In astamm/fdahotelling: Inference for Functional Data Analysis in R

Statistics for one-population tests

Let $\xi_1,\dots,\xi_n$ be a sample of $n$ random functions taking values in $L^2(T)$ where $T$ is a compact set of $\mathbb{R}^d$. Assume that these random functions share the same underlying law with mean function $\mu$. Denote by $\overline{\xi}$ the sample mean function, by $\mathcal{S}$ the sample covariance kernel and by $\widehat{V}$ the sample covariance operator. As part of the fdahotelling package, the following statistics are then available for making inference about the mean function $\mu$ [@Pini2015]:

Hotelling's $T^2$ statistic. The formal definition reads [@Secchi2013]: $$ T^2 = \max_{g \in \mathrm{Im}\widehat{V}} n \frac{ \left[ \int g(t) \left( \overline{\xi}(t) - \mu(t) \right) \mbox{d}t \right]^2 }{\int!!\int \mathcal{S}(t,s) g(t) g(s) \mbox{d}t \mbox{d}s}. $$
$L^2$-norm statistic. The formal definition reads [@Hall2002; @Hall2007; @Horvath2012]: $$ T_{L^2} = \int_T \left( \overline{\xi}(t) - \mu(t) \right)^2 \mbox{d}t. $$
Standardized $L^2$-norm statistic. The formal definition reads [@Hall2002; @Hall2007; @Horvath2012]: $$ T_{L_t^2} = \int_T \frac{ \left( \overline{\xi}(t) \phantom{}- \mu(t) \right)^2}{\mathcal{S}(t,t)} \mbox{d}t. $$
$L^1$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L^1} = \int_T \left| \overline{\xi}(t) - \mu(t) \right| \mbox{d}t. $$
Standardized $L^1$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L_t^1} = \int_T \frac{ \left| \overline{\xi}(t) - \mu(t) \right|} {\sqrt{\mathcal{S}(t,t)}} \mbox{d}t. $$
$L^\infty$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L^\infty} = \max_{t \in T} \left| \overline{\xi}(t) - \mu(t) \right|. $$
Standardized $L^\infty$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L_t^\infty} = \max_{t \in T} \frac{ \left| \overline{\xi}(t) \phantom{}- \mu(t) \right|}{\sqrt{\mathcal{S}(t,t)}}. $$

Statistics for two-population tests

Let $\xi_{11},\dots,\xi_{1n_1}$ and $\xi_{21},\dots,\xi_{2n_2}$ be two samples of respectively $n_1$ and $n_2$ random functions taking values in $L^2(T)$ where $T$ is a compact set of $\mathbb{R}^d$. Assume that, the random functions of each sample share the same underlying law with mean functions $\mu_1$ and $\mu_2$ respectively. Denote by $\overline{\xi}1$ and $\overline{\xi}_2$ the respective sample mean functions, by $\mathcal{S}{pooled}$ the sample pooled covariance kernel and by $\widehat{V}_{pooled}$ the sample pooled covariance operator. As part of the fdahotelling package, the following statistics are then available for making inference about the difference $\Delta\mu$ between the mean functions:

Hotelling's $T^2$ statistic. The formal definition reads [@Secchi2013]: $$ T^2 = \max_{g \in \mathrm{Im}\widehat{V}{pooled}} \frac{n_1n_2}{n_1+n_2} \frac{\left[ \int g(t) \left( \overline{\xi}_1(t) - \overline{\xi}_2(t) - \Delta\mu(t) \right) \mbox{d}t \right]^2 }{\int!!\int \mathcal{S}{pooled}(t,s) g(t) g(s) \mbox{d}t \mbox{d}s}. $$
$L^2$-norm statistic. The formal definition reads [@Hall2002; @Hall2007; @Horvath2012]: $$ T_{L^2} = \int_T \left(\overline{\xi}_1(t) - \overline{\xi}_2(t) - \Delta\mu(t)\right)^2 \mbox{d}t. $$
Standardized $L^2$-norm statistic. The formal definition reads [@Hall2002; @Hall2007; @Horvath2012]: $$ T_{L_t^2} = \int_T \frac{ \left( \overline{\xi}1(t) - \overline{\xi}_2(t) - \Delta\mu(t) \right)^2}{\mathcal{S}{pooled}(t,t)} \mbox{d}t. $$
$L^1$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L^1} = \int_T \left| \overline{\xi}_1(t) - \overline{\xi}_2(t) -\Delta\mu(t) \right| \mbox{d}t. $$
Standardized $L^1$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L_t^1} = \int_T \frac{ \left| \overline{\xi}1(t) - \overline{\xi}_2(t) - \Delta\mu(t) \right|}{\sqrt{\mathcal{S}{pooled}(t,t)}} \mbox{d}t. $$
$L^\infty$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L^\infty} = \max_{t \in T} \left| \overline{\xi}_1(t) - \overline{\xi}_2(t) - \Delta\mu(t) \right|. $$
Standardized $L^\infty$-norm statistic. The formal definition reads [@Hall2002]: $$ T_{L_t^\infty} = \max_{t \in T} \frac{ \left| \overline{\xi}1(t) - \overline{\xi}_2(t) - \Delta\mu(t) \right|}{\sqrt{\mathcal{S}{pooled}(t,t)}}. $$