A unified method for constructing expectation tolerance intervals

Given a random sample of size <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>n</mi> </math> </EquationSource> </InlineEquation> with mean <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\overline{X} $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mover> <mi>X</mi> <mo>¯</mo> </mover> </math> </EquationSource> </InlineEquation> and standard deviation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$s$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>s</mi> </math> </EquationSource> </InlineEquation> from a symmetric distribution <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$F(x; \mu , \sigma )=F_{0} (( x- \mu ) / \sigma ) $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mi mathvariant="italic">μ</mi> <mo>,</mo> <mi mathvariant="italic">σ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi mathvariant="italic">μ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi mathvariant="italic">σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math> </EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$F_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>F</mi> <mn>0</mn> </msub> </math> </EquationSource> </InlineEquation> known, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$X \sim F(x;\; \mu , \sigma )$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>X</mi> <mo>∼</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <mspace width="0.277778em"/> <mi mathvariant="italic">μ</mi> <mo>,</mo> <mi mathvariant="italic">σ</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> independent of the sample, we show how to construct an expansion <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$ a_n^{\prime }=\sum _{i=0}^\infty \ c_i \ n^{-i} $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msubsup> <mi>a</mi> <mi>n</mi> <mo>′</mo> </msubsup> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <mspace width="4pt"/> <msub> <mi>c</mi> <mi>i</mi> </msub> <mspace width="4pt"/> <msup> <mi>n</mi> <mrow> <mo>-</mo> <mi>i</mi> </mrow> </msup> </mrow> </math> </EquationSource> </InlineEquation> such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\overline{X} - s a_n^{\prime } > X > \overline{X} + s a_n^{\prime } $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mover> <mi>X</mi> <mo>¯</mo> </mover> <mo>-</mo> <mi>s</mi> <msubsup> <mi>a</mi> <mi>n</mi> <mo>′</mo> </msubsup> <mo>></mo> <mi>X</mi> <mo>></mo> <mover> <mi>X</mi> <mo>¯</mo> </mover> <mo>+</mo> <mi>s</mi> <msubsup> <mi>a</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> </math> </EquationSource> </InlineEquation> with a given probability <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\beta $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">β</mi> </math> </EquationSource> </InlineEquation>. The practical value of this result is illustrated by simulation and using a real data set. Copyright Springer-Verlag Berlin Heidelberg 2014