Nonparametric tolerance intervals controlling percentages in both tails and simultaneous testing of quantiles

The problem of constructing nonparametric equal-tailed tolerance intervals is considered. A method for finding minimum sample size required so that the <i>r</i>th order statistic and the largest <i>r</i>th order statistic form a <i>p</i> content- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo></mml:math> coverage equal-tailed tolerance interval is proposed. A simultaneous test for quantiles is also provided. Minimum sample sizes that are required to construct equal-tailed TIs and those to carry out hypothesis tests at a given nominal level are presented. The methods are illustrated using data collected from a new secure messaging system.