribution of W n, were Φ and φ are te probability distribution and density functions of N0,, respectively, and s n = C 0 m d wit C 0 m = m θ0 x τ E θ [ mθ0 X mθ0 X θ θ 2ν 2 2 vdv τ ] mθ0 x θ π 2 xdx ii Suppose tat Assumptions A. A.3 listed in te appendix old.. Ten te following equations old in probability wit respect to te joint distribution of W n : β n = Φl α r n κ n l α r n 2 φl α r n + o d, 2.6 β n = Φl α r n κ n l α r n 2 φl α r n + o d, 2.7

13 were r n = n C 2 n d, in wic C 2 n = 2 n xπ 2 xdx σ 2 2ν 2 2 vdv. 2.8 Assumption A.2 implies tat te random quantity C 0 m is bounded in probability. As expected, te rate of r n depends on te form of n. To simplify te following expressions, let z α be te α quantile of te standard normal distribution and d j = z 2 α c j for j =, 2, were c = 43 0µ 3 2ν 3 3σ 3 n and c 2 = µ σn 3 Let d 0 = d C 0 m. A corollary of Teorem 2.2 is given in Teorem 2.3 below. Teorem 2.3. Suppose tat te conditions of Teorem 2.2i old. Ten under H 0 l α z α + d 0 d + d 2 n d in probability, 2.20 l α z α + d 0 d + d 2 n d in probability. 2.2 Teorem 2.3 sows tat te size distortion of te proposed test is d 0 d +d 2 n d wen using te standard asymptotic normality in practice. A similar result as been obtained by Fan and Linton We sow in addition tat te bootstrap simulated critical value is approximated explicitly by z α + d 0 d + d 2 n d. As te main objective of tis paper, Section 3 below proposes a suitable selection criterion for te coice of suc tat wile te size function is appropriately controlled, te power function is maximized at suc. A closed form expression of te power function based optimal bandwidt is given. 3. Power function based bandwidt coice 2

14 We now employ te Edgewort expansions establised in Section 2 to coose a suitable bandwidt suc tat te power function β n is maximized wile te size function α n is controlled by a significance level. We tus define ew = arg max H nα β n wit H n α = { : α c min < α n < α + c min } 3. for some arbitrarily small c min > 0. We now start to discuss ow to solve te optimization problem 3.. It follows from 2.3 and 2.9 tat d µ µ3 n κ n = d 2 ν σ 3 n = c d + c 2 n d. 3.2 Let x = d. We rewrite κ n as κ n = c x + c 2 n x. Let γ n = z 2 α κ n, l α r n z α + γ n r n = z α + d n Cn 2 x + d 4 x z α + d 3 x + d 4 x, 3.3 l α s n z α + γ n s n z α + d C 0 m x + d 4 x = z α + d 0 x + d 4 x, 3.4 were d 0 = d C 0 m, d = z 2 α c, d 3 = d n C 2 n and d 4 = c 2 z 2 α n. Note tat lim n d 4 = 0. Since Assumption A.3 implies tat lim n n C 2 n = +, we tus ave lim d 3 = wen lim n C 2 n n n = Due to tis, we treat d 3 as a sufficiently large negative value wen n C 2 n is viewed as a sufficiently large positive value in te finite sample analysis of tis section. Ignoring te iger order terms i.e. terms of order ox + n x or smaller, we now re write te power and size functions β n and α n simply as functions of x = d as follows: β n Φl α r n κ n l α r n 2 φl α r n Φz α + d 3 x + d 4 x c x + c 2 n x z α + d 3 x + d 4 x 2 φ z α + d 3 x + d 4 x βx, 3.6 α n Φl α s n κ n l α s n 2 φl α s n 3

