In this paper, the estimation of the parameters of extended Marshall-Olkin inverse-Pareto (EMOIP) distribution is studied under complete and censored samples. Five classical methods of estimation are adopted to estimate the parameters of the EMOIP distribution from complete samples. These classical estimators include the percentiles estimators, maximum likelihood estimators, least squares estimators, maximum product spacing estimators, and weighted least-squares estimators. The likelihood estimators of the parameters under type-I and type-II censoring schemes are discussed. Simulation results were conducted, for various parameter combinations and different sample sizes, to compare the performance of the EMOIP estimation methods under complete and censored samples. Further, the mean square errors, asymptotic confidence interval, average interval length, and coverage percentage are calculated under the two censored schemes. The simulation results illustrate that the coverage probabilities of the confidence intervals increase to the nominal levels when the sample size increases. A real data set from the insurance field is analyzed for illustrative purposes. The data represent monthly metrics on unemployment insurance from July 2008 to April 2013 and contain 21 variables and particularly we study the variable number 11 in the data. The EMOIP model provides a better fit as compared with the inverse-Pareto distribution under complete and censored schemes. We hope that the EMOIP distribution will attract wider applications in the insurance field which contains several heavy-tailed real data.
In statistical literature, many standard distributions are used when studying real data in different applied fields, but the known standard distributions are limited comparing with various real data. There is increased interest of finding new distributions by extending the existing ability of studying the unlimited range of the real data. Marshall et al. [
In this paper, the estimation of the extended Marshall-Olkin inverse-Pareto (EMOIP) parameters is conducted using five classical estimators, including the maximum likelihood estimators (MLEs), least squares estimators (LSEs), weighted least-squares estimators (WLSEs), percentiles estimators (PCEs), and maximum product spacing estimators (MPSEs). The behavior of these estimators is addressed using extensive simulation results for small and large samples. Furthermore, we address the estimation of the EMOIP parameters under censoring schemes, including type I censoring and type II censoring schemes. We have also conducted a Monte Carlo simulation study to calculate the MLEs for the unknown parameters under type I and type II censoring schemes. Finally, we analyze real data from the insurance field for illustrative purposes.
The EMOIP model was introduced by Gharib et al. [
Gharib et al. [
Or the cumulative distribution function (CDF)
If
The MO family is one of the most common generators in the literature and has been used extensively to extend several classical distributions as well as several other families of distributions. The most notable recent works include the MO extended-Weibull [
The two-parameter IP distribution [
in which
The probability density function (PDF) of the IP distribution has the form
The CDF of the EMOIP distribution with three parameters,
in which
The PDF of the EMOIP distribution reduces to
Clearly, for
The HRF of the EMOIP model is given as
The rest of the article is organized in four sections. Section 2 describes five classical estimation methods for estimating the EMOIP parameters and examines the proposed estimators numerically via Monte Carlo simulations. Section 3 discusses the estimation under censored samples as well as provides a detailed simulation study. A real data set from the insurance science is analyzed in Section 4 for illustrative purposes. Finally, Section 5 is devoted to the conclusion.
