Introduction
The most common approach to data collection uses the concept of simple random sampling (SRS) from the population. However, sometimes sampling is difficult because the cost of collecting data is expensive or requires a long time. Therefore, cost-optimal sampling methods have received more attention from statisticians, especially when measuring the characteristic of interest is expensive and requires more time to measure it. The idea of Ranked set sampling (RSS) was first proposed by Mclntyre (1951) through his effective attempts to find a more effective estimator to estimate the production of large pasture fields in Australia.
There have been several new developments of the idea introduced by McIntyre, which have made the method applicable to a much wider scope in areas such as environmental science, reliability and quality control. The (RSS) method has become an effective alternative to (SRS), as studies presented by many researchers have proven that it is more efficient by using some statistical criteria, including variance reducing of the estimator, and thus gives more accuracy when taking samples of a smaller size than in simple random sampling .
The (RSS) can be applied in many studies when the characteristic of interest is very difficult to measure (money, time, work, and organization) but the variable under study can be easily ranked even though it cannot be easily measured. Rankings may be made on the basis of visual inspection, prior information, or other approximate methods that do not require actual measurement. There are many researches and studies that dealt with the (RSS) method, including the work were the researcher A. Wolfe showed that the (RSS) is a statistical method for collecting data that gives a more efficient estimator than simple random sampling. He also explained how to obtain (RSS) and the basic difference. Between it and the (SRS) method (Wolfe, 2004). Hassan estimated the parameters of the exponential distribution based on (RSS) and used the Bayes method and the maximum likelihood estimator and it turned out that the Bayes estimate is the best (Hassan, 2013). A study was conducted by researchers (Al-Omari et al.) to estimate the reliability function when the distributions of both stress and force are independent and follow the exponential Pareto distribution, using the (MLS) method to estimate the reliability of stress strength under (SRS) and (RSS). The performance of the estimators was compared through a simulation study. The study revealed that stress strength reliability estimates under (RSS) are more efficient than (SRS) (Al-Omari, et al, 2020)
Material and methods
2- Ranked Set Sampling (RSS)
The concept (RSS) is a statistical method to collect data by reducing the sample size and obtaining real measurements with the shortest time and least cost, that is, through which we will obtain measurements that have the most luck to represent the population, which is an alternative to simple random sampling (SRS). The steps for selecting RSS are as follows (Sabry & Al-Metwally, 2021):
- We select (m) sets randomly from the population under study, each set of size (m).
- The elements of each set in step (1) are arranged according to a predetermined property, such as ascending or descending order.
- After the arrangement, we take the smallest ordered unit from the first set and the smallest second ordered unit from the second set, and the selection process continues until the largest ordered unit is selected from the last set. The analysis includes only the specific units (m) that enter into the analysis only, and we can reverse the process by choosing the largest unit from the first set and so on to get a new group of size (m), which is called the ranked set sample.
- Repeat steps (1-3) for (c) of cycles until we get a sample of size n = mc. where y(ii)k represents the ordered unit of sequence i (i=1,2,…,m) in the i-th set in the cycle k (k=1,2,…,c) and represents a single from the sample of the ordered sets with a sample size n=mc
Step1:
Step2:
For ease, the second step will be written in the following format
[ ]
Repeating the operation to c of the cycles gives us:
3- Lomax Distribution
The Lomax distribution is known as the Pareto distribution of the second type. It is useful for modeling and analyzing survival data in medical, biological, engineering, etc. sciences. It has received a great deal of attention from statisticians due to its use in the study of reliability and life. The first to use Lomax distribution was the scientist Lomax in (1954). Let the random variable y follow the Lomax distribution with the parameters β and θ, where θ is a shape parameter and β is a scale parameter, the probability density function (pdf), the cumulative function (cdf) and the reliability function of the variable y respectively are as follows:
4- Probability Density Function for (RSS)
Let y(11)1, y(22)1, . . . , y(mm)c be a random sample drawn by the (RSS) method obtained from cycles (c) of size (m) where y(ii)k are independent random variables, and represent the ordered statistics of size mc. It has the following probability density function:
5- Estimation Methods
5-1 Maximum likelihood Estimators under RSS
