This paper deals with the Bayesian estimation of Shannon entropy for the generalized inverse exponential distribution. Assuming that the observed samples are taken from the upper record ranked set sampling (URRSS) and upper record values (URV) schemes. Formulas of Bayesian estimators are derived depending on a gamma prior distribution considering the squared error, linear exponential and precautionary loss functions, in addition, we obtain Bayesian credible intervals. The random-walk Metropolis-Hastings algorithm is handled to generate Markov chain Monte Carlo samples from the posterior distribution. Then, the behavior of the estimates is examined at various record values. The output of the study shows that the entropy Bayesian estimates under URRSS are more convenient than the other estimates under URV in the majority of the situations. Also, the entropy Bayesian estimates perform well as the number of records increases. The obtained results validate the usefulness and efficiency of the URV method. Real data is analyzed for more clarifying purposes which validate the theoretical results.

Record values are crucial in many areas of real life applications comprising data relating to weather, sports, economics and life testing studies. Reference [_{i}_{i} _{i} is an URV if _{i} > x_{j}

Let

Another record sampling scheme, known as upper record ranked set sampling, has been provided by [

Consider

For information about ranked set sampling, see [

Some researchers have considered inference about different distributions based on records. For instance, Bayesian estimators and predictions for some life distributions from record values are discussed by [

Reference [

It is seen that a very sharply peaked distribution has very low entropy, whereas if the probability is spread out, the entropy is much higher. In this sense,

To our knowledge, in the literature, there are no studies that had been performed about entropy estimation in view of URRSS. So, our interest in this study is estimating the Shannon entropy of the GIE distribution using Bayesian approach from URRSS and URV. The Shannon entropy Bayesian estimator is considered using gamma priors. The Bayesian estimator of entropy is induced related to symmetric and asymmetric loss functions. The proposed loss functions are squared error loss function (SELF), linear exponential loss function (LINEX) and precautionary loss function (PRLF). Bayesian entropy estimators under symmetric and asymmetric loss functions have complicated expressions, so we implemented the Markov Chain Monte Carlo (MCMC) technique.

The following sections are organized as follows. Formula of Shannon entropy for GIE distribution is provided in Section 2. Entropy Bayesian estimator is derived using URRSS from symmetric and asymmetric loss functions in Section 3. Based on URV, entropy Bayesian estimator for GIE distribution is discussed using the proposed loss functions in Section 4. Simulation issue and application to real data are given in Sections 5 and 6, respectively. The paper ends with some concluding remarks in Section 7.

The two-parameter GIE distribution is provided by [

The CDF of the GIE distribution is given by

Let _{1}, we use the binomial expansion as follows
_{2}, let

Also, _{3} is obtained as follows

Substituting _{1}, _{2}, and _{3} in

In this section, Bayesian estimator of the Shannon entropy for the GIE model is discussed in view of URRSS. Firstly, the Bayesian estimators of parameters must be computed in order to get the entropy Bayesian estimator. Then, entropy Bayesian estimator is obtained using

Let

Assuming that the prior of parameters

The joint posterior under the assumption that

Hence, the marginal posterior distributions of

Therefore, the Bayesian estimators of

The Bayesian estimators of

The integrals

Consequently, the Bayesian estimator of

This section provides the Bayesian estimators of

Assuming that the prior of

Consequently, expressions for the marginal posterior distributions of

Hence, Bayesian estimators of

Also, under LINEX, the Bayesian estimators of

Furthermore, considering PRLF, the Bayesian estimators of

Again, the MCMC procedure is provided to approximate the integrals

Regarding to

By similar way, the Bayesian estimator of

In this section, a simulation investigation is carried out to compare the performance of the entropy estimate of the GIE distribution based on URV and URRSS. The relative absolute bias (RAB), estimated risk (ER) and width (WD) of credible intervals for the Shannon entropy based on URV and URRSS for GIE distribution are used to evaluate the behaviour of the Bayesian estimates. In the simulation setup, the number of records are selected as