This section is devoted to estimating the parameters
Let
The MLEs of the EMOIP parameters
and
in which
The MLEs of
The percentiles estimation method is introduced by Kao et al. [
with respect to
where
and
The LSEs and WLSEs are adopted to estimate the beta parameters [
These estimates can also be obtained by solving the following nonlinear equations:
where
and
The WLSEs of the EMOIP parameters
Further, the WLSEs are obtained by solving the following equations:
in which
The maximum product of spacings (MPS) method is proposed by Cheng et al. [
The MPSEs of the EMOIP parameters
where
Alternatively, these estimates can be obtained by solving the following equations:
in which
In this section, a Monte Carlo simulation study is conducted to compare the performance of the different estimators of the unknown parameters of the EMOIP distribution. The numerical results are obtained using the Mathcad program, version 14.0, to compare the performances of different estimators with respect to their mean squared errors (MSEs). We generate 2000 samples of the EMOIP distribution for
The average values of estimates (AVEs) and MSEs of MLEs, LSEs, WLSEs, PCEs, and MPSEs are displayed in
In this section, we address the MLEs of the EMOIP parameters under type-I and type-II censored samples. We derive the approximate confidence intervals of the unknown parameters from the Fisher information matrix under type-I and type-II censored samples. Finally, we perform a simulation study to explore the behavior of the estimates. Several authors have been studied the estimation of the model parameters under different censoring schemas, such as Tomazella et al. [
In censoring of type-I, the unit
Parameters | MLEs | PCEs | LSEs | WLSEs | MPSEs |
---|---|---|---|---|---|
1.163(0.208) |
1.025(1.478) |
1.090(0.523) |
1.109(0.544) |
1.110(0.905) |
|
1.181(0.292) |
1.008(1.123) |
0.961(0.574) |
0.939(1.096) |
1.091(0.591) |
|
1.204(0.267) |
1.192(1.718) |
1.033(0.348) |
1.041(0.655) |
1.063(0.286) |
|
1.296(0.627) |
1.206(1.547) |
1.111(1.506) |
1.079(1.973) |
1.203(0.951) |
|
2.112(0.495) |
2.042(3.308) |
1.968(0.704) |
1.988(1.197) |
2.081(1.060) |
|
2.052(0.968) |
1.949(3.226) |
2.112(5.297) |
2.164(7.897) |
2.001(1.412) |
|
2.180(0.817) |
2.037(3.581) |
1.969(2.514) |
1.999(2.523) |
2.270(1.583) |
|
2.180(0.915) |
2.088(3.536) |
2.013(5.447) |
2.007(6.926) |
2.321(3.765) |
Parameters | MLEs | PCEs | LSEs | WLSEs | MPSEs |
---|---|---|---|---|---|
1.065(0.068) |
1.106(1.173) |
1.016(0.094) |
1.052(0.145) |
0.983(0.069) |
|
1.087(0.122) |
0.965(0.831) |
1.097(0.216) |
1.084(0.446) |
1.002(0.156) |
|
1.068(0.078) |
1.190(1.431) |
1.074(0.150) |
1.036(0.245) |
0.992(0.066) |
|
1.063(0.133) |
0.973(0.856) |
1.138(0.336) |
1.138(0.464) |
1.009(0.518) |
|
2.025(0.201) |
2.065(3.141) |
2.024(0.258) |
1.988(0.487) |
1.981(0.376) |
|
1.989(0.504) |
2.103(2.722) |
2.061(1.646) |
1.991(2.171) |
1.913(0.626) |
|
1.982(0.425) |
2.008(2.700) |
2.030(0.795) |
2.001(0.999) |
2.114(0.551) |
