On Progressive Type-II Censored Samples from Alpha Power Exponential Distribution
Mukhtar M. Salah
Abstract
In this paper the two-parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>α</a:mi> </a:math> -power exponential distribution is studied. We study the two-parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>α</c:mi> </c:math> -power exponential <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mfenced open="(" close=")" separators="|"> <e:mrow> <e:mi>μ</e:mi> <e:mo>,</e:mo> <e:mi>λ</e:mi> </e:mrow> </e:mfenced> </e:math> distribution with the location parameter <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:mi>μ</j:mi> <j:mo>></j:mo> <j:mn>0</j:mn> </j:math> and scale parameter <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M5"> <l:mi>λ</l:mi> <l:mo>></l:mo> <l:mn>0</l:mn> </l:math> under progressive Type-II censored data with fixed shape parameter <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" id="M6"> <n:mi>α</n:mi> </n:math> . We estimate the maximum likelihood estimators of these unknown parameters numerically since it cannot be solved analytically. We use the approximate best linear unbiased estimators <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M7"> <p:msup> <p:mrow> <p:mi>μ</p:mi> </p:mrow> <p:mrow> <p:mi>∗</p:mi> </p:mrow> </p:msup> </p:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" id="M8"> <r:msup> <r:mrow> <r:mi>λ</r:mi> </r:mrow> <r:mrow> <r:mi>∗</r:mi> </r:mrow> </r:msup> </r:math> , as an initial guesses to obtain the MLEs <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M9"> <t:mover accent="true"> <t:mi>μ</t:mi> <t:mo>^</t:mo> </t:mover> </t:math> and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M10"> <w:mover accent="true"> <w:mi>λ</w:mi> <w:mo>^</w:mo> </w:mover> </w:math> . We estimate the interval estimation of these unknowns’ parameters. Monte Carlo simulations are performed and data examples have been provided for illustration and comparison.