Estimating the Reliability Function for a Family of Exponentiated Distributions

Table 1

Simulation results for 𝑅(𝑡).

𝑡	𝑅(𝑡)	𝑛=5		𝑛=10		𝑛=20		𝑛=50
		𝑅(𝑡)	𝑅(𝑡)	𝑅(𝑡)	𝑅(𝑡)	𝑅(𝑡)	𝑅(𝑡)	𝑅(𝑡)	𝑅(𝑡)
0.40	0.9642	0.9628	0.9429	0.9726	0.9638	0.9639	0.9598	0.9656	0.964
		−0.0014	−0.0213	0.0084	−4𝑒−04	−2𝑒−04	−0.0044	0.0014	−1𝑒−04
		0.001071326	0.001390097	0.0003646232	0.0003069262	0.0002067265	0.0002223408	0.0001467748	0.0001442984
		0.1211	0.1182	0.0647	0.0668	0.0555	0.0552	0.0477	0.0476
		86.6741	92.6193	91.428	93.112	93.5019	93.8349	94.0056	94.1243
0.50	0.9391	0.8576	0.8546	0.8789	0.8769	0.9282	0.9252	0.9399	0.9386
		−0.0815	−0.0845	−0.0601	−0.0622	−0.0109	−0.0139	8𝑒−04	−5𝑒−04
		0.009229477	0.008767263	0.007167494	0.006830551	0.001421963	0.001408017	0.0005898798	0.0005745804
		0.1912	0.1524	0.2262	0.2086	0.1412	0.1371	0.0958	0.0947
		93.9509	94.0809	94.1651	94.4641	94.2804	94.6018	94.5567	94.6561
0.80	0.8330	0.8314	0.8338	0.8355	0.8374	0.8408	0.8417	0.8319	0.8325
		−0.0016	7𝑒−04	0.0025	0.0044	0.0077	0.0087	−0.0011	−5𝑒−04
		0.004528085	0.002894221	0.002485577	0.002035995	0.002223323	0.002035827	0.001419582	0.001364689
		0.2552	0.2057	0.191	0.1725	0.1796	0.1711	0.1474	0.1446
		94.1964	94.3942	94.4812	94.5182	94.6424	94.6596	94.952	94.9534
1.00	0.7474	0.729	0.7499	0.7936	0.7994	0.7279	0.7338	0.7548	0.7568
		−0.0184	0.0025	0.0461	0.052	−0.0195	−0.0137	0.0074	0.0093
		0.008217478	0.005415503	0.005849085	0.005767851	0.004407818	0.003903546	0.002428885	0.002383391
		0.3395	0.2812	0.2359	0.2142	0.2505	0.2407	0.1915	0.1884
		94.0502	94.2394	94.5787	94.6276	94.9352	94.9729	95.0586	95.0578
1.50	0.5311	0.4442	0.4933	0.5094	0.532	0.5283	0.5392	0.5256	0.53
		−0.087	−0.0378	−0.0218	9𝑒−04	−0.0028	0.0081	−0.0055	−0.0011
		0.01260328	0.00618332	0.00520108	0.004460047	0.004379108	0.004317541	0.001922681	0.001875105
		0.2803	0.2719	8 0.2721	0.2648	0.26	0.2564	0.1747	0.1738
		94.2755	94.5729	94.464	94.653	94.7342	94.7791	95.0374	95.0517
2.00	0.3535	0.4971	0.5442	0.3349	0.3573	0.3429	0.354	0.3485	0.3529
		0.1435	0.1906	−0.0187	0.0037	−0.0106	5𝑒−04	−0.005	−6𝑒−04
		0.02667172	0.0416806	0.004913515	0.004757415	0.00239676	0.002330965	0.001298761	0.001284863
		0.3125	0.2928	0.2712	0.2765	0.1923	0.1943	0.1442	0.1448
		94.3591	94.7634	94.5513	94.7147	94.7592	94.8151	95.0453	95.0583
2.50	0.2266	0.2396	0.2813	0.2699	0.2906	0.2047	0.2131	0.226	0.2295
		0.013	0.0547	0.0433	0.064	−0.0219	−0.0135	−6𝑒−04	0.0029
		0.001733384	0.004863281	0.002956535	0.005265369	0.00116556	0.0009073532	0.0004148542	0.0004315035
		0.1576	0.1725	0.1312	0.1364	0.1045	0.1075	0.0809	0.0817
		94.0777	94.3206	94.2939	94.3724	94.7371	94.7634	95.0835	95.0883

Here, the first row indicates the estimate, the second row indicates the bias, the third row indicates variance, the fourth row indicates 95% bootstrap confidence length, and the fifth row indicates the coverage percentage.