|
1.929(0.365) |
2.026(2.920) |
1.975(1.692) |
1.908(2.021) |
2.089(1.115) |
Parameters | MLEs | PCEs | LSEs | WLSEs | MPSEs |
---|---|---|---|---|---|
1.017(0.030) |
0.945(0.833) |
1.000(0.044) |
1.013(0.064) |
0.973(0.028) |
|
1.045(0.056) |
1.072(0.766) |
0.989(0.075) |
1.042(0.146) |
0.986(0.059) |
|
1.026(0.031) |
1.015(0.819) |
1.061(0.093) |
1.037(0.103) |
0.983(0.031) |
|
1.033(0.052) |
1.033(0.718) |
1.066(0.145) |
1.045(0.125) |
0.955(0.092) |
|
2.034(0.120) |
2.040(2.983) |
1.998(0.102) |
2.030(0.263) |
1.976(0.205) |
|
1.912(0.335) |
2.004(2.150) |
2.085(0.856) |
2.005(0.958) |
1.942(0.659) |
|
2.027(0.137) |
1.908(2.318) |
1.995(0.376) |
1.999(0.485) |
2.047(0.300) |
|
1.991(0.122) |
2.043(2.119) |
2.010(0.840) |
2.040(0.842) |
2.033(0.410) |
Parameters | MLEs | PCEs | LSEs | WLSEs | MPSEs |
---|---|---|---|---|---|
1.011(0.014) |
0.969(0.736) |
1.005(0.020) |
1.012(0.027) |
0.988(0.014) |
|
1.018(0.025) |
1.014(0.463) |
1.008(0.029) |
1.079(0.078) |
0.984(0.029) |
|
1.012(0.014) |
0.916(0.657) |
1.022(0.035) |
1.019(0.038) |
0.988(0.013) |
|
1.020(0.025) |
0.980(0.546) |
1.053(0.073) |
1.034(0.063) |
0.939(0.071) |
|
1.982(0.071) |
2.066(2.310) |
1.995(0.068) |
2.010(0.146) |
1.994(0.249) |
|
1.955(0.117) |
2.022(1.482) |
2.093(0.324) |
2.060(0.518) |
1.917(0.824) |
|
1.990(0.065) |
1.976(1.841) |
2.006(0.166) |
2.002(0.188) |
2.034(0.115) |
|
2.009(0.056) |
2.030(1.501) |
2.046(0.436) |
2.095(0.451) |
2.001(0.217) |
in which
Therefore,
Suppose that the fixed observation times for all units are equal to
If
The MLEs of
and
By solving the following equations simultaneously
Let
For simplicity of notation, we use
The MLEs of
and
The above three equations cannot be solved explicitly, so the numerical techniques can be employed to obtain the MLEs of the parameters, i.e.,
In this section, we conduct a simulation study to explore the performance of the MLEs of the EMOIP parameters in terms of their MSEs, asymptotic confidence interval (ACI), average interval length (AIL), and coverage percentage (CP) under type-I and type-II censored samples. We consider the values 20, 50, 100 and 200 for
All simulation results are carried out using the R software, where we generate
The simulation results, including the AVEs, MSEs, ACI, AIL, and CP, are displayed in
Par. | ||||||
---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | ||
20 | 3.0100 | 4.1150 | (0.6804, 7.0529) | 6.3725 | 94.44 | |
0.2463 | 1.6196 | (0.0049, 2.7721) | 2.7672 | 97.39 | ||
1.8714 | 3.3567 | (0.0099, 5.8824) | 5.8725 | 95.95 | ||
50 | 2.2978 | 1.1820 | (0.8544, 4.9200) | 4.0656 | 94.64 | |
0.1044 | 0.1200 | (0.0056, 0.7355) | 0.7299 | 97.68 | ||
1.9915 | 2.6562 | (0.0153, 5.2685) | 5.2532 | 97.02 | ||
100 | 2.0596 | 0.4955 | (1.0818, 3.6803) | 2.5985 | 95.87 | |
0.0790 | 0.0360 | (0.0064, 0.5862) | 0.5798 | 97.78 | ||
1.9947 | 2.1124 | (0.0199, 4.6481) | 4.6281 | 97.59 | ||
200 | 1.9404 | 0.2449 | (1.2357, 3.0412) | 1.8055 | 96.22 | |