This method is based on the concept of the likelihood function, let y1,y2,...,yn represent the measurements of a random sample drawn from a population with a probability density function f(y,θ) ; θ ∈ Ω, the likelihood function is defined as the joint distribution of those measurements. The principle of the likelihood method can be established in finding estimates of the parameters of probability distributions, which makes the likelihood function to its maximum end. Therefore, the likelihood function of the Lomax distribution under (RSS) is as follows (Aziz & Shaaban, 2021) , (Sabry, et al, 2019):
We substitute the probability density function of (RSS) into equation (1) as follows:
(4)
By taking ln for equation (4), we get the following equation:
We differentiate equation (5) with respect to θ and then equalize it to zero we get:
We differentiate equation (5) with respect to β and then equalize it to zero we get:
The equations (6 & 7) cannot be solved by ordinary mathematical methods, so they will be solved numerically using Newton-Raphson's iterative method to obtain the estimators . Therefore, the estimator of the reliability function by the maximum likelihood method under (RSS) of the Lomax distribution is as follows:
5-2 Maximum Product of Spacing Estimator (MPS) method
If y1,y2,...,yn is a random sample arranged and spaced on a regular form between its individuals and taken from a population that follows the probability distribution that has a function of a probability density function f(y,θ,β) and a cumulative function F(y), then the highest product of the spacing estimators we get from maximizing the geometric mean of the distances, and the estimate in this method increases the spacing to the maximum extent of the geometric mean Q(θ;β|y), this can be illustrated as follows (Al-Metwally & Al-Mongy, 2019), (Al-Omari, et al, 2021):
where
and , such that F(y(0)) = 0 , F(y(n+1)) = 1, so
After substituting the cdf of Lomax distribution and taking the Ln, we get:
We derive equation (11) with respect to θ and then equalize it to zero
We derive equation (11) with respect to β and then equalize it to zero
The equations (12 & 13) cannot be solved by ordinary mathematical methods, so they will be solved numerically using Newton-Raphson's iterative method to obtain the estimations . Therefore, the estimator of the reliability function by the maximum likelihood method under (RSS) of the Lomax distribution is as follows:
5-3 Least Squares Method
Swain in 1988 was the first who used to estimate beta distribution parameters based on probability theory that indicates that where is the cumulative distribution function and y(i:n) is i-th ordered statistics for the random sample . It is also one of the important methods in estimation processes, as this method depends on reducing the set of error squares that can formulate its equation as follows (Al-Omari, et al, 2021), (Taconeli & Bonat, 2019):
We substitute the cdf function of the Lomax distribution into equation (15)
We derive equation (16) with respect to θ and then equalize it to zero
We derive equation (17) with respect to β and then equalize it to zero
The equations (17 & 18) cannot be solved by ordinary mathematical methods, so they will be solved numerically using Newton-Raphson's iterative method to obtain the estimations . Therefore, the estimator of the reliability function by the maximum likelihood method under (RSS) of the Lomax distribution is as follows:
5-4 Weighted Least Squares Method
The weighted least squares (WLS) method is a basic method of estimation, which contains the weight factor (wi) and to distinguish between this method and the method of ordinary least squares based on the concept of reducing the sum of the squares of error and its shape as much as possible. Suppose yi are ordered statistics taken from a random sample of size n resulting from a continuous probability distribution. The following formula can be followed for the (WLS) method:
We substitute the cdf function of the Lomax distribution into equation (20)
We derive equation (21) with respect to θ and then equalize it to zero
We derive equation (21) with respect to β and then equalize it to zero
The equations (22 & 23) cannot be solved by ordinary mathematical methods, so they will be solved numerically using Newton-Raphson's iterative method to obtain the estimations . Therefore, the estimator of the reliability function by the maximum likelihood method under (RSS) of the Lomax distribution is as follows:
Statistical Analysis
. Experimental Side:
6-1 Simulation experiment stages:
The simulation program was written using the statistical programming language R. The program includes four basic stages to estimate the parameters and reliability function of the Lomax distribution, as follows:
The first stage: Specifying default values
The values were chosen as follows:
- The default values were chosen for the two Lomax distribution parameters, and three models were formed, as follows:
Table (1): Default values for Lomax distribution parameters
|
Model
|
|
|
|
|
1
|
0.5
|
0.5
|
θ = β
|
|
2
|
0.8
|
0.5
|
θ > β
|
|
3
|
0.5
|
1.5
|
θ < β
|
- Different sample sizes were chosen (12, 24, 36, 54,96), as follows:
Table (2): The used Sample sizes
|
Sample size
|
(m)group size
|
(k)cycles number
|
|
12
|
3
|
4
|
|
4
|
3
|
|
6
|
2
|
|
24
|
3
|
8
|
|
6
|
4
|
|
4
|
6
|
|
36
|
4
|
9
|
|
6
|
6
|
|
3
|
12
|
|
54
|
9
|
6
|
|
6
|
9
|
|
3
|
18
|
|
96
|
8
|
12
|
|
12
|
8
|
|
6
|
16
|
- Each experiment was repeated 1000 times.
