Covariogram estimation for the process contaminated with linear trend

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Discussion Overview

The discussion revolves around the estimation of the covariogram for a process contaminated with a linear trend. It involves theoretical aspects of second-order stationary processes and the implications of contamination on covariance estimation.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant defines a process Y(t) that is a combination of a zero-mean, unit variance process S(t) and a linear trend, indicating that Y(t) is not second-order stationary due to this contamination.
  • The same participant seeks assistance in demonstrating that the estimate of the covariogram R(h) converges in probability to the estimate of Rs(h) plus a specific term involving the contamination degree k and sample size n.
  • Another participant inquires about the definition of a covariogram, suggesting a need for clarification on the term.
  • A third participant provides a definition of covariogram, equating it to covariance and explaining it in the context of a spatial process.
  • One participant asks if the original poster has attempted a solution and what specific obstacles they are facing, indicating a focus on problem-solving.

Areas of Agreement / Disagreement

There is no consensus on the main question posed, as participants are at different levels of understanding regarding the covariogram and its estimation. Some definitions and clarifications are provided, but the main mathematical inquiry remains unresolved.

Contextual Notes

The discussion includes varying levels of familiarity with the concept of covariograms, and there may be assumptions about the participants' background knowledge that are not explicitly stated.

Who May Find This Useful

Researchers and students interested in statistical processes, particularly those dealing with covariance estimation in the presence of trends, may find this discussion relevant.

New_Galatea
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Let {S(t), t=1,2,...} be a zero-mean, unit variance, second-order stationary process in R^1,
and define Y(t)=S(t)+k(t-(n+1)/2), t=1,2,...,n.
Then the process Y(t) is not second-order stationary process since it is contaminated with linear trend, k – degree of contamination.

Define R(h) – covariogram for Y(t) process and
Define Rs(h) - covariogram for S(t) process.

Could you help me to show that estimate of R(h) converges in probability to estimate of Rs(h) + ((k^2) * (n^2))/12

Thank in advance
 
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What is a covariogram?
 
As I know “Covariogram” is synonym of “Covariance”.
A strict definition is following:
Let x(t) be a spatial process. Covariogram for spatial process x(t) is a function
R(t1,t2)= M[(x(t1)-Mx(t1))(x(t2)-Mx(t2))].
Here M – symbol of mean.
 
Have you attempted a solution? Is there a specific obstacle you cannot get around?
 

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