# Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

Authors: Wayan Somayasa
Journal: Journal of Applied Mathematics
 (2018)

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## Abstract

The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned valueα is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test.


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## Body

## 1. Introduction

Investigating the partial sums of least squares residuals has been shown to be reasonable and powerful tool for testing the adequacy of an assumed multivariate regression model; see Somayasa and et al. [1–4]. The development of the technique was motivated by the works proposed mainly for the purpose of detecting change in parameter as well as for detecting the existence of boundaries in univariate spatial regression; see [5–8] for references. The rejection region is constructed based on either Kolmogorov-Smirnov (KS) or Cramér-von Mises (CvM) functionals of the processes. It was shown in the literatures cited above that the limiting power function of the test appeared as a type of boundary crossing probability which has been involving shifted multidimensional Gaussian process.To understand the objective considered in this paper in more detail we present below brief review how such a kind of probability appears. LetZp≔Zii=1p be the p-dimensional set-indexed Brownian sheet defined on a probability space (Ω,B(Ω),P), say with sample paths in CpBG≔×i=1pCBG and the control measure P0, where P0 is a probability measure on G,BG, G≔Πk=1dak,bk, and ak<bk, for k=1,…,d. We refer the reader to [9, 10] for well documented notion of C(B(G)). In the literature of Gaussian process Zp is frequently called p- dimensional Gaussian white noise having P0 as the control measure, cf. [11], p. 13-14. Let W≔[f1,…,fm] and WHZp≔×i=1p[Sf1,…,Sfm], where for any g∈L2(P0,G), Sg is defined as Sg(A)≔∫AgdP0. Under mild condition, [1–3] showed after a suitable localization given to the regression function that the sequence of the partial sums of the least squares residuals obtained from the multivariate regression model (1)Yt=gt+Et,t≔t1,…,td∈Gconverges, when g∉Wp≔×i=1pW, to a p-dimensional signal plus noise model defined by (2)Y≔Σ-1/2prWHZp⊥Sg+prWHZp⊥∗Zp,Σ>0,where Σ>0 means that Σ is positive definite, and for A∈B(G), (3)prWHZp⊥SgA≔SgA-∑j=1mfj,giL2P0,Gi=1pSfjA,prWHZp⊥∗ZpA≔ZpA-∑j=1m∫GfjtdZiti=1pSfjA,provided that {f1,…,fm} builds an ONB of W in L2P0,G∩BVHG. Thereby Sg≔Sgii=1p and BVH(G) is the space of functions on G with bounded variation in the sense of Hardy. It is worth mentioning that the notion of BVH(G) is a direct extension of the definition of BVHa1,b1×a2,b2 formulated in [12] to higher dimensional space. Here the notation Zit stands for ZiΠk=1dak,tk, cf. [8]. Throughout the paper Σ-1/2prWHZp⊥Sg will be denoted by φg and prWHZp⊥∗Zp by Wf,P0, for the sake of brevity. It was established in [1–3] that Wf,P0 is a projection of Zp onto the orthogonal complement of WHZp which is a finite dimensional subspace of the so-called reproducing kernel Hilbert Space (RKHS) of Zp, denoted by HZP, given by (4)HZP≔h∣∃l=lii=1p∈L2pP0,G,hA=∫AltP0dt,with L2p(P0,G)≔×i=1pL2(P0,G). In the literatures mentioned above the process Wf,P0 is called the p-dimensional set-indexed residual partial sums limit process with the control measure P0. Hence, the process Zp itself and the p-dimensional set-indexed Brownian pillow Zp0=(Zi0)i=1p, with Zp0(A)≔Zi0(A)-P0AZi0(G), are special cases of Wf,P0 that correspond to W=[f0] and W=[f1], respectively, with f0≡0 and f1≡1. The control measure P0 appears in the process determines the design under which the experiment was constructed; see [4] for detail.It was shown by using the well-known continuous mapping theorem that the limiting power functions of sizeα KS and CvM type tests for testing the hypothesis (5)H0:g∈WpagainstH1:g∉Wpare given, respectively, by the following complicated formulas:(6)ΨWf,P0t~Wf,P0,φg≔PsupA∈BGφgA+Wf,P0ARp≥t~Wf,P0ΥWf,P0q~Wf,P0,φg≔P∫GφgA+Wf,P0ARp2dA≥q~Wf,P0,where ∥·∥Rp2 stands for the Euclidean norm, whereas t~Wf,P0 and q~Wf,P0 are constants that satisfy ΨWf,P0t~Wf,P0,0=ΥWf,P0q~Wf,P0,0=α. By the difficulty of the computation of t~Wf,P0 as well as q~Wf,P0 and the power of the test as the dimension of the experimental region and p get large, the implementation of the test in practice becomes restricted. Approximation by Monte Carlo simulation has been proposed in [1–3]. Some attempts of establishing concrete computation procedure by generalizing the principal component approach proposed, e.g., by MacNeill [5, 6] and Stute [13] for some univariate Gaussian processes on a line, have led us to incorrect result.Since analytical computation ofΨWf,P0(t~Wf,P0,φg) and ΥWf,P0(t~Wf,P0,φg) is impossible, it is the purpose of the present paper to establish approximation procedure for that functions. As suggested in [14], p. 315, and [15], p. 423-424, studying the power function is of importance to be able to evaluate the performance of the test especially their rate of decay to α. Therefore in this paper we investigate the upper and lower bounds for (6) by considering the result for the univariate Brownian sheet and Brownian pillow presented in Janssen [17] and Hashorva [18, 19]. Upper and lower bound for the power function of goodness-of-fit test involving multiparameter Brownian process have been studied by Bass [20]. The RKHS of Wf,P0 is crucial for our results. By Theorem 4.1 in [11] (factorization theorem) if there exists a family mA≔mAii=1p∈L2pP0,G:A∈BG, such that the covariance function of Wf,P0 admits the representation (7)CovWf,P0A1,Wf,P0A2=∫GmA1⊤tmA2tP0dt=mA1,mA2L2pP0,G,then the corresponding RKHS is given by (8)HWf,P0≔h∣∃l=lii=1p,hA=∫GmAitliti=1pP0dt.Furthermore, the inner product and the corresponding norm on HWf,P0 are denoted, respectively, by ·,·HWf,P0 and ·HWf,P02. For examples, the RKHS of Zp0 is given by (9)HZPo≔h∣∃l∈L2pP0,G,hA=∫AltP0dt,s.t.hG=h∅=0,with (10)h1,h2HZp0=l1,l2L2pP0,G,hHZp0=lHZp0,such that hj(A)=∫Alj(t)P0(t),j=1,2.The rest of the present paper is organized as follows. In Section2 we derive the upper and lower bounds for the power functions of KS and CvM tests by applying the Cameron-Martin translation formula of the multivariate process Wf,P0. The rate of decay of the power to α is also discussed. Alternative method of obtaining the bounds of the power function is proposed in Section 3. In Section 4 we propose Neyman-Pearson test which is a most powerful test. The comparison of the rate of decay of the obtained power to α with those of the KS and CvM tests is also investigated. Justification of the result is also studied in Section 5 by simulation. The paper is closed with a concluding remark in Section 6.