The second stage: generating data
This is a very important stage on which the subsequent steps depend, as the random variable that follows the Lomax distribution is generated by applying the inverse transformation method, as follows:
Taking the
The value of u is replaced by a generated value that follows a uniform distribution within the interval [0 , 1].
The third stage: The estimation
At this stage, the estimation process for the reliability function of the Lomax distribution is performed using the estimation methods mentioned in the theoretical aspect under the (RSS) method.
The fourth stage: the comparison stage between methods
To compare different estimation methods for the reliability function and find the best estimators, statistical criteria must be used, such as the MSE, since the method that has the lowest MSE value is considered better, such that:
since:
L: represents the number of repetitions for each experiment, which is equal to (1000).
: is an estimation of according to the used estimation method.
6-2 Experimental results using the (RSS) method.
To apply estimation methods for the reliability function of the Lomax distribution and determine the best method, which will be used in estimating the reliability function for real data in the applied aspect using the R program, the simulation results were presented in tables that included a comparison between the estimation methods for the reliability function under the (RSS) method.
Based on the equations (6, 7 & 8) for the (RSS) method, the equations (12, 13 & 24) for the MPS method, the equations (17, 18 & 19) for the OLS method, and the equations (22, 23 & 24) for the WLS method, the MSE criteria values were found for each estimation, and the results were as in the following tables.
Table (3). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 0.5 and a sample size of n = 12
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
3
|
4
|
0.05
|
0.9535
|
0.9539
|
0.9429
|
0.9454
|
0.9470
|
0.0011
|
0.0011
|
0.0010
|
0.001
|
MPS
|
|
0.5
|
0.7071
|
0.7278
|
0.6983
|
0.7026
|
0.7057
|
0.0152
|
0.0106
|
0.0111
|
0.0111
|
|
1.5
|
0.5
|
0.5173
|
0.5088
|
0.5052
|
0.5076
|
0.0226
|
0.0134
|
0.0152
|
0.0153
|
|
3.5
|
0.3536
|
0.3498
|
0.3711
|
0.3603
|
0.3615
|
0.0215
|
0.0125
|
0.0150
|
0.0146
|
|
5
|
0.3015
|
0.2887
|
0.3211
|
0.3085
|
0.3090
|
0.0196
|
0.0116
|
0.0141
|
0.0135
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0160
|
0.0098
|
0.0113
|
0.0111
|
|
6
|
2
|
0.05
|
0.9535
|
0.9507
|
0.9416
|
0.9437
|
0.9455
|
0.0020
|
0.0012
|
0.0012
|
0.0011
|
MPS
|
|
0.5
|
0.7071
|
0.7248
|
0.6948
|
0.6985
|
0.7021
|
0.0179
|
0.0106
|
0.0109
|
0.0109
|
|
1.5
|
0.5
|
0.5163
|
0.5047
|
0.5029
|
0.5049
|
0.0263
|
0.0127
|
0.0143