0.0632 | 0.0191 | (0.0071, 0.4818) | 0.4746 | 97.84 | ||
1.9023 | 1.7166 | (0.0279, 4.0166) | 3.9887 | 98.58 |
Par. | ||||||
---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | ||
20 | 2.6700 | 2.8860 | (0.7820, 6.6440) | 5.8620 | 94.59 | |
0.1825 | 0.4359 | (0.0049, 1.4160) | 1.4111 | 94.55 | ||
1.9886 | 2.9438 | (0.0107, 5.4330) | 5.4223 | 96.86 | ||
50 | 2.2599 | 1.1335 | (0.9157, 4.8435) | 3.9278 | 94.72 | |
0.1205 | 0.1179 | (0.0056, 1.0435) | 1.0379 | 95.38 | ||
1.9884 | 2.2890 | (0.0147, 4.8250) | 4.8103 | 97.36 | ||
100 | 2.0229 | 0.4666 | (1.1400, 3.6306) | 2.4905 | 95.71 | |
0.0824 | 0.0467 | (0.0062, 0.8047) | 0.7985 | 96.73 | ||
1.9867 | 1.8273 | (0.0228, 4.3354) | 4.3126 | 98.04 | ||
200 | 1.9241 | 0.2387 | (1.2205, 2.9777) | 1.7572 | 96.29 | |
0.0571 | 0.0231 | (0.0069, 0.5676) | 0.5606 | 96.83 | ||
1.9303 | 1.4338 | (0.0432, 3.7647) | 3.7214 | 98.90 |
Par. | ||||||
---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | ||
20 | 2.8432 | 3.8099 | (0.7596, 7.2906) | 6.5311 | 94.73 | |
0.2477 | 1.1857 | (0.0047, 1.9586) | 1.9539 | 94.00 | ||
1.9148 | 3.0260 | (0.0070, 5.5759) | 5.5689 | 96.36 | ||
50 | 2.2547 | 1.1372 | (0.9276, 4.8247) | 3.8971 | 94.83 | |
0.1417 | 0.1848 | (0.0055, 1.4227) | 1.4172 | 94.39 | ||
1.9922 | 2.1969 | (0.0123, 4.7968) | 4.7845 | 97.23 | ||
100 | 2.0191 | 0.4678 | (1.1482, 3.6250) | 2.4768 | 95.67 | |
0.0974 | 0.0819 | (0.0062, 1.0689) | 1.0627 | 94.54 | ||
1.9892 | 1.7853 | (0.0204, 4.2861) | 4.2657 | 98.08 | ||
200 | 1.9236 | 0.2414 | (1.2005, 2.9809) | 1.7803 | 96.31 | |
0.0616 | 0.0336 | (0.0069, 0.6479) | 0.6409 | 95.73 | ||
1.9369 | 1.4010 | (0.0421, 3.7298) | 3.6877 | 98.85 |
Par. | ||||||
---|---|---|---|---|---|---|
Avg | MSE | Asy CI | AIL | CP | ||
20 | 2.8440 | 3.8410 | (0.7600, 7.3560) | 6.5960 | 94.74 | |
0.2480 | 1.1734 | (0.0046, 2.0005) | 1.9959 | 94.08 | ||
1.9061 | 2.9982 | (0.0068, 5.5745) | 5.5677 | 96.28 | ||
50 | 2.2567 | 1.1383 | (0.9306, 4.8251) | 3.8945 | 94.82 | |
0.1505 | 0.2371 | (0.0055, 1.5395) | 1.5340 | 94.94 | ||
1.9963 | 2.2081 | (0.0114, 4.7904) | 4.7790 | 97.29 | ||
100 | 2.0179 | 0.4663 | (1.1472, 3.6458) | 2.4986 | 95.59 | |
0.1020 | 0.0944 | (0.0062, 1.1356) | 1.1294 | 95.70 | ||
1.9868 | 1.7805 | (0.0188, 4.2731) | 4.2543 | 98.05 | ||
200 | 1.9237 | 0.2412 | (1.1988, 2.9869) | 1.7881 | 96.24 | |
0.0644 | 0.0397 | (0.0069, 0.6917) | 0.6848 | 95.93 | ||
1.9371 | 1.4008 | (0.0411, 3.7248) | 3.6837 | 98.85 |
Par. | |||||||
---|---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | |||
20 | 6 | – | – | – | – | – | |
– | – | – | – | – | |||
– | – | – | – | – | |||
50 | 15 | 2.3156 | 1.1740 | (0.9078, 4.8869) | 3.9791 | 94.85 | |
0.1242 | 0.0902 | (0.0057, 0.8099) | 0.8042 | 92.80 | |||
1.6732 | 2.8923 | (0.0121, 4.7442) | 4.7321 | 96.92 | |||
100 | 30 | 2.0502 | 0.4827 | (1.1301, 3.6814) | 2.5513 | 95.50 | |
0.1011 | 0.0417 | (0.0066, 0.6317) | 0.6251 | 93.77 | |||
1.7545 | 2.3871 | (0.0169, 4.3906) | 4.3737 | 97.61 | |||
200 | 60 | 1.9358 | 0.2524 | (1.2441, 3.0681) | 1.8240 | 95.94 | |
