Given Xi, let Yi = Xi − μ with E(Yi) = 0, Cov(Yi) = . Having relaxed normality, our robustness evaluation will be based on the following general multivariate model:
Yi = Ui, (4)
whereUi ∈RpwithE(Ui)=0,Cov(Ui)=I, and ∈Rp×pis a known constant matrix with T = A, T = > 0, where Ais any positive semi-definite matrix. Model (4) contains multivariate normality as a special case and is often used for general multivariate inference; see Ahmad (2017) for details and references.
The normality-based modification in Ahmad and Ahmed (2020) is essentially based on a single extra assumption, stated as Assumption3.1below, whereas, for the present case under Model (4), we additionally need Assumptions 1–3. Letλj, j = 1, . . . , pdenote the eigenvalues of, so thatν1, . . . , νp,νj = λj/p, denote those of=/p. Further, letE(Uij3)=γ1∈RandE(Uij4)=γ2+3,γ2∈R+, be the third and fourth moments of the elements ofU, respectively.
Assumption 1limp→∞#p
j=1νj =O(1).
Assumption 2Letγ1, γ2<∞, whereγ1, γ2are defined above.
Assumption 3limp→∞t r(AA)/t r(A⊗A)=0 withAa positive semi-definite matrix, whereand⊗denote Hadamard and Kronecker products, respectively.
Assumption 1 is inevitably needed under Model (4), since the computations involve second moments of quadratic forms. Assumption 2 puts a bound on the average of the scaled eigenvalues, νj. It is simple, effective, and commonly used in high-dimensional inference; moreover, it has an interesting consequence, limp→∞#p
s=1νj2=O(1)which will be referred to in the sequel. To see practical applicability of Assumption 2 and its consequence, letbe compound symmetric, which belongs to the group of spiked covariance structures, i.e.,=(1−ρ)I+ρJ withIas identity matrix,J=11,1a vectors of 1s, andρ∈R,−1/(p−1)≤ρ ≤1.
It can be easily shown thatt r(i)=O(pi),i=1,2, which satisfies the assumption and its consequence. Finally, Assumption 3 is mild, because the numerator is a much smaller term, in terms ofp, than the denominator.
Note that, in the computations below, in Model (4) will essentially appear as 1/2, so thatAwill be representing. For example, the traces in Assumption 3 can be considered forAas well as for. In this context, with normality relaxed, Assumption 3 controls the behavior of the moments of estimators that compose the modified statistic; see Theorem1below.
3.2 Statistic, Its Limit, and Robustness
For brevity, we only focus on the charts for an individual observations case. The case of subgroup-means follows similarly. The statistic for an individual observations case,Ti2, is given in Eq. (2), which, under the normality assumption and for fixed p, follows a beta distribution which provides the upper limit given after Eq. (2).
Alternatively, as an approximation for fixedpandn→ ∞, aχp2limit can also be used. These limits, however, are not applicable for a high-dimensional setup, i.e., whenpis large, and particularly forp n, mainly due to the singularity ofin Ti2. The modified form ofTi2 in Ahmad and Ahmed (2020), valid for thep n case, is defined as
Wi = n n−1
di 2
t r(), (5)
where = /p, di = Xi/√
p, i = 1, . . . , n, · denotes the Euclidean vector norm andt r(·)is the trace operator. It is shown that, under normality and Assumption 2,
Wi−E(Wi) σWi
−→D N (0,1), (6)
asn, p→ ∞, whereE(Wi)=1+oP(1)andσW2
i, a consistent estimator ofσW2
i, is defined below; see also Theorem2and Corollary 4 in the reference mentioned above. To motivate the evaluation of Wi in (5) for robustness under the general multivariate model, (4), we first note that the limit ofWi in Ahmad and Ahmed (2020) is obtained by using the consistency oft r()forn, p → ∞and showing thatWihas the same limit as that of
Ai = n n−1
di 2
t r(). (7)
Under normality assumption, E(Ai) = 1, V ar(Ai) = 2/f, where f = [t r()]2/t r(2). This helps determine the limit of Ai, and thus that of Wi, as χf2/f whose first two moments coincide with those ofAi.