|
0.0143
|
|
3.5
|
0.3536
|
0.3493
|
0.3669
|
0.3600
|
0.3600
|
0.0250
|
0.0118
|
0.0141
|
0.0137
|
|
5
|
0.3015
|
0.2881
|
0.3170
|
0.3088
|
0.3080
|
0.0226
|
0.0111
|
0.0133
|
0.0129
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0188
|
0.0095
|
0.0108
|
0.0106
|
Table (4). Real and estimated reliability values and their associated MSE values when θ = 0.8, β = 0.5 and a sample size of n = 12
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
3
|
4
|
0.05
|
0.927
|
0.9297
|
0.9212
|
0.9231
|
0.9246
|
0.0027
|
0.0018
|
0.0016
|
0.002
|
WLS
|
|
0.5
|
0.574
|
0.6085
|
0.5969
|
0.5936
|
0.5984
|
0.0179
|
0.0127
|
0.0126
|
0.013
|
|
1.5
|
0.33
|
0.3373
|
0.3594
|
0.3523
|
0.354
|
0.021
|
0.0123
|
0.0131
|
0.012
|
|
3.5
|
0.189
|
0.1698
|
0.2104
|
0.2057
|
0.2015
|
0.016
|
0.0094
|
0.01
|
0.009
|
|
5
|
0.147
|
0.1242
|
0.1653
|
0.1619
|
0.1558
|
0.0128
|
0.0079
|
0.0084
|
0.007
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0141
|
0.0088
|
0.0092
|
0.009
|
|
6
|
2
|
0.05
|
0.927
|
0.9251
|
0.9151
|
0.9182
|
0.9196
|
0.0034
|
0.0021
|
0.0024
|
0.002
|
MPS
|
|
0.5
|
0.574
|
0.5908
|
0.5771
|
0.5838
|
0.5837
|
0.0319
|
0.0128
|
0.0143
|
0.014
|
|
1.5
|
0.33
|
0.3372
|
0.3543
|
0.3481
|
0.3482
|
0.0365
|
0.013
|
0.0146
|
0.015
|
|
3.5
|
0.189
|
0.1842
|
0.2197
|
0.2059
|
0.207
|
0.0232
|
0.01
|
0.0112
|
0.011
|
|
5
|
0.147
|
0.1383
|
0.1774
|
0.1629
|
0.1639
|
0.0171
|
0.0088
|
0.0095
|
0.009
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0224
|
0.0093
|
0.0104
|
0.01
|
Table (5). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 1.5 and a sample size of n = 12
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
3
|
4
|
0.05
|
0.97411
|
0.97645
|
0.97128
|
0.9719
|
0.9725
|
0.0001687
|
0.000177
|
0.00017
|
0.0002
|
OLS
|
|
0.5
|
0.79442
|
0.81217
|
0.78903
|
0.7912
|
0.79375
|
0.0070002
|
0.00556
|
0.00558
|
0.0056
|
|
1.5
|
0.57435
|
0.5977
|
0.58146
|
0.5821
|
0.58445
|
0.0175338
|
0.011001
|
0.01124
|
0.0119
|
|
3.5
|
0.38168
|
0.3882
|
0.39586
|
0.3929
|
0.39379
|
0.0203828
|
0.010728
|
0.01024
|
0.0111
|
|
5
|
0.30942
|
0.30472
|
0.32336
|
0.3184
|
0.31852
|
0.0192114
|
0.009614
|
0.00871
|
0.0093
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0128594
|
0.007416
|
0.00719
|
0.0076
|
|
6
|
2
|
0.05
|
0.97411
|
0.97268
|
0.96893
|
0.9702
|
0.97116
|
0.0002233
|
0.000124
|
9.7E-05
|
9E-05
|
MPS
|
|
0.5
|
0.79442
|
0.78811
|
0.76931
|
0.7721
|
0.77806
|
0.0064329
|
0.00357
|
0.00316
|
0.0028
|
|
1.5
|
0.57435
|
0.55391
|
0.54535
|
0.5376
|
0.54696
|
0.0151262
|
0.006968
|
0.00745
|
0.0064
|
|
3.5
|
0.38168
|
0.33636
|
0.35768
|
0.3356
|
0.3455
|
0.0173254
|
0.00803
|
0.01103
|
0.0093
|
|
5
|
0.30942
|
0.25478
|
0.28917
|
0.2631
|
0.27225
|
0.0156661
|