15 Φz α + d 0 x + d 4 x c x + c 2 n x z α + d 0 x + d 4 x 2 φ z α + d 0 x + d 4 x αx. 3.7 Our objective is ten to find x ew = d ew suc tat x ew = arg max x H nα βx wit H nα = {x : α c min < αx < α + c min }, 3.8 were c min is cosen as c min = α 0 for example. Finding roots of β x = 0 implies tat te leading order of te unique real root of te equation is given approximately by were t n = n C 2 n, a = as defined in Teorem 2.2ii. ew = x 2 d ew = a 2d t 3 2d cπ wit cπ = 3 2 udu n, 3.9 π 3 xdx 3, in wic Cn π xdx 2 is 2 It can also be sown tat ew is te maximizer of te power function β n at = ew suc tat β nx x= d ew < 0, 3.0 at least for sufficiently large n. Detailed derivations of 3.9 and 3.0 are given in Appendix B below. Furtermore, te coice of ew satisfies bot Assumptions A.v and A.3 tat lim n n d ew = + and lim n d n ew Cn 2 = +. Tis implies tat te coice of ew is valid to ensure lim n β n ew =. Wen bot σ 2 = µ 2 = E[e 2 ] and te marginal density function π of {X i } are unknown in practice, we propose using an estimated version of ew as follows: ĥ ew = â 2d t 3 2d n, 3. were ni= t 2 n = n Ĉ2 n wit Ĉ2 n n = nx i πx i and µ 2 2 ν 2 2 vdv ni= π 2 X â = 3 2 udu n i 3 ĉπ wit ĉπ = ni= πx n i 3, 4

16 in wic µ 2, ν 2 and π are as defined in 2.7, and n x is given by n x = wit θ and b cv being te same as in 2.7. x X i Yi ni= m θx i ni= x X i Note also tat ĥew provides an optimal bandwidt irrespectively of weter one works under te null ypotesis H 0 or under te alternative ypotesis H. In oter words, it can be used for computing not only te power under an alternative H, but also te size under H 0 in eac case. Detailed discussion about tis is given in Appendix B below. We conclude tis section by summarizing te above discussion into te following proposition; its proof is given in Appendix B below. Proposition 3.. Suppose tat Assumptions A. A.3 listed in te appendix old. Additionally, suppose tat n x is continuously differentiable suc tat lim sup nx C < and lim n inf n x D π n x x IR d n x n b d cv = in probability for some C > 0, were D π = {x IR d : πx > 0} and 2 denotes te Euclidean norm. Ten β n ĥew lim n β n ew = in probability. 3.2 As pointed out in te introduction, implementation of eac of existing nonparametric kernel tests involves eiter a single bandwidt cosen optimally for estimation purposes or a set of bandwidt values. Te proposed ĥew is cosen optimally for testing purposes. Section 4 below sows ow to implement te proposed test based on our bandwidt in practice and compares te finite sample performance of te proposed coice wit tat of some closely relevant alternatives in te literature. 4. Examples of implementation 5

17 Tis section presents two simulated examples and one real data example to illustrate te proposed teory and metods in Sections 2 and 3 as well as to make comparisons wit some closely relevant alternatives in te literature. Simulated example 4. below discusses te finite sample performance of te proposed test T n ĥew wit tat of te alternative version were te test is coupled wit a cross validation CV bandwidt coice. Simulated example 4.2 below compares our test wit some of te commonly used tests in te literature. Example 4.3 provides a real data example to sow tat te proposed test makes a clear difference. In te following finite sample study in Examples below, we consider te case were n x = c n x, in wic {c n } is a sequence of positive real numbers satisfying lim n c n = 0 and x is an unknown function not depending on n. Example 4.. Consider a nonparametric time series regression model of te form Y i = θ X i + θ 2 X i2 + c n X 2 i + X 2 i2 + e i, i n, 4. were {e i } is a sequence of Normal errors and bot X i and X i2 are time series variables generated by X i = αx i, + u i and X i2 = βx i,2 + v i, i n 4.2 wit {u i } and {v i } being i.i.d. random errors generated independently from Normal distributions as below. Under H 0, we generate a sequence of observations {Y i } wit θ = θ 2 = as te true parameters, i.e. H 0 : Y i = X i + X i2 + e i, 4.3 were {e i } is a sequence of independent and identically distributed random errors generated from N0,, and {X i } and {X i2 } are independently generated from X i = 0.5X i, + u i and X i2 = 0.5X i,2 + v i, i n 4.4 6