## 2. Rate of Decay of the Power of Tests

Our final goal in this section is to obtain an expression for the rate of decay of bothΨWf,P0t~Wf,P0,φg and ΥWf,P0q~Wf,P0,φg to the preassigned number α∈(0,1) representing the size of the test. First we derive their upper and lower bounds by generalizing the method proposed in [21] concerning bounds for the probability of shifted event; see also Theorem 7.3. in [11] for comparison. Second, we apply the technique studied in [17] to get the result. As reported in [17] and the references cited therein, they studied the upper and lower bounds for the power of signal detection test by applying Cameron-Martin density formula for a shifted measure. The rate of decay was obtained by means of mean value theorem.Throughout this work letP be the probability distributions of Wf,P0 and let Ph be a probability measure on CpBG, defined by(11)PhB≔PB-h,∀B∈CpBG.Then the Cameron-Martin density of Ph with respect to P for any h∈HWf,P0 is given by(12)dPhdPx=exp⁡Lh,x-12hHWf,P02,for almost allx∈CpBG,where L is a bilinear form, such that (13)CovLh1,Wf,P0,Lh2,Wf,P0=h1,h2HWf,P0,∀h1,h2∈HWf,P0.This general formula can be obtained by extending the formula for the univariate model presented either in [20], Theorem 5.1 of [11], and [17] or [22] to higher dimensional set-indexed Gaussian processes.The following theorem is already well known in the literatures mentioned above; however the proof is given only for the case of Gaussian random vector inRn with zero mean and identity covariance matrix (canonical Gaussian Euclidean random vector); see [21] and [11], p. 53. In this paper we present again the theorem especially for the process Wf,P0 on Cp(B(G)). Although the result for Wf,P0 is straightforward based on that of [11, 21], to give information on how the inequality for higher dimensional set-indexed Gaussian process was derived, we insist to present the proof of the theorem; see the appendix of this work.Theorem 1 (Li and Kuelbs [21] and Lifshits [11]).
LetE be any subset of Cp(B(G)) and r(E)∈R be any constant, such that r(E)=Φ-1(P(E)). Then for any h∈HWf,P0, it holds true that(14)ΦrE-Lh,hhHWf,P0≤PE-h≤ΦrE+Lh,hhHWf,P0,where Φ is the cumulative distribution function of the standard normal distribution.The following corollary which gives an expression regarding the rate of decay ofP(E-h) to P(E), for any E⊂Cp(B(G)) and h∈HHWf,P0, is an immediate implication of Theorem 1. Rate of decay describes how fast the distance between P(E-h) and P(E) vanishes, cf. [17–19].Corollary 2.
LetE be an arbitrary subset of Cp(B(G)) and r(E)∈R be any constant, such that r(E)=Φ-1(P(E)). Then under the assumption 0<Lh,h, we have, for any h∈HWf,P0, (15)PE-h-PE≤12πLh,hhHWf,P0.Proof.
