A prediction problem in $L^2 (w)$

Abstract

Abstract. For a nonnegative integrable weight function w on the unit circle T , we provide an expression for p = 2, in terms of the series coefficients of the outer function of w, for the weighted L p distance inf f T |1 − f | p wdµ, where µ is the normalized Lebesgue measure and f ranges over trigonometric polynomials with frequencies in [{. . . , −3, −2, −1}\{−n}]The problem is open for p = 2.

Extrapolation Problem for Stationary Sequences with Missing Observations

“…Let the functions determined by formulas (31), (32) be bounded. Then the functions f 0 (λ), g 0 (λ) determined by (37), (38) are the least favorable densities in the class D u v × D ε if they determine a solution to the optimization problem (27). The function h(f 0 , g 0 ) calculated by (13) is the minimax spectral characteristic of the optimal linear estimate of the functional Aξ.…”

Duals of random vectors and processes with applications to prediction problems with missing values

“…There are many situations where the cardinality of M is two or more; see Pourahmadi et al (2007), Box and Tiao (1975), Brubacher and Wilson (1976), Damsleth (1980), Abraham (1981), and there are several ad hoc methods for interpolating the missing values. For example, Brubacher and Wilson (1976) …”

Filtering of multidimensional stationary sequences with missing observations

“…In the papers by Luz & Moklyachuk (2012 results of investigation of the estimation problems for functionals which depend on the unknown values of stochastic sequences with stationary increments are described. Results of investigations of the prediction problem for stationary stochastic sequences with missing observations are presented in the papers by Bondon (2002Bondon ( , 2005, Kasahara, Pourahmadi & Inoue (2007, 2009). In papers by Moklyachuk & Sidei (2015 results of investigations of the interpolation, extrapolation and filtering problems for stationary stochastic sequences and processes with missing observations are proposed.…”