18 wit X 0 = X 02 = 0 and {u i } and {v i } are sequences of independent and identically distributed random errors and generated independently from a N0,. Under H, we are interested in two alternative models of te form H : Y i = X i + X i2 + c n Xi 2 + Xi2 2 + e i, e i N0, 4.5 wit c n being cosen as eiter c n = n 2 loglogn or c 2n = n 7 8. In te testing procedure, te parameters θ and θ 2 in te parametric model are estimated as discussed in Sections and 2. n 2 Te reasoning for te above coice of c jn is as follows. Te rate of c n = loglogn sould be an optimal rate of testing in tis kind of nonparametric kernel testing problem as discussed in Horowitz and Spokoiny 200. Te rate of c 2n = n 7 8 implies tat te optimal bandwidt ĥew in B.43 wit d = 2 is proportional to n 6. Trougout tis example, we coose as te standard normal density function. Let ĥcv be cosen by a cross validation criterion of te form ĥ cv = arg min Y i m i X i, X i2 ; 2 wit H cv = [ ] n, n 6 H cv n in wic i= m i X i, X i2 ; = nl=, i X l X i Xl2 X i2 Yl nl=, i X l X i Xl2. X i2 4.6 Let ĥ0test be te corresponding version of ĥew in B.43 and ĥ0cv be te corresponding version of ĥcv in 4.6 bot computed under H 0. Since {Y i } under H depends on te coice of c n, tus te computing of bot ĥew of B.43 and ĥcv of 4.6 under H depend on te coice of c n. Let ĥjtest be te corresponding versions of ĥew in B.43 and ĥjcv be te corresponding versions of ĥcv in 4.6 wit c n = c jn for j =, 2. In order to compare te size and power properties of T n wit te most relevant alternatives, we introduce te following simplified notation: for j =, 2, α 0 = P T n ĥ0cv > l α ĥ0cv H0, βj = P T n ĥjcv > l α ĥ0cv H, α 02 = P T n ĥ0test > l α ĥ0test H0, βj2 = P T n ĥjtest > l α ĥ0test H. 7

19 We consider cases were te number of replications of eac of te sample versions of α 0k and β jk for j, k =, 2 was M = 000, eac wit B = 250 number of bootstrapping resamples, and te simulations were done for te cases of n = 250, 500 and 750. Te detailed results at te %, 5% and 0% significance level are given in Tables , respectively. Table 4.. Simulated size and power values at te % significance level Sample Size Null Hypotesis Is True Null Hypotesis Is False n α 0 α 02 β β 2 β 2 β Table 4.2. Simulated size and power values at te 5% significance level Sample Size Null Hypotesis Is True Null Hypotesis Is False n α 0 α 02 β β 2 β 2 β Table 4.3. Simulated size and power values at te 0% significance level Sample Size Null Hypotesis Is True Null Hypotesis Is False n α 0 α 02 β β 2 β 2 β

20 Tables report compreensive simulation results for bot te sizes and power values of te proposed tests for models 4.3 and 4.4. Column 2 in eac of Tables sows tat wile te sizes for te test based on ĥ0cv are comparable wit tese given in column 3 based on ĥ0test, te power values of te test based on ĥ jtest in columns 6 and 7 are always greater tan tese given in columns 4 and 5 based on ĥjcv. Tis is not surprising, because te teory sows tat eac of ĥjtest is cosen suc tat te resulting power function is maximized wile te corresponding size function is under control by te significance level. In addition, te test based on ĥ2test is almost uniformly more powerful tan te best based on ĥtest, wic is te second most powerful test. Tis is basically because ĥ 2test is based on considering H wit c 2n = n 7 8, wic goes to zero slower tan c n = n 2 log logn, and ence te distance between te alternative and te null is biggest in te former case and terefore easier to detect. Meanwile, te last columns of Tables sow tat te test based on te bandwidt ĥ2test is still a powerful test even toug te bandwidt is proportional to n 6, wic is te same as te optimal bandwidt based on a cross validation estimation metod. Tis sows tat weter an estimation based optimal bandwidt may be used for testing depends on weter te bandwidt is cosen optimally for testing purposes. We finally want to stress tat te proposed test based on eiter ĥtest or ĥ2test as not only stable sizes even at a small sample size of n = 250, but also reasonable power values even wen te distance between te null and te alternative as been made deliberately close at te rate of n loglogn = for n = 500 for example. We can expect tat te test would ave bigger power values wen te distance is made wider. Overall, Tables sow tat te establised teory and metodology is workable in te small and medium sample case. Example 4. discusses te small and medium sample comparison results for te proposed test wit eiter testing based optimal bandwidt or estimation based CV bandwidt. Example 4.2 below considers comparing te small and medium sample 9

