1 Introduction
2 Main results
3 Simulation studies
3.1 Empirical results for unknown parameters
Sample size |
\(\hat{\alpha }_{1}\)
|
\(\hat{\alpha }_{2}\)
|
\(\hat{p}_{1}\)
|
\(\hat{p}_{2}\)
|
---|---|---|---|---|
50 | (0.0236,1.9775) | (0.3464,1.5224) | (0.2219,1.8419) | (0.0827,0.9487) |
100 | (−0.0727,0.8381) | (0.1041,0.4088) | (0.0362,0.6182) | (−0.0462,0.9545) |
200 | (0.0344,0.3609) | (0.0794,0.2350) | (−0.03262,0.9745) | (−0.1367,0.2902) |
500 | (0.0181,0.2290) | (0.0548,0.1612) | (−0.1604,0.8419) | (−0.0477,0.1436) |
Sample size |
\(\hat{\alpha }_{1}\)
|
\(\hat{\alpha }_{2}\)
|
\(\hat{p}_{1}\)
|
\(\hat{p}_{2}\)
|
---|---|---|---|---|
50 | (0.1352,2.5840) | (0.3216,3.4862) | (0.1786,3.9142) | (−0.0954,2.0482) |
100 | (0.0292,1.7485) | (−0.0842,1.6980) | (−0.0701,1.1221) | (0.0716,1.8540) |
200 | (−0.1227,0.8058) | (0.0583,0.2757) | (−0.0290,1.5073) | (−0.0824,0.3254) |
500 | (−0.0356,0.2659) | (−0.0200,0.1691) | (−0.0347,0.4000) | (0.0131,0.1199) |
Sample size |
\(\hat{\alpha }_{1}\)
|
\(\hat{\alpha }_{2}\)
|
\(\hat{p}_{1}\)
|
\(\hat{p}_{2}\)
|
---|---|---|---|---|
50 | (0.1758,1.5514) | (0.0972,1.4666) | (−0.0179,3.8898) | (0.0639,1.7230) |
100 | (0.0902,0.5844) | (0.0115,1.3228) | (0.1757,2.5644) | (−0.0125,1.5973) |
200 | (0.0215,0.5722) | (0.0361,0.4129) | (0.0602,0.5640) | (−0.0982,0.7852) |
500 | (0.0416,0.4463) | (0.0485,0.2199) | (−0.0025,0.8419) | (−0.0402,0.1436) |
Sample size |
\(\hat{\alpha }_{1}\)
|
\(\hat{\alpha }_{2}\)
|
\(\hat{p}_{1}\)
|
\(\hat{p}_{2}\)
|
---|---|---|---|---|
50 | (−0.0898,1.7002) | (0.2452,1.7146) | (0.0572,1.2197) | (−0.1446,2.2698) |
100 | (0.0773,1.3076) | (0.05041,1.3555) | (0.0303,0.7952) | (0.0696,1.9334) |
200 | (0.0570,1.7427) | (0.0363,0.3787) | (−0.0580,1.5712) | (0.0172,0.8405) |
500 | (−0.0443,0.3132) | (0.0336,0.1920) | (0.0409,0.3914) | (−0.0248,0.3106) |
3.2 Test for parameters
Sample size | λ = 1 (III) | λ = 2 (III) | λ = 1 (IV) | λ = 2 (IV) |
---|---|---|---|---|
50 | 0.076 | 0.076 | 0.084 | 0.09 |
100 | 0.066 | 0.08 | 0.08 | 0.056 |
200 | 0.058 | 0.046 | 0.056 | 0.074 |
500 | 0.052 | 0.056 | 0.044 | 0.056 |
\(\alpha _{2}\)∖ sample size | 50 | 100 | 200 | 500 |
---|---|---|---|---|
\(\alpha _{2}=0.2\)
| 0.59 | 0.608 | 0.706 | 0.842 |
\(\alpha _{2}=0.4\)
| 0.818 | 0.93 | 0.984 | 0.997 |
\(\alpha _{2}=0.6\)
| 0.808 | 0.942 | 0.996 | 0.999 |
\(\alpha _{2}=0.8\)
| 0.998 | 0.998 | 0.999 | 1 |
\(\alpha _{2}\)∖ sample size | 50 | 100 | 200 | 500 |
---|---|---|---|---|
\(\alpha _{2}=0.2\)
| 0.684 | 0.686 | 0.682 | 0.846 |
\(\alpha _{2}=0.4\)
| 0.864 | 0.936 | 0.976 | 0.999 |
\(\alpha _{2}=0.6\)
| 0.905 | 0.954 | 0.998 | 1 |
\(\alpha _{2}=0.8\)
| 0.978 | 0.997 | 0.999 | 1 |
\(\alpha _{2}\)∖ sample size | 50 | 100 | 200 | 500 |
---|---|---|---|---|
\(\alpha _{2}=0.2\)
| 0.636 | 0.604 | 0.696 | 0.824 |
\(\alpha _{2}=0.4\)
| 0.856 | 0.884 | 0.994 | 0.998 |
\(\alpha _{2}=0.6\)
| 0.964 | 0.996 | 0.999 | 0.998 |
\(\alpha _{2}=0.8\)
| 0.994 | 0.997 | 1 | 1 |
\(\alpha _{2}\)∖ sample size | 50 | 100 | 200 | 500 |
---|---|---|---|---|
\(\alpha _{2}=0.2\)
| 0.512 | 0.648 | 0.646 | 0.808 |
\(\alpha _{2}=0.4\)
| 0.816 | 0.916 | 0.992 | 0.999 |
\(\alpha _{2}=0.6\)
| 0.942 | 0.949 | 0.999 | 0.999 |
\(\alpha _{2}=0.8\)
| 0.994 | 0.997 | 1 | 1 |