We apply the technique of proving Lemma 5 of [17]. By (14) presented in Theorem 1 and by using the symmetry of Φ, it holds that(16)PE-h-PE≤ΦrE+Lh,hhHWf,P0-ΦrE=Lh,hhHWf,P0ϕηfor some mean value η∈rE,rE+Lh,h/hHWf,P0, where ϕ is the probability density function of N(0,1). Since max⁡ϕt:-∞<t<∞=1/2π, then we have (17)PE-h-PE≤Lh,hhHWf,P0ϕη≤12πLh,hhHWf,P0.Conversely, by the inequality Φr(E)-L(h,h)/hHWf,P0≤PE-h of (14), we can derive the following result: (18)ΦrE-Lh,hhHWf,P0-PE≤PE-h-PE⇔ΦrE-Lh,hhHWf,P0-ΦrE≤PE-h-PE⇔-Lh,hhHWf,P0ϕκ≤PE-h-PE,for some mean value κ∈r(E)-L(h,h)/hHWf,P0,r(E). Since Lh,h>0, by the preceding result, we get (19)PE-h-PE≥-Lh,hhHWf,P0ϕκ≥-12πLh,hhHWf,P0,which establises the proof.When the model is eitherh1+Zp, with h1∈HZp, or h2+Zp0, with h2∈HZp0, such that hj(A)=∫AljuP0du, for A∈B(G) and j=1,2, then (20)Lh1,h1=∫Gl1⊤ul1uP0du=h1,h1HZp=h1HZp2>0Lh2,h2=∫Gl2⊤ul2uP0du=h2,h2HZp=h2HZp02>0.Hence, when we are dealing with the p-dimensional set-indexed Brownian sheet and p-dimensional set-indexed Brownian pillow, Inequality (14), respectively, becomes (21)ΦrE-h1HZp≤PE-h1≤ΦrE+h1HZp,ΦrE-h2HZp0≤PE-h2≤ΦrE+h2HZp0.The corresponding rate of decays can be obtained respectively as follows: (22)PE-h1-PE≤12πh1HZp,PE-h1-PE≤12πh1HZp0.In light of the preceding results, we can state the upper and lower bounds as well as the rate of decays for the powerΨWf,P0(t~Wf,P0,φg) and ΥWf,P0t~Wf,P0,φg, when Wf,P0 is given by either Zp or Zp0. Let E be a subset of CpBG, defined by (23)E≔x∈CpBG:supA∈BGxARp≥t~Wf,P0;then for φg∈HWf,P0⊂CpBG, we get (24)E-φg=x-φg:x∈E=x-φg:supA∈BGxARp≥t~Wf,P0=x∈CpBG:supA∈BGφgA+xARp≥t~Wf,P0.Since P is the distribution of Wf,P0, then ΨWf,P0(t~Wf,P0,φg) is equivalent to (25)PE-φg=Px∈CpBG:supA∈BGφgA+xARp≥t~Wf,P0.Analogously, let (26)F≔x∈CpBG:∫GxARp2dA≥q~Wf,P0.Then ΥWf,P0(q~Wf,P0,φg)=P(F-φg). Thus by considering these two representations we have on the basis of Theorem 1 and Corollary 2 the following summary concerns the bounds for the power of the KS and CvM type tests.Corollary 3.
Suppose thatφg∈HWf,P0; then, for α∈(0,1), it holds that(27)ΦΦ-1α-φgHWf,P0≤ΨWf,P0t~Wf,P0,φg≤ΦΦ-1α+φgHWf,P0ΦΦ-1α-φgHWf,P0≤ΥWf,P0t~Wf,P0,φg≤ΦΦ-1α+φgHWf,P0.Furthermore, we have simple formulas for the rate of decay of ΨWf,P0(t~Wf,P0,φg) and ΥWf,P0t~Wf,P0,φg to α(28)ΨWf,P0t~Wf,P0,φg-α≤12πφgHWf,P0ΥWf,P0t~Wf,P0,φg-α≤12πφgHWf,P0,where in the context of model check, the norm of φg related to the process Zp and Zp0 is given by (29)φgHWf,P02=Σ-1/2prWHZp⊥SgHZp2=prWp⊥Σ-1/2gL2pP0,G2=Σ-1/2g-Σ-1/2∑j=1mfj,giL2P0,Gi=1pfjL2pP0,G2.Corollary3 says that the rate of decay or convergence of the power function to α in the case of Zp as well as Zp0 depends on the norm of the trend. A Model with small norm trend leads to faster decay. Conversely, a model with large norm trend results in slower decay. For both models, the norm can be concretely calculated. It is clear that both tests achieve their sizes as the trends vanish. Indeed the work of Samorodnitsky [23] can be incorporated in the estimation of ΨWf,P0(λ,φg), for any large real number λ. In Section 5 we demonstrate by simulation the behavior of the power functions of the KS and CvM tests as summarized in Corollary Corollary 3 to give empirical study regarding the rate of decay of the power functions.


