Abstract
This chapter discusses a few variational methods. Many statistical problems involve optimization of a functional defined over a specified function space. The theory of the calculus of variations plays a central role in functional analysis, similar to that of maxima and minima in differential calculus. The classical theory of optimization of functionals was mainly concerned with optimization of integrals involving unknown functions and their derivatives. The area of functional estimation has recently evolved as an important area of study in statistics. The classical theory of the calculus of variations was developed for solving problems in applications to mechanics. Many famous mathematicians such as Euler and Lagrange made fundamental contributions to the early development of the calculus of variations. During the past two centuries, the calculus of variations has evolved as an important branch of mathematics.
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