Let

Initialize a starting value

for

set

generate

generate

if

set

else

set

end if

end for

Loss function
Scheme
Estimate
RAB
ER
WD
4
LINEX (δ = −2)
URV
0.8382
8.20E–03
9.59E–09
0.0601
URRSS
1.0634
2.58E–01
9.53E–06
0.5085
LINEX (δ = 2)
URV
0.7993
5.43E–02
4.21E–07
0.1159
URRSS
0.8465
1.58E–03
3.55E–10
1.1973
PRLF
URV
0.9010
6.60E–02
6.23E–07
0.1051
URRSS
0.7206
1.47E–01
3.10E–06
0.3732
SELF
URV
0.8463
1.40E–03
2.78E–10
0.0024
URRSS
0.5260
3.78E–01
2.04E–05
0.7587
5
LINEX (δ = −2)
URV
0.8443
1.00E–03
1.33E–10
0.0066
URRSS
0.9265
9.62E–02
1.32E–06
0.1177
LINEX (δ = 2)
URV
0.8401
6.00E–03
5.19E–09
0.0129
URRSS
0.8257
1.00E–02
1.52E–10
0.0592
PRLF
URV
0.8513
7.30E–03
7.59E–09
0.0116
URRSS
0.8896
5.26E–02
3.96E–07
0.1412
SELF
URV
0.8510
7.00E–03
6.94E–09
0.0119
URRSS
0.8091
4.26E–02
2.60E–07
0.0970
6
LINEX (δ = −2)
URV
0.8444
9.00E–04
1.08E–10
0.0059
URRSS
0.8392
7.00E–03
7.00E–09
0.0146
LINEX (δ = 2)
URV
0.8406
5.40E–03
4.21E–09
0.0116
URRSS
0.8444
9.41E–04
1.27E–10
0.0018
PRLF
URV
0.8507
6.60E–03
6.15E–09
0.0105
URRSS
0.8569
1.39E–02
2.74E–08
0.0212
SELF
URV
0.8505
6.30E–03
5.62E–09
0.0107
URRSS
0.8496
5.31E–03
4.03E–09
0.0160
7
LINEX (δ = −2)
URV
0.8451
1.00E–04
1.33E–12
0.0007
URRSS
0.8457
6.48E–04
6.00E–11
0.0014
LINEX (δ = 2)
URV
0.8446
6.00E–04
5.19E–11
0.0013
URRSS
0.8448
3.92E–04
2.20E–11
0.0009
PRLF
URV
0.8458
7.00E–04
7.59E–11
0.0012
URRSS
0.8451
4.69E–05
3.14E–13
0.0005
SELF
URV
0.8457
7.00E–04
6.94E–11
0.0012
URRSS
0.8458
7.79E–04
8.67E–11
0.0013

Note: E–a: stands for 10^{−a}.

n | Loss function | Scheme | Estimate | RAB | ER | WD |
---|---|---|---|---|---|---|