0.007782
|
0.01175
|
0.01
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0109548
|
0.005295
|
0.0067
|
0.0057
|
Table (6). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 0.5 and a sample size of n = 24
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
3
|
8
|
0.05
|
0.9535
|
0.9493
|
0.94444
|
0.9463
|
0.94798
|
0.0011
|
0.00065
|
0.00062
|
0.00054
|
MPS
|
|
0.5
|
0.7071
|
0.7097
|
0.69458
|
0.6981
|
0.70131
|
0.0096
|
0.00602
|
0.00634
|
0.00614
|
|
1.5
|
0.5
|
0.5042
|
0.50127
|
0.4997
|
0.501
|
0.0118
|
0.0067
|
0.00754
|
0.00743
|
|
3.5
|
0.3536
|
0.3497
|
0.36384
|
0.3572
|
0.35659
|
0.0101
|
0.00598
|
0.00704
|
0.00676
|
|
5
|
0.3015
|
0.2939
|
0.31441
|
0.3062
|
0.30489
|
0.0092
|
0.00561
|
0.00668
|
0.00632
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0084
|
0.00499
|
0.00564
|
0.00544
|
|
6
|
4
|
0.05
|
0.9535
|
0.952
|
0.94542
|
0.9477
|
0.94904
|
0.0008
|
0.0007
|
0.00068
|
0.00061
|
MPS
|
|
0.5
|
0.7071
|
0.7167
|
0.69892
|
0.7039
|
0.70645
|
0.0087
|
0.00621
|
0.00647
|
0.00638
|
|
1.5
|
0.5
|
0.5095
|
0.50566
|
0.5045
|
0.5059
|
0.0111
|
0.00701
|
0.00775
|
0.00772
|
|
3.5
|
0.3536
|
0.3527
|
0.36698
|
0.3594
|
0.35945
|
0.01
|
0.00624
|
0.00727
|
0.00702
|
|
5
|
0.3015
|
0.2961
|
0.3169
|
0.3073
|
0.30681
|
0.0091
|
0.00582
|
0.00691
|
0.00656
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0079
|
0.0052
|
0.00581
|
0.00566
|
Table (7). Real and estimated reliability values and their associated MSE values when θ = 0.8, β = 0.5 and a sample size of n = 24
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
3
|
8
|
0.05
|
0.9266
|
0.93
|
0.92028
|
0.9251
|
0.92592
|
0.0012
|
0.00099
|
0.00084
|
0.00081
|
MPS
|
|
0.5
|
0.5743
|
0.5975
|
0.57493
|
0.5783
|
0.5804
|
0.0137
|
0.00855
|
0.00899
|
0.00886
|
|
1.5
|
0.3299
|
0.341
|
0.3409
|
0.3311
|
0.33266
|
0.0124
|
0.00776
|
0.00887
|
0.00871
|
|
3.5
|
0.1895
|
0.1816
|
0.2023
|
0.1862
|
0.18714
|
0.0074
|
0.00495
|
0.00561
|
0.00545
|
|
5
|
0.1469
|
0.1345
|
0.15946
|
0.1428
|
0.14354
|
0.0055
|
0.00382
|
0.00415
|
0.00399
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.008
|
0.00521
|
0.00569
|
0.00556
|
|
6
|
4
|
0.05
|
0.9266
|
0.9244
|
0.91762
|
0.9213
|
0.92254
|
0.0016
|
0.00103
|
0.001
|
0.00092
|
MPS
|
|
0.5
|
0.5743
|
0.5802
|
0.56947
|
0.575
|
0.57585
|
0.0099
|
0.00607
|
0.00677
|
0.0069
|
|
1.5
|
0.3299
|
0.3237
|
0.33903
|
0.3332
|
0.33334
|
0.0105
|
0.0064
|
0.00728
|
0.00705
|
|
3.5
|
0.1895
|
0.1763
|
0.20539
|
0.193
|
0.19208
|
0.0082
|
0.00535
|
0.00601
|
0.00543
|
|
5
|
0.1469
|
0.1346
|
0.16439
|
0.1516
|
0.15004
|
0.0068
|
0.00467
|
0.00512
|
0.00455
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0074
|
0.0047
|
0.00523
|
0.00497
|
