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An FPCA mean test uses functional principal component analysis to test a zero expectation in a Hilbert space. For independent and identically distributed observations $X_i$ with finite second moment, let $({\lambda }_j,{\varphi }_j)$ be the eigenvalues and orthonormal eigenvectors of the covariance operator. Choose a fixed positive integer $K$ and, as a sufficient separation assumption, require ${\lambda }_1>\cdots >{\lambda }_K>{\lambda }_{K+1}$ and ${\lambda }_K>0$. With consistent estimators of these eigenvalues and eigenvectors, the statistic isUnder the null hypothesis $\mathbb{E}{X}_i=0$, the central limit theorem gives joint convergence of $\sqrt{n}⟨\overline{X},{\varphi }_j⟩/\sqrt{{\lambda }_j}$ to $K$ independent standard normal random variables. Statistical consistency and Slutsky's theorem justify replacing the population eigenvectors and eigenvalues by their estimates. Consequently $T_{n,K}\stackrel{d}{\to }{\chi }_K^2$. The fixed-dimensional projected two-sample version has the same calibration.^[ref].

The positivity condition prevents division by zero; the spectral gap makes the estimated directions stable. Repeated eigenvalues can be handled through an entire retained eigenspace, provided its boundary remains separated. A choice $K=K_n$ growing with $n$ needs additional approximation theory. A fixed $K$ can miss a nonzero expectation lying entirely in the omitted orthogonal directions, so the test's simplicity comes with a choice about which deviations matter.

A functional mean test asks whether the expectation $\mu$ of a Hilbert-space-valued random variable is zero. Suppose $X_1,\dots ,X_n$ are independent and identically distributed in a Hilbert space with a countable dense subset $H$, with $\mathbb{E}\Vert X_1{\Vert }^2<\infty$. Write $C$ for their covariance operator and $\overline{X}=n^{-1}{\sum }_i{X}_i$. Under the null hypothesis $\mu =0$, the Hilbert-space central limit theorem giveswhere $G$ is a centered Gaussian random element in $H$, the ${\lambda }_j$ are the eigenvalues of $C$, and the $Z_j$ are independent standard normal random variables. The second assertion follows by the continuous mapping theorem. Since ${\sum }_j{\lambda }_j=\mathbb{E}\Vert X_1-\mu {\Vert }^2<\infty$, the nonnegative series is finite almost surely. Limits given by Gaussian random elements for functional sample means and the corresponding weighted squared-norm laws also extend to suitable dependent observations.^[ref]

The eigenvalues determine the calibration: the limit is generally a weighted sum of random variables with a chi-squared distribution, rather than a single ordinary chi-squared distribution. The squared norm measures deviations across all covariance operator directions. An FPCA mean test instead retains finitely many eigenvectors and divides their squared coordinates by their eigenvalues, producing a different statistic with a simpler limiting calibration.

A functional time series is a time series whose observation at each time is a whole function, often modeled as a random element of the Hilbert space $L^2[0,1]$. For example, $X_j(s)$ might be the temperature profile within day $j$, with $s$ marking time of day. Dependence across days is different from dependence between nearby positions within one curve.

For a simple model, let $X_j(s)=A_j+B_{j}s$, where $(A_j,B_j)$ is a stationary bivariate time series. Its mean curve is $\mu (s)=\mathbb{E}{A}_j+s\mathbb{E}{B}_j$. The covariance between curves at lag $h$ is a kernel $C_h(s,t)=\mathrm{Cov}(X_0(s),X_h(t))$, retaining both within-curve coordinates.

Estimating $\mu$ by $n^{-1}{\sum }_{j=1}^n{X}_j$ resembles scalar averaging, but its uncertainty depends on serial covariance operators rather than only the variance of one curve. Treating every densely measured point as an independent observation can severely overstate precision.^[ref]

Lajos Horváth works on mathematical statistics, time series and econometrics. His University of Utah profile identifies these research interests.^[ref]

His joint paper with Piotr Kokoszka and Ron Reeder studies comparisons of mean curves in a functional time series. Its Hilbert space formulation yields Gaussian limits and a test based on the distance between two estimated mean functions, retaining dependence within each sample.^[ref]

Piotr Kokoszka studies functional data analysis, time series and spatial statistical methods. His Colorado State University homepage describes inference for observations that can themselves be curves.^[ref]

With Lajos Horváth and Ron Reeder, he developed estimation and statistical hypothesis testing for mean curves under temporal dependence. Averaging many curves requires a covariance accounting for correlation between successive observations; treating the curves as independent can misstate uncertainty.^[ref]

Ron Reeder coauthored Estimation of the Mean of Functional Time Series and a Two-Sample Problem by Lajos Horváth and others with Lajos Horváth and Piotr Kokoszka. Their problem treats successive daily curves as observations of a functional time series. The paper develops a normal approximation for their mean and a comparison of two mean curves that allows dependence across days.^[ref]

Let $X_1,\dots ,X_n$ and $Y_1,\dots ,Y_n$ be two independent samples of independent and identically distributed Hilbert-space-valued random variables, with finite second moments and covariance operators $C_X,C_Y$. To test equality of their expectations, use the pooled covariance operator $C=(C_X+C_Y)/2$. This choice works even when the two population covariance operators differ: under the null hypothesis, the covariance operator of $\sqrt{n/2}(\overline{X}-\overline{Y})$ is exactly $C$.

Let $({\hat{\lambda }}_j,{\hat{\varphi }}_j)$ estimate the eigenvalues and orthonormal eigenvectors of $C$, for example from the average of the two within-sample empirical covariance operators. For fixed $K$, positive retained eigenvalues, a separated retained eigenspace and suitable statistical consistency of these estimates,The proof applies the Hilbert-space central limit theorem to the difference of sample means, then standardizes its first $K$ coordinates and uses Slutsky's theorem. A sufficient simple-eigenvalue condition is ${\lambda }_1>\cdots >{\lambda }_K>{\lambda }_{K+1}$ with ${\lambda }_K>0$.^[ref]

For unequal sample sizes $n,m$, the scaling becomes $nm/(n+m)$ and the relevant population covariance operator is $\{m{C}_X+n{C}_Y\}/(n+m)$. Thus a sample-size-weighted pooled estimate cannot automatically be reused for unequal sizes and unequal covariance operators: its weights must match the variance of the difference being tested.