4 | LINEX (δ = −2) | URV | 1.3421 | 1.20E–02 | 5.31E–08 | 0.1316 |

URRSS | 1.6688 | 2.29E–01 | 1.93E–05 | 0.8824 | ||

LINEX (δ = 2) | URV | 1.2583 | 7.40E–02 | 2.00E–06 | 0.2550 | |

URRSS | 1.3556 | 2.03E–03 | 1.52E–09 | 0.0184 | ||

PRLF | URV | 1.4823 | 9.10E–02 | 3.07E–06 | 0.2334 | |

URRSS | 2.0788 | 5.30E–01 | 1.04E–04 | 1.1534 | ||

SELF | URV | 1.4763 | 8.70E–02 | 2.78E–06 | 0.2357 | |

URRSS | 2.2136 | 6.30E–01 | 1.46E–04 | 1.4540 | ||

5 | LINEX (δ = −2) | URV | 1.3567 | 1.00E–03 | 5.31E–10 | 0.0132 |

URRSS | 1.3475 | 8.00E–03 | 2.37E–08 | 0.0488 | ||

LINEX (δ = 2) | URV | 1.3482 | 8.00E–03 | 2.08E–08 | 0.0258 | |

URRSS | 1.3616 | 2.40E–03 | 2.13E–09 | 0.01491 | ||

PRLF | URV | 1.3707 | 9.10E–02 | 3.04E–08 | 0.0232 | |

URRSS | 1.3448 | 1.00E–02 | 3.65E–08 | 0.0919 | ||

SELF | URV | 1.3701 | 9.00E–03 | 2.78E–08 | 0.0239 | |

URRSS | 1.3912 | 2.40E–02 | 2.16E–07 | 0.1017 | ||

6 | LINEX (δ = −2) | URV | 1.3480 | 8.00E–03 | 5.13E–10 | 0.0132 |

URRSS | 1.3567 | 1.00E–03 | 5.34E–10 | 0.0065 | ||

LINEX (δ = 2) | URV | 1.3394 | 1.40E–02 | 7.14E–08 | 0.0884 | |

URRSS | 1.3639 | 4.04E–03 | 6.03E–09 | 0.0208 | ||

PRLF | URV | 1.4233 | 4.80E–02 | 3.04E–08 | 0.0232 | |

URRSS | 1.3573 | 1.00E–03 | 2.22E–10 | 0.0044 | ||

SELF | URV | 1.2667 | 9.00E–03 | 2.78E–08 | 0.0239 | |

URRSS | 1.3607 | 2.00E–03 | 1.06E–09 | 0.0067 | ||

7 | LINEX (δ = −2) | URV | 1.3582 | 0.00E+00 | 5.31E–12 | 0.0013 |

URRSS | 1.3579 | 0.00E+00 | 4.34E–11 | 0.0009 | ||

LINEX (δ = 2) | URV | 1.3573 | 1.00E–03 | 2.08E–10 | 0.0026 | |

URRSS | 1.3578 | 3.93E–04 | 5.71E–11 | 0.0103 | ||

PRLF | URV | 1.3596 | 1.00E–03 | 3.04E–10 | 0.0023 | |

URRSS | 1.3582 | 0.00E+00 | 6.34E–12 | 0.0010 | ||

SELF | URV | 1.3595 | 1.00E–03 | 2.78E–10 | 0.0024 | |

URRSS | 1.3582 | 0.00E+00 | 7.55E–12 | 0.0009 |

Note: E–a: stands for 10^{−a}.

Loss function | Scheme | Estimate | RAB | ER | WD | |
---|---|---|---|---|---|---|

4 | LINEX (δ = −2) | URV | 3.3481 | 1.78E–02 | 6.84E–07 | 0.1290 |

URRSS | 3.8466 | 1.69E–01 | 6.20E–05 | 0.6574 | ||

LINEX (δ = 2) | URV | 3.1869 | 3.12E–02 | 2.11E–06 | 0.2583 | |

URRSS | 3.4590 | 5.15E–02 | 5.74E–06 | 1.0862 | ||

PRLF | URV | 3.4135 | 3.77E–02 | 3.07E–06 | 0.2336 | |

URRSS | 3.2859 | 1.13E–03 | 2.78E–09 | 0.5921 | ||

SELF | URV | 3.4089 | 3.62E–02 | 2.84E–06 | 0.2415 | |

URRSS | 3.2191 | 2.14E–02 | 9.94E–07 | 0.8571 | ||

5 | LINEX (δ = −2) | URV | 3.2733 | 5.00E–03 | 5.31E–08 | 0.1316 |

URRSS | 3.2774 | 3.73E–03 | 3.00E–08 | 0.0917 | ||

LINEX (δ = 2) | URV | 3.1896 | 3.04E–02 | 2.00E–06 | 0.2550 | |

URRSS | 3.3409 | 1.56E–02 | 5.25E–07 | 0.0686 | ||

PRLF | URV | 3.4135 | 3.77E–02 | 3.07E–06 | 0.2334 | |

URRSS | 3.2916 | 6.02E–04 | 7.83E–10 | 0.0150 | ||

SELF | URV | 3.4075 | 3.58E–02 | 2.78E–06 | 0.2357 | |

URRSS | 3.3371 | 1.44E–02 | 4.50E–07 | 0.0793 | ||

6 | LINEX (δ = −2) | URV | 3.2880 | 5.00E–04 | 5.31E–10 | 0.0132 |

URRSS | 3.2835 | 1.87E–03 | 7.54E–09 | 0.0146 | ||

LINEX (δ = 2) | URV | 3.2794 | 3.10E–03 | 2.08E–08 | 0.0258 | |

URRSS | 3.2950 | 1.65E–03 | 5.91E–09 | 0.0108 | ||

PRLF | URV | 3.3019 | 3.70E–03 | 3.04E–08 | 0.0232 | |

URRSS | 3.2905 | 2.65E–04 | 1.52E–10 | 0.0080 | ||

SELF | URV | 3.3014 | 3.60E–03 | 2.78E–08 | 0.0239 | |

URRSS | 3.2874 | 6.84E–04 | 1.01E–09 | 0.0131 | ||

7 | LINEX (δ = −2) | URV | 3.2895 | 0.00E+00 | 1.33E–12 | 0.0007 |

URRSS | 3.2897 | 3.12E–05 | 2.10E–12 | 0.0008 | ||

LINEX (δ = 2) | URV | 3.2916 | 5.94E–04 | 7.64E–10 | 0.0120 | |

URRSS | 3.2902 | 1.66E–04 | 5.96E–11 | 0.0150 | ||

PRLF | URV | 3.2902 | 2.00E–04 | 7.59E–11 | 0.0012 | |

URRSS | 3.2903 | 2.21E–04 | 1.06E–10 | 0.0016 | ||

SELF | URV | 3.2902 | 2.00E–04 | 6.94E–11 | 0.0012 | |

URRSS | 3.2896 | 7.49E–08 | 1.21E–17 | 0.0005 |

Note: E–a: stands for 10^{−a}.