21 performance of te proposed test associated wit te optimal bandwidt wit some closely related nonparametric tests available in bot te econometrics and statistics literature. Example 4.2. Consider a linear model of te form Y i = α 0 + β 0 X i + e i, i n = 250, 4.7 were {X i } is a sequence of independent random variables sampled from N0, 25 distribution truncated at its 5t and 95t percentiles, and {e i } is sampled from one of te tree distributions: i e i N0, 4; ii a mixture of Normals in wic {e i } is sampled from N0,.56 wit probability 0.9 and from N0, 25 wit probability 0.; and iii te Type I extreme value distribution scaled to ave a variance of 4. Te mixture distribution is leptokurtic wit a variance of 0.39, and te Type I extreme value distribution is asymmetrical. Tis is te same example as used in Horowitz and Spokoiny 200 for te comparison wit some of te commonly used tests in te literature, suc as te Andrews test proposed in Andrews 997, te HM test proposed in Härdle and Mammen 993, te HS test proposed in Horowitz and Spokoiny 200 and te empirical likeliood EL test proposed in Cen, Härdle and Li To compute te sizes of te test, coose α 0 = β 0 = as te true parameters and ten generate {Y i } from Y i = + X i + e i under H 0, and generate {Y i } from Y i = + X i + 5 φ X i τ τ + ei under H, were τ = or 0.25, and φ is te density function of te standard normal distribution. Te kernel function used ere is x = 5 6 x2 2 I x. Coose c n = 5τ and x = φx τ for te corresponding forms in.2. For j =, 2, let c jn = 5τ j and j x = φx τj wit τ = and τ 2 = Let ĥinew be te corresponding version of ĥew of B.43 based on c jn, j x for j =, 2. In order to make a fair comparison, we use te same number of te bootstrap resamples of M = 99, te same number of replications of M = 000 under H 0 and 20

22 M = 250 under H as in Table of Horowitz and Spokoiny 200. In Table 4.4 below, we add te size and power values to te last two columns for bot te EL test and te proposed test T n ĥinew of tis paper. Te oter parts of te table are obtained and tabulated similarly to Table of Horowitz and Spokoiny 200. Table 4.4. Simulated size and power values at te 5% significance level Probability of Rejecting Null Hypotesis Andrews HM HS EL T n ĥnew Distribution τ Test Test Test Test Test Null Hypotesis Is True Normal Mixture Extreme Null Hypotesis Is False Normal Mixture Extreme Normal Mixture Extreme Table 4.4 sows tat te proposed test as better power properties tan any of te commonly used tests, wile te size values are comparable wit tose of te competitors. Te results furter support te power based bandwidt selection procedure proposed in Sections 2 and 3. As discussed in te supplemental material, te proposed teory and metodology for model. can be applied to an extended model of te form Y i = mx i + e i wit e i = σx i ɛ i, i n, 4.8 2

23 were σ satisfying inf x IR d σx > 0 is unknown nonparametrically and {ɛ i } is a sequence of i.i.d. random errors wit zero mean and finite variance. In addition, {ɛ i } and {X j } are assumed to be independent for all j i n. A special case of model 4.8 is discussed in Example 4.3 below. Example 4.3. Tis example examines te ig frequency seven day Eurodollar deposit rate sampled daily from June 973 to 25 February 995. Tis provides us wit n = 5505 observations. Let {X i : i =, 2,, n = 5505} be te set of Eurodollar deposit rate data. Figures 4. and 4.2 below plot te data values and te conventional nonparametric kernel density estimator πx = n cv x Xi i= cv respectively, were x = 2π e x2 2 and cv is te conventional normal reference based bandwidt given by cv =.06 n 5 i n i=x X 2 wit X = X i. 4.9 n i= Note tat b cv of 2.7, ĥcv of 4.6 and cv of 4.9 are normally different from eac oter. In te case were {X i } follows an autoregressive model, tey can be cosen te same. Tus, tey are cosen te same in tis example. It as been assumed in te literature see, for example, Aït Saalia 996; Fan and Zeng 2003; Arapis and Gao 2006 tat te Eurodollar data set {X i } may be modeled by a nonlinear time series model of te form Y i = µx i + σx i ɛ i, i n, 4.0 were Y i = X i+ X i, σ > 0 is unknown nonparametrically, and ɛ Λ i N 0, Λ, in wic Λ is te time between successive observations. Since we consider a daily data set, tis gives Λ =

24 Eurodollar Interest Rate Year Figure 4.: Seven day Eurodollar deposit rate, June, 973 to February 25, 995. density function spot rate, r Figure 4.2: Nonparametric kernel density estimator of te Eurodollar rate. On te question of weter tere is any nonlinearity in te drift function µ, existing studies ave provided no definitive answer. For example, Aït Saalia 996, and Arapis and Gao 2006 sow tat tere is some evidence of supporting nonlinearity in te drift on te one and. On te oter and, existing studies, suc as 23