## 3. Alternative Approaches

In this section other formulas for the upper and lower bounds of the power of KS and CvM tests involving thep-dimensional set-indexed Brownian sheet and pillow models are derived. Our results are obtained by generalizing the approach proposed in that studied in [18, 19] who confined the investigation to one-dimensional Kolmogorov type boundary noncrossing probability involving the so-called univariate ordinary Brownian sheet and pillow.To simplify the notation we restrict the attention to the case of two-dimensional experimental regionG=[a1,b1]×[a2,b2]⊂R2.Theorem 4.
Let the ONB{f1,…,fm} of W be in L2(P0,G)∩BVH(G) and let φg=Σ-1/2prWHZp⊥Sg, such that prW⊥Σ-1/2g are constant on the boundary of G. Then for the Zp model it holds that(30)1-1-αLZp≤ΨZpt~Zp,φg≤1-1-ΨZp12t~Zp,0UZpwhere (31)LZp≔exp⁡2t~Zp∑i=1pΔGWi-12φgHZp2UZp≔exp⁡-2t~Zp∑i=1pΔGWi-12φgHZp2,where ΔGWi≔Wi(b1,b2)-Wi(b1,a2)-Wi(a1,b2)+Wi(a1,a2) and Wi is the ith component of prWp⊥Σ-1/2g, which is given by (32)Wi≔∑k=1pσik∗gk-∑j=1mfj,∑k=1pσik∗gkL2P0,Gi=1pfj∈BVHGwith σik∗ denoting the (i,k)th element of Σ-1/2, say, for i,k=1,…,p. Furthermore, for the Zp0 model, we have(33)1-1-αLZp0≤ΨZp0t~Zp0,φg≤1-1-ΨZp012t~Zp0,0UZp0,where (34)LZp0≔exp⁡-12φgHZp02≕UZp0.Proof.
By using a rule for the probability of complement, we get for theZp model (35)ΨZpt~Zp,φg=1-Px∈CpBG:supA∈BGφgA+xARp<t~Zp=1-Px∈CpBG:φgA+xARp<t~Zp,∀A∈BG,where by using transformation of variables, it can be further expressed as (36)Px∈CpBG:φgA+xARp<t~Zp,∀A∈BG=∫CpBG1φgA+xARp<t~Zp,∀A∈BGPdx=∫CpBG1yARp<t~Zp,∀A∈BGPφgdy.Next, Cameron-Martin formula (12) for the p-dimensional set-indexed Brownian sheet implicates (37)∫CpBG1yARp<t~Zp,∀A∈BGPφgdy=∫CpBG1yARp<t~Zp,∀A∈BG×exp⁡∫GprWp⊥Σ-1/2g⊤tdyt-12φgZp2Pdy=∫CpBG1yARp<t~Zp,∀A∈BG×exp⁡∑i=1p∫GWitdyit-12φgZp2Pdy.Since yARp<t~Zp means -t~Zp<yiA<t~Zp, then under the indicator 1yARp<t~Zp0,∀A∈BG we get by recalling integration by parts formula on G, cf. [24, 25] and the assumption that Wi is constant throughout the boundary of G; for the Zp model we get (38)∫CpBG1yARp<t~Zp,∀A∈BG×exp⁡∑i=1p∫GWitdyit-12φgHZp2Pdy≤exp⁡2t~Zp∑i=1pΔGWi-12φgHZp2PsupA∈BGZpHRp<t~Zp=exp⁡2t~Zp∑i=1pΔGWi-12φgHZp21-ΨZpt~Zp,0.Thus, the lower bound in (30) is established. To prove the upper bound, we start with the following inequality: (39)ΨZpt~Zp,φg≤1-PφgA+ZpARp≤12t~Zp,∀A∈BG.By applying the similar technique as that used in deriving the preceding result and the implication (40)yARp≤12t~Zp⇒-t~Zp≤yiA≤t~Zp∀i=1,…,p,under the indicator 1yA≤1/2t~Zp,∀A∈B(G) we have, by the integration by parts, the following inequality:(41)PφgA+ZpARp≤12t~Zp,∀A∈BG=∫CpBG1yA≤12t~Zp,∀A∈BG×exp⁡∑i=1p∫GWitdyit-12φgHZp2Pdy≥∫CpBG1yA≤12t~Zp,∀A∈BG×exp⁡-2t~Zp∑i=1pΔGWi-12φgHZp2Pdy=exp⁡-2t~Zp∑i=1pWib1,b2-12φgHZp21-ΨZp12t~Zp,0,completing the proof for the Zp model. To prove the lower and upper bounds (33) for the Zp0 model, we