The ER of entropy estimates under SELF and LINEX based on URRSS is smaller than that of the corresponding under URV at

The ER of entropy estimates under LINEX (

The ER of entropy estimates based on URSS is smaller than the corresponding under URV at

The ER of entropy estimates based on URRSS is smaller than the corresponding under URV at

The WD of entropy estimates based on URV is smaller than the corresponding under URRSS at

In general, as

As the true value

In this section, a real data set is analysed for illustrative purposes. The suggested data represent the lifetimes of steel specimens tested at different stress levels (for more details see [

The extracted records from a part of this data are presented as

60 | 100 | 141 | 173 |

83 | 128 | 143 | 218 |

140 | 186 | 194 | 288 |

318 | 394 | ||

585 |

Based on the above record data, it can be shown that URRSS of size

From

True entropy | (θ,β) | Loss function | Scheme | Estimate | RAB | ER | WD |
---|---|---|---|---|---|---|---|

0.8452 | (4, 2) | LINEX (δ = −2) | URV | 0.667 | 0.211 | 6.33E–06 | 0.267 |

URRSS | 0.874 | 0.035 | 1.71E–07 | 0.219 | |||

LINEX (δ = 2) | URV | 0.894 | 0.058 | 4.75E–07 | 0.115 | ||

URRSS | 0.787 | 0.069 | 6.76E–07 | 0.119 | |||

PRLF | URV | 0.891 | 0.054 | 4.16E–07 | 0.168 | ||

URRSS | 0.772 | 0.086 | 1.06E–06 | 0.208 | |||

SELF | URV | 0.799 | 0.054 | 4.17E–07 | 0.176 | ||

URRSS | 0.952 | 0.126 | 2.27E–06 | 0.220 | |||

1.3584 | (2, 2) | LINEX (δ = −2) | URV | 1.342 | 0.012 | 5.31E–08 | 0.132 |

URRSS | 1.301 | 0.042 | 6.57E–07 | 0.156 | |||

LINEX (δ = 2) | URV | 1.258 | 0.074 | 2.00E–06 | 0.255 | ||

URRSS | 1.373 | 0.011 | 4.11E–08 | 0.117 | |||

PRLF | URV | 1.343 | 0.011 | 4.74E–08 | 0.118 | ||

URRSS | 1.398 | 0.029 | 3.20E–07 | 0.143 | |||

SELF | URV | 1.409 | 0.037 | 5.17E–07 | 0.138 | ||

URRSS | 1.338 | 0.015 | 8.05E–08 | 0.119 | |||

3.2896 | (0.5, 2) | LINEX (δ = −2) | URV | 3.347 | 0.017 | 6.61E–07 | 0.114 |

URRSS | 3.274 | 0.005 | 4.80E–08 | 0.180 | |||

LINEX (δ = 2) | URV | 3.243 | 0.014 | 4.31E–07 | 0.124 | ||

URRSS | 3.234 | 0.017 | 6.16E–07 | 0.159 | |||

PRLF | URV | 3.362 | 0.022 | 1.06E–06 | 0.218 | ||

URRSS | 3.310 | 0.006 | 8.01E–08 | 0.250 | |||

SELF | URV | 3.278 | 0.003 | 2.49E–08 | 0.178 | ||

URRSS | 3.244 | 0.014 | 4.11E–07 | 0.106 |

This paper provides Bayesian estimation of the Shannon entropy for the generalized inverse exponential distribution using URRSS and URV shemes. The entropy Bayesian estimators are considered using gamma prior functions for symmetric (SELF) and asymmetric (LINEX and PRLF) loss functions. In order to obtain the Bayesian estimators, we employed Markov Chain Monte Carlo method based on Metropolis-Hastings algorithm. The performance of the entropy estimates for the GIE distribution is investigated in terms of their relative absolute bias, estimated risk and the width of credible intervals. From simulation results, it turns out that, the entropy Bayesian estimator approaches the true value as the number of record increases. Generally, the entropy and ERs are directly proportional, that is; if the real value of entropy increases, the ERs increase. The WD of Bayes credible intervals for estimated values of entropy URRSS is smaller than the corresponding estimated values based on URV for all loss functions for most values of record values in the majority of the cases. A data real example has been considered to illustrate the applicability of the proposed methodology for the considered record schemes.

The authors are grateful to the Editor and anonymous reviewers for their valuable comments and suggestions.