Effective and flexible depth-based inference for functional parameters

Abstract

For hypothesis testing of functional parameters, given a functional statistic $T_n$ and a functional depth $D$ with respect to the distribution $P_n$ of $T_n$, we propose the depth value $DT_n \equiv D(T_n;P_n)$ as a test statistic, which we refer to as a depth statistic. In practice, its sampling distribution is approximated by a resampling method such as bootstrap. While achieving accurate sizes, a test based on the proposed depth statistic produces stronger power, as it remains sensitive even to subtle variations arising from complex functional patterns in the alternatives. Moreover, it is broadly applicable to a broad range of inference problems for functional parameters, including two-sample tests, analysis of variance, regression, etc. We provide its theoretical guarantee under mild assumptions along with examples of bootstrap methods and functional depths that satisfy these conditions. Its effectiveness is thoroughly investigated through numerical studies under two popular frameworks: (i) two-sample functional mean tests and (ii) mean response inference for function-on-function regression. The proposed depth statistic is illustrated with two data examples: Canadian weather and German electricity prices datasets.

Figures (5)

• Figure 1: The test statistic $T_{\mathrm{two},\bm{n}} = \bar{X}_1-\bar{X}_2$ (green), bootstrap statistics $T_{\mathrm{two},\bm{n}}^* = \bar{X}_1^* - \bar{X}_2^*$ (grey), and the true mean difference $\theta = \mu_1-\mu_2$ (red, cubic alternative). The black horizontal line at zero stands for the true mean difference in the null hypothesis $H_0:\mu_1=\mu_2$. The $n_k=25$ functional observations in each group $k$ are generated as described in \ref{['ssec_5_1']} under equal covariance scenario when the functional principal components are of NN-type. The details about the two-sample bootstrap procedure can be found in \ref{['ssec_2test']}.
• Figure 2: The curves with different shapes considered under strongest alternatives.
• Figure 3: Empirical rejection rates for testing $H_0:\mu(X_0) = \mathsf{E}[Y]$ in FoFR models, when $n=50$, $a_X = a_\varepsilon =2.5$, $b=1.5$, and the FPC scorse are of NN-type.
• Figure 4: The differences between the temperature curves averaged over two groups of years 1962--1981 and 1982--2011. The horizontal line at zero is colored in gray.
• Figure 5: Regressor (wind power in-feed, kWh) and response (electricity prices) curves in gray. Twenty randomly selected curves are marked in black. Three artificial new regressors $\{X_{0,l}: l \in \{\rm cub, sin, peak\}\}$ are indicated with red circles (cubic), blue triangles (sine), and green squares (peak).

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Remark 1
• Example 1: Functional ANOVA
• Example 2: Functional regression
• Theorem 3
• Theorem 4
• Theorem 5
• Example 3
• proof : Proof of \ref{['thmDepthStat']}