Modeling functional data: a test procedure

Abstract

The paper deals with a test procedure able to state the compatibility of observed data with a reference model, by using an estimate of the volumetric part in the small-ball probability factorization which plays the role of a real complexity index. As a preliminary by-product we state some asymptotics for a new estimator of the complexity index. A suitable test statistic is derived and, referring to the U-statistics theory, its asymptotic null distribution is obtained. A study of level and power of the test for finite sample sizes and a comparison with a competitor are carried out by Monte Carlo simulations. The test procedure is performed over a financial time series.

A: Appendix—theoretical results

1.1 A.1 Proof of Proposition 1 and some further details

The proof is based on similar arguments as in Corollary 5.1 in Ferraty et al. (2012).

1.1.1 Bias

Compute the mean of the estimator:

$$\begin{aligned} \mathbb {E}\left[ \widehat{\phi }_{n}\left( h\right) \right]= & {} \mathbb {E} \left[ \dfrac{1}{n\left( n-1\right) }\sum _{i=1}^{n}\sum _{j\ne i}\mathbb {I} _{\left\{ \left\| X_{i}-X_{j}\right\| \le h\right\} }\right] \\= & {} \dfrac{1}{n\left( n-1\right) }\sum _{i=1}^{n}\sum _{j\ne i}\mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right] \\= & {} \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right] . \end{aligned}$$

Using the law of total expectation, one has

$$\begin{aligned} \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right] =\mathbb {E}\left[ \ \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }|X_{2}\right] \ \right] = \mathbb {E}\left[ \varphi _{X_{2}}\left( h\right) \right] . \end{aligned}$$

Thanks to (H2) and the constraint (2) it follows

$$\begin{aligned} \mathbb {E}\left[ \varphi _{X_{2}}\left( h\right) \right] =\left( \mathbb {E}\left[ f\left( X_{2}\right) \right] +o\left( 1\right) \right) \phi \left( h\right) =\phi \left( h\right) +o\left( \phi \left( h\right) \right) . \end{aligned}$$

(7)

Combining the results one gets

$$\begin{aligned} \mathbb {E}\left[ \widehat{\phi }_{n}\left( h\right) \right] =\phi \left( h\right) +o\left( \phi \left( h\right) \right) \end{aligned}$$

(8)

that allows to conclude that the estimator is unbiased when $h\rightarrow 0$ and $n\rightarrow \infty$.

1.1.2 Variance

By using classical results on U-statistics (see e.g. Lehmann 1999, Theorem 6.1.1) it follows

$$\begin{aligned} Var \left( \widehat{\phi }_{n}\left( h\right) \right) =\dfrac{4\left( n-2\right) }{n\left( n-1\right) }\sigma _{1}^{2}\left( h\right) +\dfrac{2}{ n\left( n-2\right) }\sigma _{2}^{2}\left( h\right) \end{aligned}$$

where

• $\sigma _{1}^{2}\left( h\right) = Var \left( \mathbb {E} \left[ \mathbb {I} _{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} } |X_{2}\right] \right) = Var \left( \varphi _{X_{2}}\left( h\right) \right)$,

• $\sigma _{2}^{2}\left( h\right) = Var \left( \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right)$.

Consider the second term $\sigma _{2}^{2}\left( h\right)$, one has

$$\begin{aligned} Var \left( \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right) =\mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right] \left( 1-\mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right] \right) . \end{aligned}$$

Since (see Eq. (7))

$$\begin{aligned} \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }\right] =\mathbb {E}\left[ \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }|X_{2}\right] \right] = \mathbb {E}\left[ \varphi _{X_{2}}\left( h\right) \right] =\phi \left( h\right) +o\left( \phi \left( h\right) \right) \end{aligned}$$

it follows

$$\begin{aligned} \sigma _{2}^{2}\left( h\right) =\left( \phi \left( h\right) +o\left( \phi \left( h\right) \right) \right) \left( 1-\phi \left( h\right) -o\left( \phi \left( h\right) \right) \right) =\phi \left( h\right) +o\left( \phi \left( h\right) \right) . \end{aligned}$$

About the first term $\sigma _{1}^{2}\left( h\right)$, one has

$$\begin{aligned} Var \left( \varphi _{X_{2}}\left( h\right) \right) =\mathbb {E}\left[ \varphi _{X_{2}}^{2}\left( h\right) \right] -\mathbb {E}^{2}\left[ \varphi _{X_{2}}\left( h\right) \right] \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}^{2}\left[ \varphi _{X_{2}}\left( h\right) \right] =\phi ^{2}\left( h\right) +o\left( \phi ^{2}\left( h\right) \right) . \end{aligned}$$

