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My data $\{X_1,...X_N \}$ are assumed exchangeable with known correlation $\rho$. I want a "t-test" (or similar) of mean $E(\bar{X})=0$ in the R language.

More generally, i seek an easy and standard way to correct for known correlation structure.

Obviously from the description above I'm not talking about the usual "dependent t-tests" (pair t-tests).

Secondary questions: a. What if $\rho$ is known up to distribution (i.e. a prior on $\rho$)? Intuitively it seems impossible that one could proceed without some prior distribution assumptions on $\rho$ but I'm happy to be corrected. b. are there standard tests of variance (not mean) in this setting described above?

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    $\begingroup$ what do you mean by exchangeable with a known correlation? $\endgroup$
    – rep_ho
    Commented Apr 20, 2021 at 10:24
  • $\begingroup$ Further, could you provide more information about your problem and data? $\endgroup$
    – rep_ho
    Commented Apr 20, 2021 at 10:33
  • $\begingroup$ @rep_ho thanks for your question. I have added a hyperlink to exchangability. My problem is: test the zero-mean null hypothesis for a sample with the described correlation structure. $\endgroup$
    – JRC
    Commented Apr 20, 2021 at 14:54
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    $\begingroup$ you might get a better answer if you provide more details about your dataset and experimental design and so on. You can see a definition for t-test for correlated data with known correlation here arxiv.org/pdf/1609.08905.pdf eq 1. although used in a different context. The idea is to adjust the t-statistic used in the t-test by the cor. coefficient. You can also read the primary Nadeu and Bengio paper, but this is IMO unreadable. I don't know about R implementation, but the procedure is simple enough to implement yourself $\endgroup$
    – rep_ho
    Commented Apr 20, 2021 at 17:31
  • $\begingroup$ @rep_ho Thanks! The Nadeu and Bengio paper was perfect! If you move this to be an answer, I'd accept :+) I'm new to cross Validated to apologies if my style is unusual :+) $\endgroup$
    – JRC
    Commented Apr 21, 2021 at 12:08

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t-test for correlated data with known correlation was proposed by Nadeau and Bengio: Nadeau, Claude and Bengio, Yoshua. Inference for the generalization error.Machine Learning, 52(3):239–281, 2003, which is described possibly more clearly in Corani et al. https://arxiv.org/pdf/1609.08905.pdf eq 1.

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