A \(k\)-stage procedure for estimating the mean vector of a multivariate normal population. (English) Zbl 1434.62172
This paper considers a \(k\)-stage sequential estimation of the mean vector \(\mu\) of a normal distribution. The \(k\)-stage sequential procedure is a generalization of W. Liu [J. Stat. Plann. Inference 65, No. 1, 109–127 (1997; Zbl 0915.62068)] for the multivariate normal population \(N(\mu, \sigma^2\Sigma)\), where the variance \(\sigma^2\) is unknown and the matrix \(\Sigma\) is known. The authors evaluate the regret in minimum risk point estimation and prove that the regret is negative. The authors also obtain the second-order approximation of confidence region estimation. These theoretical results are verified through synthetic and Fisher’s Iris data sets.
Reviewer: Kazuho Watanabe (Toyohashi)
MSC:
62L12 | Sequential estimation |
62F10 | Point estimation |
62H12 | Estimation in multivariate analysis |
62F25 | Parametric tolerance and confidence regions |
Keywords:
confidence region estimation; \(k\)-stage procedure; minimum risk; multivariate normal population; point estimation; regret; second-order approximationsCitations:
Zbl 0915.62068Software:
MVNReferences:
[1] | Anscombe, F. J., Sequential Estimation,, Journal of Royal Statistical Society, Series B, 15, 1-21 (1953) · Zbl 0050.36301 |
[2] | Chow, Y. S.; Robbins, H., On the Asymptotic Theory of Fixed Width Sequential Confidence Intervals for the Mean,, Annals of Mathematical Statistics, 36, 457-462 (1965) · Zbl 0142.15601 |
[3] | Fisher, R. A., The Use of Multiple Measurements in Taxonomic Problems,, Annals of Eugenics, 7, 179-188 (1936) |
[4] | Hall, P., Asymptotic Theory of Triple Sampling for Sequential Estimation of a Mean,, Annals of Statistics, 9, 1229-1238 (1981) · Zbl 0478.62068 |
[5] | Korkmaz, S.; Goksuluk, D.; Zararsiz, G., Mvn: An R Package for Assessing Multivariate Normality,, R Journal, 6, 151-162 (2014) |
[6] | Liu, W., A k-Stage Sequential Sampling Procedure for Estimation of Normal Mean,, Journal of Statistical Planning and Inference, 65, 109-127 (1997) · Zbl 0915.62068 |
[7] | Mukhopadhyay, N., Sequential Methods in Estimation and Prediction (1975), Indian Statistical Institute: Indian Statistical Institute, Calcutta |
[8] | Robbins, H., Probability and Statistics, Sequential Estimation of the Mean of a Normal Population, 235-245 (1959), Uppsala: Almquist and Wiksell, Uppsala · Zbl 0095.13005 |
[9] | Srivastava, M. S., On Fixed-Width Confidence Bounds for Regression Parameters and Mean Vector,, Journal of Royal Statistical Society B, 29, 132-140 (1967) · Zbl 0152.37002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.