Monotone function, such as growth function and cumulative distribution function, is often a study of interest in statistical literature. In this dissertation, we propose a nonparametric least-squares method for estimating monotone functions induced from stochastic processes in which the starting time of the process is subject to interval censoring. We apply this method to estimate the mean function of tumor growth with the data from either animal experiments or tumor screening programs to investigate tumor progression. In this type of application, the tumor onset time is observed within an interval. The proposed method can also be used to estimate the cumulative distribution function of the elapsed time between two related events in human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome (AIDS) studies, such as HIV transmission time between two partners and AIDS incubation time from HIV infection to AIDS onset. In these applications, both the initial event and the subsequent event are only known to occur within some intervals. Such data are called doubly interval-censored data. The common property of these stochastic processes is that the starting time of the process is subject to interval censoring. A unified two-step nonparametric estimation procedure is proposed for these problems. In the first step of this method, the nonparametric maximum likelihood estimate (NPMLE) of the cumulative distribution function for the starting time of the stochastic process is estimated with the framework of interval-censored data. In the second step, a specially designed least-squares objective function is constructed with the above NPMLE plugged in and the nonparametric least-squares estimate (NPLSE) of the mean function of tumor growth or the cumulative distribution function of the elapsed time is obtained by minimizing the aforementioned objective function. The theory of modern empirical process is applied to prove the consistency of the proposed NPLSE. Simulation studies are extensively carried out to provide numerical evidence for the validity of the NPLSE. The proposed estimation method is applied to two real scientific applications. For the first application, California Partners' Study, we estimate the distribution function of HIV transmission time between two partners. In the second application, the NPLSEs of the mean functions of tumor growth are estimated for tumors with different stages at diagnosis based on the data from a cancer surveillance program, the SEER program. An ad-hoc nonparametric statistic is designed to test the difference between two monotone functions under this context. In this dissertation, we also propose a numerical algorithm, the projected Newton-Raphson algorithm, to compute the non– and semi-parametric estimate for the M-estimation problems subject to linear equality or inequality constraints. By combining the Newton-Raphson algorithm and the dual method for strictly convex quadratic programming, the projected Newton-Raphson algorithm shows the desired convergence rate. Compared to the well-known iterative convex minorant algorithm, the projected Newton-Raphson algorithm achieves much quicker convergence when computing the non- and semi-parametric maximum likelihood estimate of panel count data.

(i) We prove that intermediate Banach spaces A, B with respect to arbitrary Hilbert couples H, K are exact interpolation iff they are exact K-monotonic, i.e. the condition f0∊A and the inequality K(t,g0;K)≤K(t,f0;H), t>0 imply g0∊B and ||g0||B≤||f0||A (K is Peetre's K-functional). It is well-known that this property is implied by the following: for each ρ>1 there exists an operator T : H→K such that Tf0=g0, and K(t,Tf;K)≤ρK(t,f;H), f∊H0+H1, t>0.Verifying the latter property, it suffices to consider the "diagonal" case where H=K is finite-dimensional. In this case, we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem, it is shown that the statement remains valid when substituting ρ=1. (ii) A new proof is given to a theorem of W. F. Donoghue which characterizes certain classes of functions whose domain of definition are finite sets, and which are subject to certain matrix inequalities. The result generalizes the classical Löwner theorem on monotone matrix functions, and also yields some information with respect to the finer study of monotone functions of finite order. (iii) It is shown that with respect to a positive concave function ψ there exists a function h, positive and regular on ℝ+ and admitting of analytic continuation to the upper half-plane and having positive imaginary part there, such that h≤ψ≤ 2h. This fact is closely related to a theorem of Foiaş, Ong and Rosenthal, which states that regardless of the choice of a concave function ψ, and a weight λ, the weighted l2-space l2(ψ(λ)) is c-interpolation with respect to the couple (l2,l2(λ)), where we have c≤√2 for the best c. It turns out that c=√2 is best possible in this theorem; a fact which is implicit in the work of G. Sparr. (iv) We give a new proof and new interpretation (based on the work (ii) above) of Donoghue's interpolation theorem; for an intermediate Hilbert space H* to be exact interpolation with respect to a regular Hilbert couple H it is necessary and sufficient that the norm in H* be representable in the form ||f||*= (∫[0,∞] (1+t-1)K2(t,f;H)2dρ(t))1/2 with some positive Radon measure ρ on the compactified half-line [0,∞]. (v) The theorem of W. F. Donoghue (item (ii) above) is extended to interpolation of tensor products. Our result is related to A. Korányi's work on monotone matrix functions of several variables.