25 Capman and Pearson 2000, and Fan and Zeng 2003, suggest tat nonlinearity may just be caused by estimation biases wen using nonparametric kernel estimation. To furter discuss weter te assumption on linearity in te drift is appropriate for te given set of data, we apply our test to propose testing H 0 : µx = µx; θ 0 = β 0 α 0 x versus H : µx = β α x+c n x 4. for some θ j = α j, β j Θ for j = 0, and c n = parameter space in IR 2 and x is a continuous function. n log logn, were Θ is a It can be sown tat te proposed test in Section 2 as an asymptotically equivalent version of te form: T n = nj= ni=,i j ê j X i X j 2 n j= ni= ê 2 j 2 Xi X j ê i ê 2 i, 4.2 were ê i = Y i β α X i, in wic α, β is te pair of te conventional least squares estimators minimizing n i= Yi β α X i 2. As pointed out in te literature Arapis and Gao 2006, T n is independent of te structure of te conditional variance σ 2. Te kernel function used is te standard normal density function given by x = 2π e x2 2. Let test be te corresponding version of B.43. It as been sown in Appendix B below tat test = â 2 t 3 2 n, 4.3 were t n and â are te same as in B.43, in wic ĉπ becomes ĉπ = n π 2 X i σ 6 X i i= n 3 πx i σ 4 2 X i i= 4.4 wit σ 2 X i = nu= X i X u ê 2 u cv. nv= X i X v cv 24

26 Let L = T n test and L 2 = T n cv. To apply te test L j for eac j =, 2 to test H 0, we propose te following procedure for computing te p value of L j : Compute ê i = Y i β α X i and ten generate a sequence of bootstrap resamples {ê i } given by ê i = σx i ɛ i, were {ɛ i } is a sequence of i.i.d. bootstrap resamples generated from N0, Λ and σ 2 is defined as above. Generate Ŷ i = β α X i + ê i. Compute te corresponding version L j of L j for eac j =, 2 based on {Ŷ i }. Repeat te above steps M = 000 times to find te bootstrap distribution of L j and ten compute te proportion tat L j < L j for eac j =, 2. Tis proportion is a simulated p value of L j. Our simulation results return te simulated p values of p = 0.02 for L and p 2 = for L 2. Wile bot of te simulated p values suggest tat tere is no enoug evidence of rejecting te linearity in te drift at te 5% significance level, te evidence of accepting te linearity based on L is stronger tan tat based on L 2. As our test T n test involves no estimation biases, te process of computing te simulated p values is quite robust. We terefore believe tat tis improved test furter reinforces te findings of Capman and Pearson 2000 and Fan and Zang 2003 tat tere is no definitive answer to te question weter te sort rate drift is actually nonlinear. 5. Conclusion Tis paper as addressed te issue of ow to appropriately coose te bandwidt parameter wen using a nonparametric kernel based test. Bot te size and power properties of te proposed test ave been studied systematically. Te establised teory and metodology as sown tat a suitable bandwidt can be optimally cosen after appropriately balancing te size and power functions. Furtermore, te new 25

27 metodology as resulted in a closed form representation for te leading term of suc an optimal bandwidt in te finite sample case. Existing results see, for example, Li and Wang 998; Li 999; Fan and Linton 2003; Gao 2007 sow tat tis kind of nonparametric kernel test associated wit a large sample critical value may not ave good size and power properties. Our small and medium sample studies in bot te simulated and real data examples ave sown tat te performance of suc a test can be significantly improved wen it is coupled wit a power based optimal bandwidt as well as a bootstrap simulated critical value. It is pointed out tat te establised teory and metodology as various applications in providing solutions to some oter related testing problems, in wic nonparametric metods are involved. Future extensions also include dealing wit cases were bot X i and e i may be strictly stationary time series. Appendix A Tis appendix lists te necessary assumptions for te establisment and te proofs of te main results given in Section 2. A.. Assumptions Assumption A.. i Assume tat {e i } is a sequence of i.i.d. continuous random errors wit E[e ] = 0, 0 < σ 2 = E[e 2 ] = σ2 < and E[e 6 ] <. ii We assume tat {X i } is strictly stationary and α mixing wit mixing coefficient αt being defined by αt = sup{ P A B P AP B : A Ω s, B Ω s+t} C α α t A. for all s, t, were 0 < C α < and 0 < α < are constants, and Ω j i denotes te σ field generated by {X k : i k j}. iii We also assume tat {X s } and {e t } are independent for all s t n. Let π be te marginal density suc tat 0 < π 3 xdx <, and π τ,τ 2,,τ l be te joint probability density of X +τ,..., X +τl l 4. Assume tat π τ,τ 2,,τ l for all l 4 do exist and are continuous and bounded. 26

