In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$X_{1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mn>1</mn> </msub> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$X_{2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mn>2</mn> </msub> </math> </EquationSource> </InlineEquation> be two independent gamma random variables with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X_{i}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> having shape parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$r_{i}>0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>r</mi> <mi>i</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> and scale parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\lambda _{i}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="italic">λ</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$i=1,2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$X^{*}_{1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi>X</mi> <mn>1</mn> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$X^{*}_{2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi>X</mi> <mn>2</mn> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> </math> </EquationSource> </InlineEquation> be another set of independent gamma random variables with <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$X^{*}_{i}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi>X</mi> <mi>i</mi> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> </math> </EquationSource> </InlineEquation> having shape parameter <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$r_{i}^{*}>0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> <mo>></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation> and scale parameter <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$\lambda _{i}^{*}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi mathvariant="italic">λ</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$i=1,2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>. Denote by <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$$X_{2:2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mn>2</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$$X^{*}_{2:2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mn>2</mn> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> </math> </EquationSource> </InlineEquation> the corresponding maxima, respectively. It is proved that, among others, if <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$$(r_{1},r_{2})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> majorize <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$$(r_{1}^{*},r_{2}^{*})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>r</mi> <mrow> <mn>1</mn> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>r</mi> <mrow> <mn>2</mn> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$$(\lambda _{1},\lambda _{2})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="italic">λ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi mathvariant="italic">λ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> weakly majorize <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$$(\lambda _{1}^{*},\lambda _{2}^{*})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="italic">λ</mi> <mrow> <mn>1</mn> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi mathvariant="italic">λ</mi> <mrow> <mn>2</mn> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>, then <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$$X_{2:2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mn>2</mn> </mrow> </msub> </math> </EquationSource> </InlineEquation> is stochastically larger that <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$$X^{*}_{2:2}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msubsup> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mn>2</mn> </mrow> <mrow> <mrow/> <mo>∗</mo> </mrow> </msubsup> </math> </EquationSource> </InlineEquation> in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature. Copyright Springer-Verlag Berlin Heidelberg 2014
