Bayesian Analysis of the Survival Function and Failure Rate of Weibull Distribution with Censored Data

Table 4

MSEs and Abs. Biases (in parenthesis) for the failure rate 𝐹(𝑡) using GNP.

		𝛼=0.5		𝛼=1.5
𝑛		𝛽=0.8	𝛽=1.2	𝛽=0.8	𝛽=1.2
		𝑎=5, [3]		𝑎=5, [3]
25	BS	1.60882 (0.60581)	0.75601 (0.74247)	0.26918 (0.22161)	0.08242 (0.24192)
		[3.50808 (0.66123)]	[0.70983 (0.72011)]	[0.21814 (0.22270)]	[0.08227 (0.24568)]
	BL (𝑐=1)	3.10223 (0.66497)	0.66435 (0.70130)	0.24086 (0.21740)	0.07698 (0.23560)
		[0.97145 (0.64334)]	[0.73049 (0.73445)]	[0.46015 (0.22941)]	[0.08183 (0.24475)]
	BL (𝑐=−1)	5.07183 (0.70026)	0.72658 (0.71728)	0.17702 (0.22059)	0.07997 (0.24249)
		[1.89290 (0.65885)]	[0.72855 (0.73625)]	[0.28853 (0.22911)]	[0.07864 (0.23852)]
	BL (𝑐=2)	2.57595 (0.67277)	0.70089 (0.71957)	0.19602 (0.21443)	0.08062 (0.24190)
		[1.42447 (0.65551)]	[0.73245 (0.73581)]	[0.26676 (0.22149)]	[0.08261 (0.24604)]
	BL (𝑐=−2)	2.47065 (0.68677)	0.68712 (0.70818)	0.29206 (0.22263)	0.07838 (0.24087)
		[2.90974 (0.65568)]	[0.69822 (0.71461)]	[0.31298 (0.23181)]	[0.08179 (0.24609)]
	BG (𝑘=1)	5.09142 (0.63923)	0.68482 (0.71454)	0.32629 (0.21537)	0.07963 (0.24117)
		[1.59876 (0.64264)]	[0.71224 (0.72443)]	[0.39187 (0.22194)]	[0.08104 (0.24269)]
	BG (𝑘=−1)	1.11653 (0.60663)	0.73329 (0.72535)	0.28561 (0.21867)	0.07938 (0.24085)
		[2.05799 (0.65800)]	[0.67003 (0.70302)]	[0.28235 (0.28235)]	[0.07783 (0.24017)]
	BG (𝑘=2)	3.26989 (0.63433)	0.70409 (0.72102)	0.14943 (0.21824)	0.08079 (0.24219)
		[3.59616 (0.65300)]	[0.70879 (0.71412)]	[0.20827 (0.22005)]	[0.08084 (0.24227)]
	BG (𝑘=−2)	1.56123 (0.63166)	0.68200 (0.71084)	0.22126 (0.21289)	0.07983 (0.24229)
		[1.82844 (0.64736)]	[0.71757 (0.72167)]	[0.16709 (0.21693)]	[0.07978 (0.24356)]
50	BS	0.99284 (0.55903)	0.53895 (0.64125)	0.21602 (0.18856)	0.06088 (0.21759)
		[1.20392 (0.55531)]	[0.53767 (0.64578)]	[0.23509 (0.18781)]	[0.05879 (0.21411)]
	BL (𝑐=1)	0.85858 (0.55004)	0.54626 (0.65313)	0.21497 (0.18676)	0.06021 (0.21590)
		[0.84299 (0.54806)]	[0.54644 (0.65612)]	[0.18170 (0.18565)]	[0.06051 (0.21811)]
	BL (𝑐=−1)	0.90621 (0.56868)	0.55652 (0.65539)	0.17717 (0.18777)	0.06052 (0.21565)
		[0.70963 (0.54405)]	[0.53659 (0.64556)]	[0.19359 (0.18475)]	[0.06063 (0.21566)]
	BL (𝑐=2)	0.84116 (0.54422)	0.55124 (0.65304)	0.19584 (0.18896)	0.06132 (0.21808)
		[1.05227 (0.55656)]	[0.55016 (0.65747)]	[0.19113 (0.18416)]	[0.06118 (0.21996)]