start with the equality (42)1-ΨZp0t~Zp0,φg=∫CpBG1yARp<t~Zp0,∀A∈BG×exp⁡∑i=1p∫GWitdyit-12φgHZp02Pdy.Next by the integration by parts and the assumption that φg∈HZp0 and Wi are constant on the boundary of G, we have under 1yARp<t~Zp0,∀A∈B(G) and the fact yi(b1,b2)=yib1,a2=yia1,b2=yia1,a2=0 that (43)∫GWitdyit<t~Zp0∫GdWit,s=0.Hence, 1-ΨZp0t~Zp0,φg≤exp⁡-1/2φgHZp021-α, establishing the lower bound in (33). The similar argument as that used in the case of Zp model can be applied in deriving the upper bound of ΨZp0t~Zp0,φg as follows:(44)1-ΨZp0t~Zp0,φg=∫CpBG1yARp<t~Zp0,∀A∈BG×exp⁡∑i=1p∫GWitdyit-12φgHZp02Pdy≥∫CpBG1yARp≤12t~Zp0,∀A∈BG×exp⁡∑i=1p∫GWitdyit-12φgHZp02Pdy≥exp⁡-12φgHZp021-ΨZp012t~Zp0,φg,establishing the proof.Now we can derive other formulas for the rate of decay ofΨZp(t~Zp,φg) and ΨZp0(t~Zp0,φg) to α by applying the similar method as that utilized in deriving the formula in Corollary 3. However by Theorem 4 we lead to computationally more complicated results.Corollary 5.
Under the condition of Theorem7, it holds true that(45)-eρ2tZp∑i=1pΔGWi+12φgHZp2≤ΨZpt~Zp,φg-α≤eζ2tZp∑i=1pΔGWi+12φgHZp2,for some mean values (46)ζ∈-2tZp∑i=1pΔGWi-12φgHZp2+ln⁡1-ΨZp12t~Zp,0;ln⁡1-α,ρ∈ln⁡1-α;2tZp∑i=1pΔGWi-12φgHZp02+ln⁡1-α.In particular, if the mean values ρ and ζ are taken to be the same, then (47)ΨZpt~Zp,φg-α≤eζ2tZp∑i=1pΔGWi+12φgHZp2.Proof.
From Inequality (30), we have, by applying the mean value theorem, (48)ΨZpt~Zp,φg-α≤exp⁡ln⁡1-α-exp⁡-2tZp∑i=1pΔGWi-12φgHZp2+ln⁡1-ΨZp12t~Zp,0=eζln⁡1-α1-ΨZp1/2t~Zp,0+2tZp∑i=1pΔGWi+12φgHZp2≤eζ2tZp∑i=1pΔGWi+12φgHZp2,for some mean value ζ laid in the interval (49)-2tZp∑i=1pΔGWi-12φgHZp2+ln⁡1-ΨZp12t~Zp,0;ln⁡1-α.Conversely, based on the lower bound formula (30), we get (50)ΨZpt~Zp,φg-α≥exp⁡ln⁡1-α-exp⁡2tZp∑i=1pΔGWi-12φgHZp2+ln⁡1-α=-exp⁡2tZp∑i=1pΔGWi-12φgHZp2+ln⁡1-α-exp⁡ln⁡1-α=-eρ2tZp∑i=1pΔGWi-12φgZp2≥-eρ2tZp∑i=1pΔGWi+12φgHZp2for some mean value ρ within the interval(51)ln⁡1-α;2tZp∑i=1pΔGWi-12φgHZp02+ln⁡1-α.Thus it can be concluded that ΨZpt~Zp,φg-α is laid in the following closed interval: (52)-eρ2tZp∑i=1pΔGWi+12φgHZp2;eζ2tZp∑i=1pΔGWi+12φgHZp2.In particular, if the mean values ρ and ζ are taken to be the same, then (53)ΨZpt~Zp,φg-α≤eζ2tZp∑i=1pΔGWi+12φgHZp2,establishing the proof.Analogously, from (33), we get (54)ΨZp0t~Zp0,φg-α≤exp⁡ln⁡1-α-exp⁡ln⁡1-ΨZp012t~Zp0,0-12φgHZp02=eτln⁡1-α1-ΨZp01/2t~Zp0,0+12φgHZp02≤eτexp⁡12φgHZp02,for some τ∈ln1-ΨZp01/2t~Zp0,0-1/2φgHZp02;ln⁡1-α. Conversely, (55)ΨZp0t~Zp0,φg-α≥exp⁡ln⁡1-α-exp⁡ln⁡1-α-12φgHZp02=eιexp⁡12φgHZp02,for some ι∈ln⁡1-α-1/2φgHZp02,ln⁡1-α. Particularly, for ι=τ, we get (56)ΨZp0t~Zp0,φg-α≤eτexp⁡12φgHZp02.Thus, we proved the following corollary.Corollary 6.
Under the condition of Theorem7, we have (57)eιexp⁡12φgHZp02≤ΨZp0t~Zp0,φg-α≤eτexp⁡12φgHZp02,for some τ and ι specified above. If ι and τ are chosen to be the same, then (58)ΨZp0t~Zp0,φg-α≤eτexp⁡12φgHZp02.


