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Mean and Variance Estimation by Kriging

Tomasz Suslo

TL;DR

The paper addresses estimating the mean and variance of a random variable via kriging under non-asymptotic conditions. It develops a coordinate-dependent least-squares approach and derives practical estimators for the correlation structure using the semi-variogram $\gamma(h)$ and covariance $C(h)$, including $\hat{C}_n(h)$, $\hat{\gamma}_n(h)$, and $\hat{\rho}_n(h)$ with monotonicity constraints. A Central Limit Theorem for frozen-in-time stationary processes is established, enabling finite-sample inference through standardized statistics and KS testing, illustrated on the Warsaw Stock Market Index. The work provides a framework for finite-sample kriging-based mean/variance estimation and model-validation in time-dependent, locally stationary contexts.

Abstract

The aim of the paper is to derive the numerical least-squares estimator for mean and variance of random variable. In order to do so the following questions have to be answered: (i) what is the statistical model for the estimation procedure? (ii) what are the properties of the estimator, like optimality (in which class) or asymptotic properties? (iii) how does the estimator work in practice, how compared to competing estimators?

Mean and Variance Estimation by Kriging

TL;DR

The paper addresses estimating the mean and variance of a random variable via kriging under non-asymptotic conditions. It develops a coordinate-dependent least-squares approach and derives practical estimators for the correlation structure using the semi-variogram and covariance , including , , and with monotonicity constraints. A Central Limit Theorem for frozen-in-time stationary processes is established, enabling finite-sample inference through standardized statistics and KS testing, illustrated on the Warsaw Stock Market Index. The work provides a framework for finite-sample kriging-based mean/variance estimation and model-validation in time-dependent, locally stationary contexts.

Abstract

The aim of the paper is to derive the numerical least-squares estimator for mean and variance of random variable. In order to do so the following questions have to be answered: (i) what is the statistical model for the estimation procedure? (ii) what are the properties of the estimator, like optimality (in which class) or asymptotic properties? (iii) how does the estimator work in practice, how compared to competing estimators?

Paper Structure

This paper contains 3 sections, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Correlation function estimator
  3. Central Limit Theorem

Figures (1)

  • Figure 1: Long-lived asymmetric index profile, Warsaw's Stock Market Index (WIG) from 11 X 1999 up to 19 III 2001 (359 close quotes) the numerical least-squares estimator of mean at $j \ge n+1$ (black dots) compared for frozen model to the asymptotic generalized least-squares estimator of mean (grey line). The dashed vertical line represents $j=n=115$.