28 iv Assume tat te univariate kernel function is a symmetric and bounded probability density function. In addition, we assume te existence of bot 3, te tree time convolution of wit itself, and 2 2, te two time convolution of 2 wit itself. v Te bandwidt parameter satisfies bot lim n = 0 and lim n n d =. Assumption A.2. i Let H 0 be true. Ten for any sufficiently small ε > 0 and some B L > 0 were θ 0 is te same as defined in.2. lim P n θ θ 0 > B L < ε, n ii Let H be true. Ten for any sufficiently small ε 2 > 0 and some B 2L > 0 were θ is te same as defined in.2. lim P n θ θ > B 2L < ε 2, n iii Tere exist some absolute constants ε 3 > 0 and 0 < B 3L < suc tat te following lim n P n θ θ > B 3L W n < ε 3 olds in probability, were θ is as defined in te Simulation Procedure above Teorem 2.. iv Let m θ x be differentiable wit respect to θ and m θx [ θ be continuous in bot x and mθ0 X θ. In addition, E mθ0 X τ ] θ θ is a positive definite matrix, and Assumption A.3. 0 < 2 nxπ 2 xdx <. m θ x 2 0 < θ=θ0 π 2 xdx <. θ i Let { n x} be a sequence of continuous functions suc tat ii Let C 2 n satisfy lim n n d C 2 n = and lim n n C 6 n = 0, were C 2 n = 2 n xπ 2 xdx σ 2ν 2, 2 2 vdv in wic ν 2 = E [ π 2 X ] <. Assumptions A. A.3 are standard and justifiable conditions. Some detailed justifications are given in Appendix C below. 27

29 A.2. Tecnical lemmas Recall tat using lim n n d = d µ µ3 n κ n = d 2 ν σn 3 c d + c 2 n d = c d + c 2 c n d c d. A.2 In order to establis some useful lemmas witout including non essential tecnicality, we introduce te following simplified notation: a ij = n Xi X j, L n = a ij e i e j, d σ n i= j=, i π 3 3 udu ρ = π 3 2 udu vdv 2 d. A.3 We need te following lemmas; teir proofs are given in Appendix C below. Lemma A.. Suppose tat te conditions of Teorem 2.2i old. Ten for any sup x IR P L n x Φx + ρ x 2 φx = O Recall L n = n i= nj=, i e i a ij e j as defined in A.3 and let T n = d 2 nσ n + d 2 nσ n + 2 d 2 nσ n i= j=, i i= j=, i i= j=, i ê i X i X j ê j = d 2 nσ n i= j=, i d. e i X i X j e j [ ] [ ] X i X j mx i m θx i mx j m θx j [ ] e i X i X j mx j m θx j A.4 L n + S n + D n, A.5 were S n = d 2 ni= [ ] [ ] nj=, i nσ n X i X j mx i m θx i mx j m θx j and D n = 2 d 2 nσ n i= j=, i [ ] e i X i X j mx j m θx j. A.6 Define L n, S n and D n as te corresponding versions of L n, S n and D n involved in A.5 wit X i, Y i and θ being replaced by X i, Y i and θ respectively. 28