0.0772 | 0.0249 | (0.0072, 0.5066) | 0.4994 | 94.51 | |||
1.7605 | 1.9302 | (0.0246, 3.8966) | 3.8719 | 98.67 |
Par. | |||||||
---|---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | |||
20 | 12 | 2.8281 | 3.5921 | (0.7649, 7.1116) | 6.3467 | 94.34 | |
0.1957 | 0.5230 | (0.0048, 1.4233) | 1.4185 | 93.95 | |||
1.8058 | 3.1219 | (0.0079, 5.4631) | 5.4551 | 96.32 | |||
50 | 30 | 2.2477 | 1.1111 | (0.9331, 4.8277) | 3.8946 | 94.74 | |
0.1430 | 0.1339 | (0.0055, 1.1069) | 1.1014 | 94.98 | |||
1.9132 | 2.4029 | (0.0129, 4.7335) | 4.7206 | 97.46 | |||
100 | 60 | 2.0115 | 0.4656 | (1.1482, 3.6038) | 2.4556 | 95.57 | |
0.0988 | 0.0583 | (0.0063, 0.8366) | 0.8303 | 95.17 | |||
1.9227 | 1.9465 | (0.0195, 4.2839) | 4.2644 | 98.15 | |||
200 | 120 | 1.9192 | 0.2417 | (1.2156, 2.9851) | 1.7695 | 96.18 | |
0.0649 | 0.0269 | (0.0070, 0.5969) | 0.5899 | 96.17 | |||
1.8875 | 1.5137 | (0.0373, 3.7293) | 3.6920 | 98.87 |
Par. | |||||||
---|---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | |||
20 | 18 | 2.8391 | 3.8100 | (0.7566, 7.3584) | 6.6018 | 94.73 | |
0.2413 | 1.0568 | (0.0047, 1.8411) | 1.8364 | 94.32 | |||
1.9022 | 3.0562 | (0.0072, 5.5011) | 5.4939 | 96.34 | |||
50 | 45 | 2.2526 | 1.1356 | (0.9291, 4.8266) | 3.8975 | 94.91 | |
0.1503 | 0.2081 | (0.0055, 1.4383) | 1.4328 | 94.82 | |||
1.9902 | 2.2318 | (0.0120, 4.8070) | 4.7950 | 97.18 | |||
100 | 90 | 2.0163 | 0.4632 | (1.1478, 3.6146) | 2.4668 | 95.67 | |
0.1008 | 0.0914 | (0.0062, 1.0643) | 1.0581 | 95.31 | |||
1.9873 | 1.7925 | (0.0194, 4.2929) | 4.2735 | 98.11 | |||
200 | 180 | 1.9220 | 0.2403 | (1.2008, 2.9647) | 1.7640 | 96.29 | |
0.0628 | 0.0346 | (0.0070, 0.6931) | 0.6861 | 96.57 | |||
1.9345 | 1.4069 | (0.0398, 3.7310) | 3.6912 | 98.83 |
Par. | |||||||
---|---|---|---|---|---|---|---|
AVEs | MSEs | ACI | AIL | CP | |||
20 | 20 | 2.8440 | 3.8410 | (0.7600, 7.3560) | 6.5960 | 94.74 | |
0.2480 | 1.1734 | (0.0046, 2.0005) | 1.9959 | 94.08 | |||
1.9061 | 2.9982 | (0.0068, 5.5745) | 5.5677 | 96.28 | |||
50 | 50 | 2.2567 | 1.1383 | (0.9306, 4.8251) | 3.8945 | 94.82 | |
0.1505 | 0.2371 | (0.0055, 1.5395) | 1.5340 | 94.94 | |||
1.9963 | 2.2081 | (0.0114, 4.7904) | 4.7790 | 97.29 | |||
100 | 100 | 2.0179 | 0.4663 | (1.1472, 3.6458) | 2.4986 | 95.59 | |
0.1020 | 0.0944 | (0.0062, 1.1356) | 1.1294 | 95.70 | |||
1.9868 | 1.7805 | (0.0188, 4.2731) | 4.2543 | 98.05 | |||
200 | 200 | 1.9237 | 0.2412 | (1.1988, 2.9869) | 1.7881 | 96.24 | |
0.0644 | 0.0397 | (0.0069, 0.6917) | 0.6848 | 95.93 | |||
1.9371 | 1.4008 | (0.0411, 3.7248) | 3.6837 | 98.85 |
The results show that the MLEs of the EMOIP parameters under type-I and type-II schemes are asymptotically unbiased and consistent. As expected, the performance of the estimates under complete samples is better than those under type-I and type-II censored samples in terms of the MSEs. Furthermore, the AIL decreases as