	BL (𝑐=−2)	0.83699 (0.55410)	0.54021 (0.64881)	0.19281 (0.18479)	0.06111 (0.21948)
		[0.87199 (0.55693)]	[0.52738 (0.63962)]	[0.18451 (0.18884)]	[0.06072 (0.21715)]
	BG (𝑘=1)	1.08699 (0.55039)	0.55381 (0.65826)	0.17302 (0.18223)	0.06119 (0.21882)
		[0.95041 (0.55106)]	[0.54985 (0.65467)]	[0.24233 (0.18548)]	[0.06097 (0.21819)]
	BG (𝑘=−1)	1.12507 (0.55889)	0.55956 (0.66129)	0.25057 (0.19005)	0.06023 (0.21461)
		[1.01425 (0.55157)]	[0.55221 (0.65488)]	[0.11878 (0.18655)]	[0.06018 (0.21560)]
	BG (𝑘=2)	1.17354 (0.55312)	0.57334 (0.67120)	0.17699 (0.18661)	0.06244 (0.22011)
		[1.00135 (0.56841)]	[0.54657 (0.64917)]	[0.10609 (0.18484)]	[0.05985 (0.21521)]
	BG (𝑘=−2)	1.18804 (0.56841)	0.56341 (0.66432)	0.13249 (0.18605)	0.06024 (0.21510)
		[1.15358 (0.55573)]	[0.54699 (0.65363)]	[0.14781 (0.18359)]	[0.06057 (0.21754)]
100	BS	0.66670 (0.53230)	0.48857 (0.63609)	0.05482 (0.17604)	0.05524 (0.21483)
		[0.44568 (0.52663)]	[0.49065 (0.64062)]	[0.06864 (0.17410)]	[0.05323 (0.21001)]
	BL (𝑐=1)	0.47245 (0.53712)	0.48375 (0.63740)	0.04677 (0.16927)	0.05397 (0.21152)
		[0.48126 (0.52985)]	[0.49223 (0.64333)]	[0.06145 (0.17697)]	[0.05467 (0.21400)]
	BL (𝑐=−1)	0.50560 (0.52798)	0.48800 (0.63826)	0.06408 (0.17864)	0.05512 (0.21358)
		[0.48709 (0.52132)]	[0.47258 (0.62645)]	[0.06531 (0.17577)]	[0.05552 (0.21507)]
	BL (𝑐=2)	0.44645 (0.52047)	0.48540 (0.63612)	0.06605 (0.17422)	0.05524 (0.21454)
		[0.80817 (0.52984)]	[0.48774 (0.63795)]	[0.06651 (0.17786)]	[0.05648 (0.21780)]
	BL (𝑐=−2)	0.50473 (0.52856)	0.49342 (0.64368)	0.05039 (0.17666)	0.05434 (0.21327)
		[0.52768 (0.53450)]	[0.49421 (0.64037)]	[0.07692 (0.17494)]	[0.05561 (0.21493)]
	BG (𝑘=1)	0.44440 (0.51305)	0.49572 (0.64148)	0.07150 (0.17498)	0.05361 (0.21112)
		[0.59699 (0.52223)]	[0.49308 (0.64049)]	[0.05309 (0.17608)]	[0.05444 (0.21287)]
	BG (𝑘=−1)	0.49145 (0.52292)	0.49925 (0.64372)	0.05013 (0.17478)	0.05340 (0.21109)
		[0.46108 (0.52107)]	[0.49309 (0.63951)]	[0.04829 (0.17618)]	[0.05185 (0.20729)]
	BG (𝑘=2)	0.50315 (0.52624)	0.50124 (0.64500)	0.05319 (0.17324)	0.05761 (0.22003)
		[0.45561 (0.52818)]	[0.49235 (0.64167)]	[0.05073 (0.17732)]	[0.05492 (0.21381)]
	BG (𝑘=−2)	0.55240 (0.52938)	0.51051 (0.65583)	0.04795 (0.17732)	0.05482 (0.21406)
		[0.42269 (0.52581)]	[0.48926 (0.63954)]	[0.06453 (0.17769)]	[0.05630 (0.21743)]

GNP: generalised noninformative prior, BG: general entropy loss function, BL: LINEX loss function, BS: squared error loss function.