30 Lemma A.2. Suppose tat te conditions of Teorem 2.2i old. Ten sup x IR P L n x Φx + ρ x 2 φx = O P d. A.7 Lemma A.3. Suppose tat te conditions of Teorem 2.2i old. Ten under H 0 E [S n ] = O d and E [D n ] = o d, A.8 E [Sn] = O P d and E [Dn] = o P d, A.9 E [S n ] E [Sn] = O P d and E [D n ] E [Dn] = o P d in probability wit respect to te joint distribution of W n, were E [ ] = E[ W n ]. Lemma A.4. Suppose tat te conditions of Teorem 2.2ii old. Ten under H A.3. Proof of Teorem 2.: lim E [S E [D n ] n] = and lim n n E [S n ] = 0. A.3.. Proof of 2.0: Recall from 2.8 and A.5 A.6 tat were σ 2 n, σ 2 n and σ 2 n A.0 A. T n = L n + S n + D n σn + o P d, A.2 σ n T n = L n + Sn + Dn σn σ n + o P d, A.3 are as defined in 2.4, 2.7 and 2.9 respectively. In view of Assumption A.2 and Lemmas A. A.3, we may ignore any terms wit orders iger tan d and ten consider te following approximations: T n = L n + E [S n ] + o P d and T n = L n + E [Sn] + o P d. A.4 Let s = E[S n ] and s = E [S n]. We ten apply Lemmas A. and A.2 to obtain tat uniformly over x IR, P T n x P T n x = P L n x s + o P d A.5 = Φx s ρx s 2 φx s + o d and = P L n x s + o P d = Φx s ρx s 2 φx s + o P d. 29

31 A.3.2. Teorem 2.2i follows consequently from A.0 and A.5. Proof of 2.: In view of te definition tat P T n l α conclusion from Teorem 2.i tat P te proof of P T n lα convergence teorem. A.4. Proof of Teorem 2.2 It follows from Lemmas A. A.4 tat = α and te T n lα P T n lα = O P d, A.6 = α + O d follows unconditionally from te dominated α n = P T n l α H 0 = P L n l α S n + o P S n H 0 = P L n l α S n + o P S n H 0, A.7 αn = P T n lα H 0 = P L n lα S n + o P S n H 0 = P L n lα S n + o P S n H 0, A.8 β n = P T n l α H = P L n l α S n + o P S n H = P L n l α S n + o P S n H, A.9 βn = P T n lα H = P L n lα S n + o P S n H = P L n l α S n + o P S n H. A.20 Using Assumptions A.2iv and A.3, a Taylor expansion of m θ at θ 0 implies tat for sufficiently large n S n = C 0 m d + o P under H 0 and A.2 S n = n Cn 2 d + o P under H A.22 old in probability, were Cn 2 is as defined in Teorem 2.2ii. Te proof of Teorem 2.2 ten follows from A.5 and A.7 A.22. A.5. Proof of Teorem 2.3: Te proof follows from tat of Teorem 2.2. Te details are given in Appendix C below. 30

32 Appendix B B.. Derivation of 3.9 in te submission In te sequel we work wit te approximate power function βx, and discuss ow to find a maximum for tis function. Straigtforward calculations imply tat te first derivative may be written as 0 β x = φz α + d 3 x + d 4 x x 6 ϑ i x i, i=0 B. were te coefficients are given by ϑ 0 = c 2 n d 4 4, ϑ = 3z α d 3 4c 2 n, ϑ 2 = 2c 2 n d 3 d 3 4 d 2 4c 2 n 4 3z 2 α + d 4 4c, ϑ 3 = 3z α d 3 d 2 4c 2 n c 2 n z α d 4 5 z 2 α + 3z α d 3 4c, ϑ 4 = 2d 3 d 4 c 2 n + d 2 4c + d 2 4c 2 3zα, 2 ϑ 5 = 3z α d 2 3d 4 c 2 n + d 3 z α d 4 + 3z α d 2 4c z α d 4 c, ϑ 6 = c z 2 α + d 3 + 2d 4 c d 2 3c 2 n 2 + 3z 2 α 2d 3 3d 4 c 2 n, ϑ 7 = 3z α d 3 3c 2 n 3z α d 2 3d 4 c + z α d 3 c 5 z 2 α, ϑ 8 = c 2 d 4 3n 2c d 3 3d 4 c d 2 33z 2 α 4, ϑ 9 = 3c z α d 3 3, ϑ 0 = c d 4 3. B.2 Due to te complexity of te expressions of β x, it is very difficult to find suc an x 0 explicitly. Numerically, owever, it may be possible to find x ew = d ew suc tat x ew = arg max βx wit H nα = {x : α c min < αx < α + c min } B.3 x H nα wen c min is cosen as c min = α 0 for example. We now discuss ow to get to an explicit expression of an optimal bandwidt, by maximizing te power function over a subset of H n α. Note tat te minimal conditions tat lim n = 0 and lim n n d = imply tat tere is some large integer N suc tat for any arbitrarily large but finite c max, c max and n d c max for any n N. B.4 3

