By Essam K. AL-Hussaini, Mohammad Ahsanullah (auth.)
This booklet comprises solely new effects, to not be came upon in different places. additionally, extra effects scattered in different places within the literature are in actual fact awarded. numerous recognized distributions corresponding to Weibull distributions, exponentiated Burr kind XII distributions and exponentiated exponential distributions and their houses are established. research of actual in addition to well-simulated info are analyzed. a couple of inferences in response to a finite mix of distributions also are offered.
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Extra resources for Exponentiated Distributions
NÀr X 1 X Cj1 Cj1 ; S3 ¼ ; rþb1 þ1 1 T1jrþb j1 ¼0 j2 ¼0 Tj1 ;j2 1 ð2:3:13Þ T1j1 ¼ T0j1 À ln Gðz0 Þ; ð2:3:14Þ Tj1 ;j2 ¼ T0j1 À ðj2 þ 1Þ ln Gðz0 Þ: ð2:3:15Þ For proof, see AL-Hussaini (2010a). This development shall be called ‘standard Bayes method’ (SBM). 2 Basic Properties, Estimation and Prediction … 28 Remarks 1. In the complete sample case, the Bayes estimator ^aSEL ¼ b2 À 1 Pnþb n i¼1 ln GðXi Þ , based on the SEL function, agrees with the result obtained by AL-Hussaini (2010b). 2. In the complete sample case, ^aML and ^aSEL coincide for non-informative prior of a (the case in which b1 ¼ b2 ¼ 0).
Two-sample scheme In the case of two-sample scheme, we have two independent samples of sizes n and m. The informative sample consists of the ﬁrst r order statistics X1 \ Á Á Á \Xr of a random sample of size n. The future sample is assumed to consist of the order statistics Y‘ ; ‘ ¼ 1; . . ; m. It is also assumed that all observations are drawn from the same population whose CDF is H ðxÞ ¼ ½GðxÞa . Derivations of the estimators and predictive interval of the future observable Y‘ ; ‘ ¼ 1; . . 2) by f ðy‘ jhÞ / ½Hðy‘ jhÞ‘À1 ½1 À Hðy‘ jhÞmÀ‘ hðy‘ jhÞ: Proceeding as in the one-sample case, we ﬁnally obtain the estimators of a; bi ; ði ¼ 1; .
J2 ! 2 Basic Properties, Estimation and Prediction … 36 Therefore, ^ LNX ðx0 Þ ¼ À 1 ln R j eÀj Z Z1 AwðbÞp2 ðbÞ b ! Cj1 a rþb1 À1 exp½ÀaT0j1 ðbÞ j1 ¼0 0 1 X jj2 eaj2 ln½Gðx0 Þ j2 ¼0 nÀr X j2 ! dadb Z nÀr X 1 X 1 jj2 Cj1 ¼ À ln AeÀj wðbÞp2 ðbÞ j j2 ! j ¼0 j ¼0 1 Z1 2 b arþb1 À1 eÀa½T0j1 ðbÞÀj2 ln Gðx0 Þ dadb 0 Z nÀr X 1 X 1 jj2 Cj1 Cðr þ b1 Þ Àj ¼ À ln Ae wðbÞp2 ðbÞ dadb j j ! 23). The Bayes estimator of kH ðx0 Þ, at some x0 , is given by ^kLNX ðx0 Þ ¼ R R1 Àjkðx Þ 0 À j1 ln e pða; bj xÞdbda.
Exponentiated Distributions by Essam K. AL-Hussaini, Mohammad Ahsanullah (auth.)