When we replace normality with Model (4),E(Ai) = 1 remains same, using E( di 2)= [(n−1)/n]t r(), butV ar( di 2)differs. The following lemma, the proof of which follows easily using Theorem3, collects the moments of di 2under Model (4):
Lemma 1 For di 2defined above, we have, under Model (4),
E( di 2)= n−1 n t r() V ar( di 2)=
n−1 n
25
2t r(2)+M1 p2 6
Cov( di 2, dj 2)= 1 n2
5
2t r(2)+M1
p2 6
,
∀i=j,i, j=1, . . . , n, whereM1is defined in Theorem3.
Note that, the moments in Lemma1reduce to those in Ahmad and Ahmed (2020) under normality whenγ2 =0⇒M1 =0. Further, these moments help us define the moments of the traces involved in the limit of Wi, and particularly f given above, which in turn justifies the use of Assumption 3 involving the trace operator with Hadamard product. In this context, we exploit the equivalence of the limits ofAi andWi and consider di 2/t r(), where the scaling trace factor will make the terms involving Hadamard product vanish under the assumption. Finally, this last point will further help us, using the covariance part in Lemma1, obtain the multivariate limit of the vector of di 2. To approach this multivariate limit, write Aiin (7) as
Ai = n
n−1ai, (8)
where ai = di 2/t r(),i = 1, . . . , n. Asai are correlated, we are essentially seeking the distribution of the vectora=(a1, . . . , an)T. From Lemma1, it follows
thatE(ai)= [n/(n−1)]and it holds without Model (4) or any assumption. Further, forn → ∞and fixed p,E(ai) → ∞,V ar(ai) → 2/f andCov(ai, aj) → 0, without needing any assumptions. Now, when we let n, p → ∞, the so-called high-dimensional setup, then, under Assumption 2 and its consequence, f is uniformly bounded inp (using the moments in Theorem1below) so that, under Assumptions 1–3, the convergence forai, and therefore also that ofAi, or the vector A=(A1, . . . , An), may hold conveniently. For this, we use Lemma1for Eq. (8) and note, fori=j,i, j =1, . . . , n, that
E(Ai)=1 V ar(Ai)= 2
f 1
1+ M1
[t r()]2 p s=1
νj2 2
Cov(Ai, Aj)= 2
f · 1
(n−1)2 1
1+ M1 [t r()]2
p s=1
νj2 2
,
where νj are the eigenvalues of ; see the assumptions above. Now, under the consequence of Assumption 2 discussed above, limp→∞#p
s=1νj2 in V ar(Ai) andCov(Ai, Aj)is uniformly bounded where the fractional term involving M1 vanishes under Assumption 3. Note that, this vanishing limit can also be obtained by replacing Assumption 3 witht r(AA)/p2→0 asp→ ∞. But, we also need to assume a simultaneous rate of convergence ofpandn, i.e.,p/n→ c∈(0,∞) as n, p → ∞. Assumption 3, for which the denominator implies t r(A⊗A) = t r(⊗)= [t r()]2, helps us avoid any such(n, p)-relationships.
This argument implies that, even under Model (4), moments ofAi, and later of Wi, behave similarly as under normality so that a limit similar to that in Ahmad and Ahmed (2020) may be obtained. For the entire vectorA, we can now write
E(A)=1, Cov(A)= 2
fIn[1+O(1)] + [(Jn−In)O(n−2)], (9) whereIn is identity matrix, Jn = 1n1Tn with1n a vector of 1s, so that, using the limits forAi,
n,plim→∞Cov(A)= 2
fIn[1+o(1)], (10) under the assumptions. AsCov(A)is a diagonal (in fact, a spherical) matrix in the limit,Ai are asymptotically independent and a limit ofAfollows by the central limit theorem. Note that, for such vectors with correlated elements, the essential requirement for multivariate limit is that the covariances, Cov(Ai, Aj), or more precisely, the corresponding correlations, converge to the same fixed constant. This limit, in our case, is 0, makingCov(A)a diagonal matrix.