In this section, we analyze a real data set for illustrative purpose. The following data set represents monthly metrics on unemployment insurance from July 2008 to April 2013 as reported by the department of Labor, Licensing and Regulation, State of Maryland, USA. The data set contains 21-variable and here we consider the variable number 11 in the data file which is available at:
Scheme | Type-I Censoring | Type-II Censoring | ||
---|---|---|---|---|
Fraction of censoring | Fraction of censoring | |||
I | (58, 57) | (58, 18) | ||
II | (58, 66.4) | (58, 35) | ||
III | (58, 102.7) | (58, 53) | ||
Complete |
These methods will be compared using the Akaike’s information criterion (AIC), (BIC) Bayesian information criterion (BIC), and Negative log-likelihood criterion (NLC). The EMOIP and IP distributions are fitted to the given data set under the considered schemes.
The ML estimates of the parameters of the EMOIP and IP distributions along with their standard errors (SEs), AIC, BIC, and NLC were reported in
Scheme | Model | AIC | BIC | NLC | |||
---|---|---|---|---|---|---|---|
I | EMOIP | 21.7401 |
15.7283 |
0.0102 |
213.166 | 216.153 | 103.583 |
IP | 45.2062 |
1.8753 |
228.199 | 230.190 | 12.099 | ||
II | EMOIP | 22.6840 |
17.4891 |
0.0040 |
333.760 | 338.426 | 163.880 |
IP | 42.7601 |
1.6610 |
377.512 | 380.622 | 186.756 | ||
III | EMOIP | 34.7061 |
11.2465 |
0.0033 |
476.846 | 482.700 | 235.423 |
IP | 45.3645 |
1.3538 |
548.188 | 552.090 | 272.094 | ||
Complete | EMOIP | 41.2813 |
9.0897 |
0.0038 |
526.502 | 532.631 | 260.251 |
IP | 42.9429 |
1.4453 |
602.601 | 606.687 | 299.300 |
Scheme | Model | AIC | BIC | NLC | |||
---|---|---|---|---|---|---|---|
I | EMOIP | 31.0400 |
8.9209 |
0.0253 |
198.515 | 201.186 | 96.257 |
IP | 41.7591 |
2.1110 |
208.674 | 210.455 | 102.337 | ||
II | EMOIP | 30.0781 |
13.6080 |
0.0027 |
332.935 | 337.601 | 163.467 |
IP | 42.4001 |
1.6742 |
377.365 | 380.476 | 186.682 | ||
III | EMOIP | 44.697 |
8.3233 |
0.0039 |
486.395 | 492.306 | 240.197 |
IP | 41.7660 |
1.4781 |
558.778 | 562.719 | 277.389 | ||
Complete | EMOIP | 30.8834 |
12.9921 |
0.0031 |
535.684 | 541.865 | 264.842 |
IP | 44.0252 |
1.4049 |
613.669 | 617.790 | 304.834 |
In this paper, the Extended Marshall-Olkin Inverse Pareto (EMOIP) parameters are estimated using five classical estimation methods from complete samples, including the maximum likelihood, percentiles, least squares, maximum product spacings, and weighted least-squares. Furthermore, the EMOIP parameters are also estimated using the maximum likelihood estimation under type-I and type-II censored samples. Monte Carlo simulations are conducted to compare and explore the performance of the different estimators of the EMOIP parameters under complete and censored samples. Based on our study, the maximum likelihood is the best performing method in terms of its mean squared errors. We conduct another Monte Carlo simulation study to calculate the maximum likelihood estimators for the model parameters under type-I and type-II censoring schemes. Finally, we analyze a real data set to validate our results.