Quadratic forms of the empirical processes for the two-sample problem for functional data

Abstract

The use of quadratic forms of the empirical process for the two-sample problem in the context of functional data is considered. The convergence of the family of statistics proposed to a Chi-squared limit is established under metric entropy conditions for smooth functional data. The applicability of the proposed methodology is evaluated in simulations and real data examples.

Appendix: Proof of results

Proof of Proposition 1:

For each random function X on J, and $L_g\in \mathcal{H}$, by the Cauchy–Schwarz inequality, we have

$$\begin{aligned} |L_g(X)|\le \sqrt{\int X^2(t)\hbox {d}t\int g^2(t)\hbox {d}t}\le \sqrt{M^2\int F^2(t)\hbox {d}t}. \end{aligned}$$

(20)

by hypothesis. Next, let $g^*_1,g^*_2,\dots ,g^*_l$ be a minimal set of functions such that, for every $g\in \mathscr {G}$, there exists $j\le l$ for which $\Vert g-g^*_j\Vert _{2,J}\le \epsilon$. Let $Q^*$ be a probability measure on $\mathcal{X}$. Then,

$$\begin{aligned} {Q}^*(L_g-L_{g^*_j})^2={Q}^*\left( \int X(t)(g-g^*_j)(t)\hbox {d}t \right) ^2\le M^2\epsilon ^2, \end{aligned}$$

(21)

again by the Cauchy–Schwarz inequality, and independently of the particular ${Q}^*$. It follows that, for an appropriate choice of a positive constant $\gamma$,

$$\begin{aligned} N_2(C\epsilon ,\mathcal{H})\le N_2(\epsilon \gamma ,\mathscr {G},\lambda ) \end{aligned}$$

and the result follows. $\square$

Proof of Proposition 2:

Under the null hypothesis of equality of distributions, the covariance matrices at the limiting vector of functions, $C({X,\mathbf {g}_\infty })$ and $C({Y,\mathbf {g}_\infty })$, are the same. We are writing $\mathbf {g}_\infty$, as in the statement of Proposition 1, for the vector of the $g_{j,\infty }$, $j\le k$. Now, by Pollard’s Uniform Entropy Condition that holds for $\mathcal{H}$, the Donsker property holds for the inner product class $\mathcal{H}$. This means that the empirical processes $\nu _X(L_g)$ and $\nu _Y(L_g)$, both indexed in $\mathscr {G}$, converge uniformly to a limiting Gaussian process and, by Dudley’s asymptotic equicontinuity condition and the assumed convergence of the functions in the vector $\tilde{\mathbf {g}}$,

$$\begin{aligned} \nu _X(L_{\tilde{\mathbf {g}}})\mathop {\rightarrow }\limits ^{\mathrm{(p)}}\nu _X(L_{\mathbf {g}_\infty }) \text{ and, } \text{ likewise, } \nu _Y(L_{\tilde{\mathbf {g}}})\mathop {\rightarrow }\limits ^{\mathrm{(p)}}\nu _Y(L_{\mathbf {g}_\infty }). \end{aligned}$$

(22)

Let $\tilde{C}({X,\mathbf {g}_\infty })$ be the sample covariance of the vectors $(L_{g_{1,\infty }}(X_i),\dots ,$ $L_{g_{k,\infty }}(X_i))$, $i\le m$, and define similarly $\tilde{C}({Y,\mathbf {g}_\infty })$ for the Y sample. It is clear that, under the null hypothesis, both $\tilde{C}({X,\mathbf {g}_\infty })$ and $\tilde{C}({Y,\mathbf {g}_\infty })$ are consistent estimators of $C({X,\mathbf {g}_\infty })$. Using the independence of the processes $\nu _X(L_g)$ and $\nu _Y(L_g)$, it follows that

$$\begin{aligned} \tilde{C}({\mathbf {g}_\infty })=\frac{\alpha ^2+\beta ^2}{m+n-2}\big ((m-1) \tilde{C}({X,\mathbf {g}_\infty })+(n-1)\tilde{C}({Y,\mathbf {g}_\infty })\big ), \end{aligned}$$

(23)

is a consistent estimator of the covariance matrix of the vector

$$\begin{aligned} \varphi =\alpha \, \nu _X(L_{\mathbf {g}_\infty })-\beta \,\nu _Y(L_{\mathbf {g}_\infty }). \end{aligned}$$

Since $L_{\mathbf {g}_\infty }$ is a fixed set of functionals, from the usual k-dimensional Central Limit Theorem and Slutzky’s theorem, it follows that

$$\begin{aligned} \varphi ^t \tilde{C}({\mathbf {g}_\infty })^{-1}\varphi \mathop {\rightarrow }\limits ^{\mathrm{(d)}}\chi ^2_k \end{aligned}$$

(24)

In view of (22), the same limit is obtained the if we replace $\varphi$ by $\gamma$ in (24). Thus, by Slutzky’s theorem again, it only remains to show that $\tilde{C}({X,\tilde{\mathbf {g}}})$ converges pointwise, in probability, to $C({X,\mathbf {g}_\infty })$. But using inequalities (20) and (21), it is easy to see that the covariance matrix $C(X,\tilde{\mathbf {g}})$ is a continuous function of the vector $\tilde{\mathbf {g}}$, with respect to the norm of $\mathcal{L}^2(J)$. Thus, by the triangle inequality, it suffices to have a uniform law of large numbers for the class

$$\begin{aligned} \mathcal{H}^{(2)}=\{L_g\,L_f:\> g,f\in \mathcal{G}\} \end{aligned}$$

(25)

and for the class $\mathcal{H}$ as well. Now, let $Q^*$ be a probability law on $\mathcal{X}$ and $g,g',f,f'$ functions in $\mathscr {G}$. Then, using Proposition 1, we get

$$\begin{aligned} Q^*|L_g\,L_f-L_{g'}\,L_{f'}|\le & {} Q^*(|L_g-L_{g'}||L_{f'}|)+Q^*(|L_f-L_{f'}||L_{g}|)\nonumber \\\le & {} C(Q^*(|L_g-L_{g'}|)+Q^*(|L_f-L_{f'}|)), \end{aligned}$$

a for the constant C in that Proposition. It follows that,

$$\begin{aligned} N_1(\epsilon ,\mathcal{H}^{(2)},Q^*)\le N^2_1 \left( \frac{\epsilon }{2C},\mathcal{H},Q^*\right) \le N^2_2 \left( \frac{\epsilon }{2C},\mathcal{H},Q^*\right) , \end{aligned}$$

and since the covering number $N_2({\epsilon }/{2C},\mathcal{H})$ satisfies Pollard’s uniform entropy condition (16), the same will hold for $\sup _{Q^*}N_1(\epsilon ,\mathcal{H}^{(2)},Q^*)$ (squaring the covering number does not affect the entropy condition), and this is more that enough for a Uniform Law of Large Numbers for $\mathcal{H}^{(2)}$. The argument for $\mathcal{H}$ is simpler and omitted, and the proof of Proposition 2 is complete. $\square$