## 4. Comparison to Neyman-Pearson Test

Our aim in this section is to establish nonrandomized Neyman-Pearson test for the hypothesis defined in the preceding section. It is well known in the literatures of test theory that Neyman-Pearson test constitutes a most powerful (MP) test for simple hypotheses; see, e.g., Theorem 3.2.1 in [15]. If some criterion is satisfied, the test can be extended to a uniformly most powerful (UMP) test for composite hypotheses. In this section the behavior of the power function including the rate of decay to α will be investigated and compared to those of KS and CvM type tests studied in the preceding section.LetV be a linear subspace generated by a set of known and orthonormal regression functions f1,…,fm,fm+1,…,fq⊂L2P0,G∩BVHG including W=[f1,…,fm]. In this section we consider the hypothesis H0:g∈Wp against H1:g∈Vp instead of H0:g∈Wp against H1:g∉Wp. The former is actually the common frame work of model check for multivariate regression in which one is testing whether or not g∈Wp while observing g∈Vp; see [26] for reference. Suppose there exist g1∈Wp and g2∈Vp∩Wp⊥, such that g=g1⊕g2. It is enough to consider the simple hypotheses(59)H0:g2≡0vs.H1:g2≡f0,for somef0∈Vp∩Wp⊥.Hence the p-dimensional set-indexed partial sums process of the residuals is given by Y=Wf,P0, when H0 is true; otherwise Y=Σ-1/2prWHZp⊥Sf0+Wf,P0.The following theorem presents an MP test of sizeα for testing (59). Here we exhibit again the application of Cameron-Marin density formula of the shifted measure Pφf0 with respect to P, for φf0=Σ-1/2prWHZp⊥Sf0. Recently, [4] investigated the asymptotic optimality of a test for the mean vector in multivariate regression by means of Neyman-Pearson test.Theorem 7.
Supposeφf0∈HWf,P0. Neyman-Pearson test of size α for testing (59) will reject H0, if and only if (60)Lφf0,Y≥Φ-11-αφf0HWf,P0.Furthermore, suppose ΓWf,P0:Vp→0,1 is the corresponding power function of the test. Then the value of the power, evaluated at any f∈Vp, is given by (61)ΓWf,P0f=ΦLφf0,φfφf0HWf,P0-Φ-11-α,f∈Vp∩Wp⊥,and otherwise, ΓWf,P0(f)=α.Proof.
Letχ0(Y) and χ1(Y) be the density of Pφf0 with respect to P under H0 and H1, respectively. By Theorem 3.2.1 in [15], an MP test of size α for testing (59) will reject H0, if and only if χ0(Y)/χ1(Y)≤k, for a constant k such that (62)Pω∈Ω:χ0Yωχ1Yω≤kH0=α.Since χ0(Y)=1 and χ1(Y)=exp⁡Lφf0,Y-1/2φf0HWf,P02, then by recalling the fact Lφf0,Wf,P0~N0,φf0HWf,P02, we get (63)Pω∈Ω:χ0Yωχ1Yω≤kH0=α⇔Pexp⁡-Lφf0,Y+12φf0HWf,P02≤kH0=α⇔PLφf0,Y≥-ln⁡k+12φf0HWf,P02H0=α⇔PLφf0,Wf,P0≥-ln⁡k+12φf0HWf,P02=α⇔PN0,1≥-ln⁡k+1/2φf0HWf,P02φf0HWf,P0=α⇔-ln⁡k+12φf0HWf,P02=Φ-11-αφf0HWf,P0,establishing the rejection region of the test. Next, we compute the power function for any f∈Vp∩Wp⊥. By the definition of ΓWf,P0 and by the symmetry of Φ, we have (64)ΓWf,P0f=PLφf0,Y≥Φ-11-αφf0HWf,P0g2≡f=PLφf0,φf+Wf,P0≥Φ-11-αφf0HWf,P0=PLφf0,Wf,P0≥Φ-11-αφf0HWf,P0-Lφf0,φf=1-ΦΦ-11-α-Lφf0,φfφf0HWf,P0=ΦLφf0,φfφf0HWf,P0-Φ-11-α.The