Since

$$\begin{aligned} \varphi _{X_{2}}^{2}\left( h\right)= & {} \mathbb {E}^{2}\left[ \mathbb {I} _{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }|X_{2}\right] \le \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }^{2}|X_{2}\right] \\= & {} \mathbb {E}\left[ \mathbb {I}_{\left\{ \left\| X_{2}-X_{1}\right\| \le h\right\} }|X_{2}\right] =\varphi _{X_{2}}\left( h\right) \end{aligned}$$

then

$$\begin{aligned} \mathbb {E}\left[ \varphi _{X_{2}}^{2}\left( h\right) \right] \le \mathbb {E} \left[ \varphi _{X_{2}}\left( h\right) \right] =\phi \left( h\right) +o\left( \phi \left( h\right) \right) . \end{aligned}$$

Concluding, there exist two finite positive constants $c_{1}$ and $c_{2}$ depending on h, such that

$$\begin{aligned} Var \left( \widehat{\phi }_{n}\left( h\right) \right) \le \dfrac{4\left( n-2\right) }{n\left( n-1\right) }c_{1}\left( h\right) +\dfrac{2}{n\left( n-2\right) }c_{2}\left( h\right) \end{aligned}$$

and then

$$\begin{aligned} Var \left( \widehat{\phi }_{n}\left( h\right) \right) =O\left( \dfrac{1}{n} \right) . \end{aligned}$$

(9)

1.1.3 Asymptotic distribution

Using classical asymptotic results on U-statistics (see e.g. Lehmann 1999, theorems 3.3.1 and 6.1.2) since $0<\sigma _{1}^{2}\left( h\right) <\infty$ and $0<\sigma _{2}^{2}\left( h\right) <\infty$ (thanks to (H3) and results above), one gets, for $h\rightarrow 0$, and $n\rightarrow \infty$,

$$\begin{aligned} \dfrac{\widehat{\phi }_{n}\left( h\right) -\mathbb {E}\left[ \widehat{\phi }_{n} \left( h\right) \right] }{\sqrt{ Var \left( \widehat{\phi }_{n}\left( h\right) \right) }}\overset{d}{\longrightarrow }\mathcal {N}\left( 0,1\right) . \end{aligned}$$

(10)

It is worth to noting that combining (10) with (8) we get, as $n\rightarrow \infty$,

$$\begin{aligned} \dfrac{\widehat{\phi }_{n}\left( h\right) -\phi \left( h\right) }{\sqrt{ Var \left( \widehat{\phi }_{n}\left( h\right) \right) }}\overset{d}{\longrightarrow }\mathcal {N}\left( 0,1\right) . \end{aligned}$$

(11)

1.2 A.2 Proof of Propositions 2, 4, 5 and 6

For what concerns Proposition 2, the result is a consequence of asymptotic normality of the estimator $\widehat{\phi }_{n}$ and its unbiasness (8). In particular, for any $k=1,\dots ,m$, under the marginal null hypothesis $H_{0}^{k}$, when $h\rightarrow 0$ from (11) it follows:

$$\begin{aligned} \dfrac{\left( \widehat{\phi }_{n}\left( h\right) -\phi _{0}\left( h\right) \right) ^{2}}{ Var \left( \widehat{\phi }_{n}\left( h\right) \right) }\overset{ d}{\longrightarrow }\mathcal {\chi }^{2}\left( 1\right) \ \ \ \ \ \text { as } n\rightarrow \infty . \end{aligned}$$

About the statements in Propositions 4, 5 and 6, the estimators consistency, their asymptotic unbiasness and normality, together with consistency of the jackknife variance estimators allow to invoke Slustky’s Theorem and lead to the results (see e.g. Maesono 1995).

1.3 Proof of Proposition 3

Recalling that a multiple test is consistent w.r.t. the marginal alternatives $H_{1}^{1},\dots ,H_{1}^{m}$ if each marginal test is consistent (see e.g. Alt 2005), then we have to prove that, under each $H_{1}^{k}$ one has

$$\begin{aligned} D_{k}^{2}\longrightarrow +\infty \ \ \ \ \ \text { in probability as } n\rightarrow \infty . \end{aligned}$$

Observe that for any k,

$$\begin{aligned} D_{k}^{2}=\left( \dfrac{\widehat{\phi }_{n}\left( h\right) -\phi _{1}\left( h\right) }{ Var \left( \widehat{\phi }_{n}\left( h\right) \right) }+\dfrac{\phi _{1}\left( h\right) -\phi _{0}\left( h\right) }{ Var \left( \widehat{\phi } _{n}\left( h\right) \right) }\right) ^{2}=\left( A_{n}+B_{n}\right) ^{2}. \end{aligned}$$

On the one hand, under each $H_{1}^{k}$, thanks to (11), the sequence of random variables $A_{n}$ is bounded in probability. On the other hand, under each $H_{1}^{k}$, thanks to (9), the deterministic sequence $B_{n}$ diverges with n. The conclusion follows immediately.