On Wald’s sequential test for vector mean under multivariate normal distribution and correlated data

Abstract

In this article Wald’s sequential probability ratio test (SPRT) is implemented for multivariate normal distribution, for independent and autocorrelated data and known covariance matrix. The methodology based on residuals from the vector autoregressive moving average (VARMA) class models is presented for autocorrelated data. In this approach the average sample size required to take a decision in the sequential test is based on the Mahalanobis distance between the vectors of the intercept constants with respect to the error covariance matrix. Monte Carlo simulations were performed considering different scenarios for bivariate normal distribution. For fixed probabilities of type I and II errors, the results showed that the estimated average sample sizes to stop the sequential test were a little larger than those expected by Wald’s theory for autocorrelated and independent data. Under independence assumption the SPRT estimated sample sizes were also smaller than the sample sizes required by Hotelling’s $T^2$ test. It was shown that the omission of the correlation structure of the data strongly affects the type I and II errors of the sequential test. An example in the quality control field is presented using real data from a pig iron production process and the multivariate VAR(1) model.

Keywords:

• Autocorrelated data
• Monte Carlo simulation
• multivariate normal distribution
• statistical quality control
• VAR model
• Wald’s sequential test

SUBJECT CLASSIFICATIONS:

• 62L10
• 62F03
• 62P30

Additional information

Funding

This work was partially supported by the Brazilian institution: CNPq–National Council for Scientific and Technological Development. The authors are very thankful for the support.