last formula results in ΓWf,P0(f)=α, when f vanishes. The proof of the theorem is complete.The test presented in Theorem7 depends on the choice of f specified under H1. For example, if we consider H1:g2≡f1, for some f1∈Vp∩Wp⊥, then H0 is rejected at level α, if (65)Lφf1,Y≥Φ-11-αφf1HWf,P0.This means that the test cannot be extended as a uniformly most powerful (UMP) test for the composite alternative H1:g2∈Vp∩Wp⊥. It is also not a UMP test for more specific one-sided alternatives H1:g2>0 or H1:g2<0.As discussed in the preceding section, we are also interested in investigating the rate of decay ofΓWf,P0(f) to α=ΓWf,P0(0). Toward this topic the result of Theorem 7 leads us to the following important corollary. The proof is left since it can be handled by using the similar technique as in the proof of Corollary 3.Corollary 8.
Letf be an element of Vp∩Wp⊥, such that φf∈HWf,P0 and L(φf0,φf)>0. Then for every presigned α∈(0,1), it holds that(66)ΓWf,P0f-α≤12πLφf0,φfφf0HWf,P0.In the case Lφf0,φf<0, we have (67)ΓWf,P0f-α≤12πL-φf0,φfφf0HWf,P0,where L-φf0,φf≔-Lφf0,φf>0.Corollary8 states that how fast the power function ΓWf,P0(f) decays to α depends on some value determined by L(φf0,φf) whose structure is influenced by the type of Wf,P0. For comparison study suppose that the simple hypothesis (59) is tested using the KS or CvM type test. Then by virtue of Corollary 2, we have (68)ΨWf,P0t~Wf,P0,φf-α≤12πLφf,φfφfHWf,P0ΥWf,P0t~Wf,P0,φf-α≤12πLφf,φfφfHWf,P0.Thus, in contrast to Corollary 8, the rate of decay of the KS and CvM type tests does not depend on f0 at all. Consequently, compared to Neyman-Pearson test, KS and CvM test cannot detect whether to take f larger or less than f0 in order to have faster or slower decay.The result presented throughout this section will become more visible when we look at the model involvingp-dimensional set-indexed Brownian sheet or pillow. For example suppose we observe the model Y=Σ-1/2prWHZp⊥Sg2+Zp0, for testing (59). Then H0 is rejected at level α, if (69)Lφf0,Y≥Φ-11-αΣ-1/2prWp⊥f0L2pP0,G,where for the p-dimensional set-indexed Brownian pillow; we have (70)Lφf0,Y=Σ-1/2prWp⊥f0L2pP0,G2+∫GΣ-1/2prWp⊥f0⊤dZp0,Furthermore, we get, for fixed f0∈Vp∩Wp⊥, (71)Lφf0,φfφf0HZp0=φf0,φfHZp0φf0HZp0=Σ-1/2prWHZp⊥Sf0,Σ-1/2prWHZp⊥SfHZp0φf0HZp0=prWp⊥Σ-1/2f0,prWp⊥Σ-1/2fL2pP0,GprWp⊥Σ-1/2f0L2P0,G≤prWp⊥Σ-1/2f0L2pP0,GprWp⊥Σ-1/2fL2pP0,GprWp⊥Σ-1/2f0L2pP0,G≤Σ-1/2RpprWp⊥RpfL2pP0,G,where the first inequality appears by Holder’s inequality, whereas the second follows by the fact that Σ-1/2 and prWp⊥ represent continuous linear transformations on L2p(P0,G); therefore they are uniformly bounded, cf. [27], p. 26-27. Since ·L2p(P0,G) is continuous on the closed subset Vp∩Wp⊥, then ·L2p(P0,G) is bounded on Vp∩Wp⊥. Thus there exists M>0, such that M is the uniform upper bound for ΓZp0f-α. It is clear that M is also the uniform upper bounds of ΨZp0t~Zp0,φf-α as well as ΥZp0t~Zp0,φf-α.


