Supplementary Material for “Time-to-event Analysis with Unknown Time Origins via Longitudinal Biomarker Registration” Tianhao Wang, Sarah J. Ratcliffe, and Wensheng Guo ∗ November 29, 2021 Introduction In this supplementary material we provide the proofs of model identifiability and the the- orems of the paper. We denote Pf = ∫ f (x)dP (x), P n f = 1 n ∑ n i=1 f (X i ) and G n f = √ n(P n − P )f . We use the symbol . to denote that the left hand side is bounded above by a constant times the right hand side, & to denote that the left hand side is bounded below by a constant times the right hand side, and ≍ to denote that both . and & apply. We use |X | to denote the absolute value of X if X is a scalar, or the square-root of the largest eigenvalue of XX ⊤ if X is a vector or matrix. Let K = {τ 1 , ..., τ K } be a set of partition points of [0, 1] with max 1<j ≤K |τ j − τ j −1 | = O(K −1 ). Let S (K,p) denote the space of polynomial splines of order p ≥ 1 with the knots sequence K as defined in the Definition 4.1 of Schumaker (1981). Let K µ = {τ 1 , ..., τ Kμ } be a set of partition points of [0, 1] with max 1<j ≤K μ |τ j − τ j −1 | = O(K −1 µ ); and K ψ = {t 1 , ..., t K ψ } denote a set of partition points of [a, b] with max 1<j ≤K ψ |t j − t j −1 | = O(K −1 ψ ). Define S (K µ ,p µ ) and S (K ψ ,p ψ ) similarly as S (K,p). For notation consistency, let K β = K and p β = p. Define B β = S (K β ,p β ), and the sieve spaces B µ n = S (K µ ,p µ ), G ψ n = S (K ψ ,p ψ ), * Tianhao Wang is Assistant Professor, Department of Neurological Sciences, and Faculty S- tatistician, Rush Alzheimer’s Disease Center, Rush University Medical Center, Chicago, IL 60612 (tianhao wang@rush.edu); Sarah J. Ratcliffe is Professor, Division of Biostatistics, Departmen- t of Public Health Sciences, University of Virginia School of Medicine, Charlottesville, VA 22908 (sratcliffe@virginia.edu); and Wensheng Guo is Professor, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA 19104 (wguo@pennmedicine.upenn.edu). This research was supported by National Institutes of Health grants R01–GM104470 and R01–DK117208. The authors are grateful to the Editor, Associate Editor, and three referees for their insightful comments that greatly improved the paper. 1