33 In our finite sample analysis, we ten define a new interval of te form ] H n = [ cmax d n, c max. B.5 Since Teorem 2.2i sows tat lim n α n = α olds in probability under te minimal conditions lim n = 0 and lim n n d =, we ave H n H n α at least for sufficiently large n. In order to represent ew explicitly we consider, quite naturally, solving te optimization problem: ew = arg max H n β n. B.6 To keep te notation simple, we still use te notation ew even wen H n may not be identical to H n α. We impose te very natural condition tat lim n n d = and obtain te following approximations: σn 2 = 2µ 2 2ν 2 2 udu σ0, 2 κ n = σn 3 d 4µ 3 2 ν µ n d d σ0 3 4µ3 2 ν B.7 Let a = 43 0µ 3 2 ν3 3σ 3 0 = udu cπ B.8 wit cπ = π 3 xdx π 2 xdx 3. We ten ave κ n a d. Let b = z 2 α a. From te condition lim n n d following simplified versions: were a 2 = b nc 2 n. l α r n = z α + γ n r n z α + b ncn 2 x = and using B.7 and B.8, we ten get te z α + a 2 x, B.9 β n Φl α r n κ n l α r n 2 φl α r n Φz α + a 2 x a x z α + a 2 x 2 φz α + a 2 x βx, B.0 32

34 Te natural condition lim n n d = implies tat we may take c 2 = 0, and subsequently d 4 = 0. Tis simplifies te expression of β x substantially. As a result, we ave ϑ i = 0 for 0 i 5. Tus, wit te following new coefficients: θ 0 = ϑ 6 = nc 2 n, θ = ϑ 7 = z α 5 z 2 αa a 2, θ 2 = ϑ 8 = 3z 2 α 4a a 2 2, θ 3 = ϑ 9 = 3z α a a 3 2, θ 4 = ϑ 0 = a 4 2a, B. equation B. simplifies to β x = φz α + a 2 x θ 0 + θ x + θ 2 x 2 + θ 3 x 3 + θ 4 x 4. B.2 Since φ is nonnegative, te equation β x = 0 is equivalent to θ 0 + θ x + θ 2 x 2 + θ 3 x 3 + θ 4 x 4 = 0. B.3 We find te zeros of equation B.3 by using existing results for a general quartic equation see for example, ttp://matworld.wolfram.com/quarticequation.tml. Note tat equation B.3 can be written as x 4 + r 3 x 3 + r 2 x 2 + r x + r 0 = 0 wit r i = θ i θ 4, i = 0,, 2, 3. B.4 Let x = u 4 r 3, p 2 = r r2 3, p = r 2 r 2r r3 3, p 0 = r 0 4 r r r 2r r4 3. B.5 We ten may eliminate x 3 from B.4 to obtain a standard equation of te form u 4 + p 2 u 2 + p u + p 0 = 0. B.6 Existing results immediately imply tat te zeros can be represented by u = 2 A + A 2, u 2 = 2 A A 2, B.7 u 3 = 2 A + A 3, u 4 = 2 A A 3, B.8 33

35 were A = p 2 + y, A 2 = A 3 = A 2 2p 2 2p A for A 0 2p y 2 4p 0 for A = 0 A 2 2p 2 + 2p A for A 0, 2p 2 2 y 2 4p 0 for A = 0, B.9 in wic y is a real root of te cubic equation y 3 + q 2 y 2 + q y + q 0 = 0, B.20 were q 2 = p 2, q = 4p 0 and q 0 = 4p 2 p 0 p 2. A real root of equation B.20 is y = q B + B 2, B.2 in wic B = R c + /3 D c and B 2 = R c /3 D c B.22 wit R c = 9q 2q 27q 0 2q2 3, Q c = 3q q and D c = Q 3 c + R 2 c. B.23 In order to evaluate te four roots in B.7 and B.8, note first of all tat te quantities R c and D c can be re expressed as R c = 72p 0p p 2 + 2p and Q c = 2p 0 p 2 2. B.24 9 For eac of te quantities involved we now provide te dominant terms, in view of te fact tat t n θ 0 = nc 2 n can be viewed as sufficiently large wen assuming tat t n as n. Note tat for finding te dominant term in te quantity B + B 2, some caution is needed since certain terms cancel out eac oter, and te dominant term comes from te second order terms in te quantities B and B 2. Indeed, we ave B = /3 Dc + R /3 c { /3 Dc + Dc 3 } R c Dc, B.25 34