## 5. Simulation Study

In this section we investigate the behavior ofΨZp0(t~Zp0,φf) and ΥZp0(q~Zp0,φf) with respect to their lower and upper bounds derived in Corollary 3. The simulated model is represented by the trend plus noise process (72)φgA+Z20A,A∈B0,1×0,1where Z20 is the two-dimensional Brownian pillow and (73)φgA=Σ-1/2∫Aρt+s,0⊤λ2dt,ds,A∈B0,1×0,1,ρ∈R.Such a model appears as the limit process of the two-dimensional set-indexed partial sums processes of the residuals of two variate regression model (74)Y1,Y2⊤=g1,g2⊤+ε1,ε2⊤for testing whether or not a constant model holds true. That is, we test the hypothesis that (75)H0:g1,g2⊤∈W2vs.H1:g1,g2⊤∉W2,where W=f1, with f1t,s=1, for t,s∈G≔0,1×0,1. For fixed n≥1, the n×n×2 arrays of observation are generated from the model (76)Ylk1Ylk2=2+ρln+ρkn3+0ln+0kn+εlk1εlk2,1≤l,k≤n,according to an experimental design given by a regular lattice with n×n points on G. Let f2(t,s)=t and f3(t,s)=s, for (t,s)∈G; then we equivalently have (77)Ylk1Ylk2=2f1ln,kn+ρf2ln,kn+ρf3ln,kn3f1ln,kn+0f2ln,kn+0f3ln,kn+εlk1εlk2,for 1≤l,k≤n. Hence, if ρ=0, then the observations are from the model assumed under H0. Otherwise, they support H1. In this simulation, the random vector (εlk1,εlk2)⊤ is generated independently from the two-dimensional centered normal distribution with the covariance matrix given by (78)Σ=6.26-0.50-0.506.26so that we have after some computations (79)Σ-1=0.160770.012840.012840.16077,Σ-1/2=0.400640.016030.016030.40064.Now, the norm of φg for the matrix Σ can be computed concretely as (80)φgHZ20=∫0,1×0,1ρ2t+s,0Σ-1t+s,0dtds=ρ0.16077∫0,1×0,1t2+2ts+s2dtds=0.43309ρ.The simulation results using a sample of size50×50 with 1000 runs are exhibited in Table 1 and Figure 1 for α=0.05. The figures presented in Table 1 represent the values of the power functions of the KS and CvM tests together with the associated values of the lower (L) and upper (U) bounds evaluated at each given value of ρ utilizing the formulas given in Corollary 3, where in this case (81)L=ΦΦ-1α-0.43309ρ,U=ΦΦ-1α+0.43309ρ.It is shown that the values of L are never exceeding the corresponding powers. Likewise, the values of U are also never preceding those of the corresponding power functions as suggested by the theory. Figure 1 presents the graphs of L (dotdash line), U (dotted line), ΨZp0(t~Zp0,φg) (smoothed line), and ΥZp0(q~Zp0,φg) (dashed line) scattered together in one panel. It can be seen that the curves of the power functions are laid within a band formed by the paired curve of L and U as they should be.Table 1
The numerical upper and lower bounds forΨZp0t~Zp0,φg and ΥZp0q~Zp0,φg. The size of the test is α=0.05.
ρ L Ψ Z p 0 t ~ Z p 0 , φ g Υ Z p 0 t ~ Z p 0 , φ g U 0 0.04990 0.06100 0.05500 0.04990 1 0.01886 0.04900 0.05100 0.11280 2 0.00602 0.05600 0.06500 0.21809 3 0.00162 0.05700 0.07100 0.36483 4 0.00037 0.07400 0.07600 0.53486 5 6.9e-05 0.07500 0.09700 0.69867 6 1.1e-05 0.10600 0.13000 0.82988 7 1.5e-06 0.12600 0.14600 0.91724 8 1.6e-07 0.15700 0.21000 0.96561 9 1.5e-08 0.14700 0.19600 0.98787 10 1.2e-09 0.20300 0.27200 0.99639 11 7.3e-11 0.21000 0.28300 0.99909 12 3.9e-12 0.27500 0.35000 0.99981 13 1.7e-13 0.29600 0.39900 0.99997 14 6.4e-15 0.32800 0.45500 0.99999Figure 1
Upper (dotted line) and lower (dashed line) bounds forΨZp0(t~Zp0,φf) (solid line), with α=0.05.


## 6. Concluding Remark

We have established the upper and lower bounds for the boundary crossing probability involving multivariate trend plus noise model. Our results give important contributions not only in the area of statistics, but also in other disciplines such as in finance mathematics and in statistical physics, where such probability model is also frequently encountered. It is important to note that the Cameron-Martin translation formula is valid if the trend function laid in the RKHS of the corresponding Gaussian process. In practice this is not always the case. Therefore further research must be conducted to be able to handle the problem appears in such situation.



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*Source: 2071861-2018-09-02.xml*