Error bounds in divided difference expansions. A probabilistic perspective. (English) Zbl 1091.65023
The authors discuss error bounds for the remainder term and also for the leading term in the divided difference expansion of a function. These complete the results of M. S. Floater [J. Approximation Theory 122, 1–9 (2003; Zbl 1016.41016)]. Using a probabilistic representation of divided difference in terms of mathematical expectation of \(n\)-th derivative of the given function, having as argument a random variable equals to a linear combination of uniform order statistics, a probabilistic Taylor’s formula is proved. This formula is an extension of the classical Taylor’s formula with remainder in integral form.
Based on probabilistic Taylor’s formula, the authors derive exact values, as well as upper and lower bounds for the variance of the standardized random variable, which enter in the remaider term and in leading term in divided difference expansion. A probabilistic interpretation of the results, is given.
Based on probabilistic Taylor’s formula, the authors derive exact values, as well as upper and lower bounds for the variance of the standardized random variable, which enter in the remaider term and in leading term in divided difference expansion. A probabilistic interpretation of the results, is given.
Reviewer: Iulian Coroian (Baia Mare)
MSC:
| 65D25 | Numerical differentiation |
| 41A80 | Remainders in approximation formulas |
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 62J10 | Analysis of variance and covariance (ANOVA) |
| 65C60 | Computational problems in statistics (MSC2010) |
| 62G30 | Order statistics; empirical distribution functions |
Keywords:
divided difference; numerical differentiation; uniform order statistics; modulus of smoothness; error bounds; probabilistic Taylor’s formula; varianceCitations:
Zbl 1016.41016References:
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