Abstract
The Bernstein-type approximation for smooth functions is proposed and studied. We propose the Bernstein-type approximation with definitions that directly apply the binomial distribution and the multivariate binomial distribution. The Bernstein-type approximations generalize the corresponding Bernstein polynomials, by considering definitions that depend on a convenient approximation coefficient in linear kernels. In the Bernstein-type approximations, we study the uniform convergence and the degree of approximation. The Bernstein-type estimators of smooth functions of population means are also proposed and studied.
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