It follows from the above arguments that a limit of Wi under Model (4), similar to (6), follows if consistent and efficient estimators of the traces involved inf are defined for Model (4) under a high-dimensional asymptotic setup. The estimators in Ahmad and Ahmed (2020) are indeed non-parametrically defined and hence applicable under Model (4) as well. These estimators, oft r(),t r(2)and [t r()]2, respectively, are defined as
E1=t r() (11)
E2=η{(n−1)(n−2)t r(2)+ [t r()]2−nQ} (12) E3=η{2t r(2)+(n2−3n+1)[t r()]2−nQ}, (13) whereη=(n−1)/[n(n−2)(n−3)]andQ=#n
i=1qi2/(n−1)withqi = di 2. For an equivalentU-statistics formulation of the estimators, justifying their non- parametric nature, see Sect.8. Thus, to use them in the present context of robustness, we need efficient and consistent moments of these estimators under Model (4). They are given in the following theorem, proved in Sect.8, which reduce to Theorem 3 in Ahmad and Ahmed (2020) under normality.
Theorem 1 The estimators, E1, E2, and E3, defined in Eqs. (11)–(13), are unbi- ased fort r(),t r(2)and[t r()]2, respectively, with
V ar(E1)= 2
n−1t r(2)+M1 V ar(E2)= 4
P (n) 3
a(n)t r(4)+b(n)[t r(2)]2+2c(n)M1+d(n){6M2+M3}
−2e(n)M4
4 V ar(E3)= 4
P (n) 3
4t r(4)+f (n)[t r(2)]2+g(n)t r(2)[t r()]2+d(n)M12 +h(n)M1t r(2)+k(n)M1[t r()]2−2e(n)[M5+M4]4
, wherea(n)=2n3−12n2+21n−5,b(n)=n2−6n+11,c(n)=(n−1)(n−3)2, d(n) = (n−2)(n−3)/2,e(n) = n−3,f (n) = n2−6n+10,g(n) = (n− 2)(n−3)(2n−3),h(n)=(n−3)(2n−5),k(n)=(n−1)(n−2)(n−3),M1= γ1t r(AA),M2=γ1t r(A2A2),M3=γ12t r(AA)2,M4=γ22t r[(AA)A2], M5=γ22t r(AAA2). Further,V ar(Ei)and likewiseCov(Ei, Ej)are uniformly bounded byO(1/n),i, j=1,2,3,,i=j.
From Theorem1, the variances and covariances are uniformly bounded inpwhere the bounds only depend onn. This important consequence will help us arrive at the limit of the test statistic conveniently, which in turn ensuresf= E3/E2as a consistent estimator off, implying a consistent estimator of the test statistic, i.e.,
2/f. In summary, we have the following theorem, the proof of which is sketched in Sect.9. Note that, following the arguments around Eqs. (6) and (8), using the consistency oft r(), it immediately follows thatE(Wi) = 1+oP(1) → 1 for n, p→ ∞, same as under normality.
Theorem 2 GivenWiin Eq. (5), Model (4) and Assumptions 2–3. Then, asn, p→
∞
Wi−E(Wi) σWi
−→D N (0,1),
whereE(Wi) = 1,σW2
i = 2/fwithf= E3/E2 a consistent estimator off = [t r()]2/t r(2).
Although, Theorem2 deals with a univariate limit, it follows from the moments ofain Eq. (9) and the arguments around it that the multivariate limit of the vector W= (W1, . . . , Wn) can also be conveniently obtained, through a similar limit of A = (A1, . . . , An), so that the required limit in Theorem 2 follows simply as a marginal projection. In fact,Cov(A), forn, p→ ∞, has the same limit,[2/f]I[1+ o(1)], as that ofain Eq. (10). WithCov(Ai, Aj)=0, makingAi’s asymptotically independent and the variances uniformly bounded inp, the limit ofa, eventually of A, follows as
9f/2(A−E(A))−→D Nn(0,I),
as n, p → ∞. Likewise, the limit of Wfollows by replacingf with its (n, p)- consistent estimator,E2/E3. Theorem2extends the use of modifiedT2statistic for statistical control to a general model covering normality as a special case.
A very similar approach, with precisely the same limit, holds for the phase II chart of future observation and also for both types of charts for subgroup-means as well. In fact, as shown in Ahmad and Ahmed (2020), the convergence of the limit in case of subgroup-means is relatively better because the statistics are composed of averages. To avoid repetition, we shall not discuss these cases here, but they can be approached following the same steps as above.