Confidence intervals for the means of the selected populations

Consider an experiment in which p independent populations πi with corresponding unknown means θi are available, and suppose that for every 1 ≤ i ≤ p, we can obtain a sample Xi1, . . . , Xin from πi. In this context, researchers are sometimes interested in selecting the populations that yield the largest sample means as a result of the experiment, and then estimate the corresponding population means θi. In this paper, we present a frequentist approach to the problem and discuss how to construct simultaneous confidence intervals for the means of the k selected populations, assuming that the populations πi are independent and normally distributed with a common variance σ2. The method, based on the minimization of the coverage probability, obtains confidence intervals that attain the nominal coverage probability for any p and k, taking into account the selection procedure.

[1]  Jiunn T. Hwang.  EMPIRICAL BAYES ESTIMATION FOR THE MEANS OF THE SELECTED POPULATIONS , 2016 .

[2]  Marina Schmid,et al.  Design And Analysis Of Experiments For Statistical Selection Screening And Multiple Comparisons , 2016 .

[3]  Aditi Kar Gangopadhyay,et al.  Estimation of the Mean of the Selected Population , 2015 .

[4]  Robert Tibshirani,et al.  Post‐selection point and interval estimation of signal sizes in Gaussian samples , 2014, 1405.3340.

[5]  S. Geer,et al.  On asymptotically optimal confidence regions and tests for high-dimensional models , 2013, 1303.0518.

[6]  Dennis L. Sun,et al.  Exact post-selection inference, with application to the lasso , 2013, 1311.6238.

[7]  Noah Simon,et al.  On Estimating Many Means, Selection Bias, and the Bootstrap , 2013, 1311.3709.

[8]  Mohammad Ahsanullah,et al.  An Introduction to Order Statistics , 2013 .

[9]  Zhigen Zhao,et al.  Empirical Bayes false coverage rate controlling confidence intervals , 2012 .

[10]  B. Efron.  Tweedie’s Formula and Selection Bias , 2011, Journal of the American Statistical Association.

[11]  Cun-Hui Zhang,et al.  Confidence Intervals for Low-Dimensional Parameters With High-Dimensional Data , 2011 .

[12]  Y. Benjamini,et al.  False Discovery Rate–Adjusted Multiple Confidence Intervals for Selected Parameters , 2005 .

[13]  J. T. Gene Hwang,et al.  SHARP SIMULTANEOUS INTERVALS FOR THE MEANS OF SELECTED POPULATIONS WITH APPLICATION TO MICROARRAY DATA ANALYSIS , 2003 .

[14]  Shanti S. Gupta,et al.  Multiple decision procedures - theory and methodology of selecting and ranking populations , 1979, Classics in applied mathematics.

[15]  C. Lele.  Admissibility Results in Loss Estimation , 1993 .

[16]  J. H. Venter,et al.  Estimation of the mean of the population selected from K populations , 1991 .

[17]  S. Gupta,et al.  On Finding the Largest Normal Mean and Estimating the Selected Mean , 1989 .

[18]  Harold B. Sackrowitz,et al.  A Decision Theory Formulation for Population Selection Followed by Estimating the Mean of the Selected Population , 1988 .

[19]  E. Samuel-Cahn,et al.  Evaluating the chosen population: a Bayes and minimax approach , 1986 .

[20]  E. Samuel-Cahn,et al.  Estimation of the Mean of a Selected Negative Exponential Population , 1984 .

[21]  John S. Carson,et al.  Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations , 1981 .

[22]  L. Brown.  A Heuristic Method for Determining Admissibility of Estimators--With Applications , 1979 .

[23]  Edward J. Dudewicz,et al.  Procedures for Fixed-Width Interval Estimation of the Largest Normal Mean , 1976 .

[24]  J. Berger.  Admissibility Results for Generalized Bayes Estimators of Coordinates of a Location Vector , 1976 .

[25]  K. M. Lal Saxena,et al.  A Single-Sample Procedure for the Estimation of the Largest Mean , 1976 .

[26]  K. L. Saxena,et al.  Interval Estimation of the Largest Mean of k Normal Populations with Known Variances , 1969 .

[27]  A. Cohen,et al.  ESTIMATION OF THE LARGER OF TWO NORMAL MEANS , 1968 .

[28]  Irwin Guttman,et al.  A Bayesian Approach to Some Best Population Problems , 1964 .

[29]  Shanti S. Gupta,et al.  On a Statistic which Arises in Selection and Ranking Problems , 1957 .

[30]  R. Bechhofer.  A Single-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with known Variances , 1954 .