Table (8). Real and estimated reliability values and their associated MSE values when θ = 0.8, β = 1.5 and a sample size of n = 24
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
3
|
8
|
0.05
|
0.9837
|
0.9847
|
0.9801
|
0.982
|
0.9818
|
9.05E-05
|
0.00014
|
7.6E-05
|
9.5E-05
|
MPS
|
|
0.5
|
0.866
|
0.8772
|
0.8529
|
0.8604
|
0.86
|
0.003629
|
0.00362
|
0.00251
|
0.003
|
|
1.5
|
0.7071
|
0.7299
|
0.6997
|
0.7069
|
0.7071
|
0.009544
|
0.00625
|
0.00539
|
0.00608
|
|
3.5
|
0.5477
|
0.5707
|
0.5502
|
0.5547
|
0.5547
|
0.011957
|
0.00604
|
0.00622
|
0.0065
|
|
5
|
0.4804
|
0.4991
|
0.4866
|
0.4899
|
0.4893
|
0.011595
|
0.00562
|
0.0061
|
0.00617
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.007363
|
0.00434
|
0.00406
|
0.00437
|
|
6
|
4
|
0.05
|
0.9837
|
0.9849
|
0.9828
|
0.9832
|
0.9838
|
4.76E-05
|
2.5E-05
|
2.4E-05
|
2.2E-05
|
MPS
|
|
0.5
|
0.866
|
0.878
|
0.8641
|
0.8663
|
0.8698
|
0.001848
|
0.00095
|
0.00093
|
0.00094
|
|
1.5
|
0.7071
|
0.7304
|
0.713
|
0.714
|
0.7188
|
0.004533
|
0.00252
|
0.00262
|
0.00277
|
|
3.5
|
0.5477
|
0.5694
|
0.5643
|
0.5618
|
0.566
|
0.006011
|
0.00394
|
0.00424
|
0.00442
|
|
5
|
0.4804
|
0.4969
|
0.5013
|
0.4968
|
0.5005
|
0.006591
|
0.0045
|
0.00486
|
0.00496
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.003806
|
0.00239
|
0.00254
|
0.00262
|
Table (9). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 0.5 and a sample size of n = 36
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
9
|
4
|
0.05
|
0.9535
|
0.9524
|
0.9465
|
0.9481
|
0.9494
|
0.0005
|
0.0005
|
0.0004
|
0.0004
|
MPS
|
|
0.5
|
0.7071
|
0.7131
|
0.6966
|
0.7002
|
0.7027
|
0.0062
|
0.0044
|
0.0045
|
0.0044
|
|
1.5
|
0.5
|
0.5051
|
0.4999
|
0.4994
|
0.5002
|
0.0073
|
0.0046
|
0.005
|
0.005
|
|
3.5
|
0.3536
|
0.3505
|
0.3604
|
0.3556
|
0.3547
|
0.0061
|
0.0041
|
0.0045
|
0.0044
|
|
5
|
0.3015
|
0.2949
|
0.3105
|
0.3042
|
0.3027
|
0.0057
|
0.0038
|
0.0043
|
0.0041
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0051
|
0.0035
|
0.0037
|
0.0036
|
|
6
|
6
|
0.05
|
0.9535
|
0.9524
|
0.9473
|
0.949
|
0.9503
|
0.0004
|
0.0004
|
0.0004
|
0.0004
|
MPS
|
|
0.5
|
0.7071
|
0.7121
|
0.699
|
0.7022
|
0.7052
|
0.005
|
0.0041
|
0.0042
|
0.0042
|
|
1.5
|
0.5
|
0.505
|
0.5024
|
0.5011
|
0.5025
|
0.006
|
0.0044
|
0.0048
|
0.0048
|
|
3.5
|
0.3536
|
0.3525
|
0.3626
|
0.3569
|
0.3565
|
0.0051
|
0.0038
|
0.0043
|
0.0041
|
|
5
|
0.3015
|
0.2978
|
0.3125
|
0.3053
|
0.3042
|
0.0047
|
0.0035
|
0.004
|
0.0038
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0042
|
0.0032
|
0.0035
|
0.0035
|
Table (10). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 1.5 and a sample size of
n = 36
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
9
|
4
|
0.05
|
0.9837
|
0.9862
|
0.9839
|
0.9849
|
0.9851
|
4.7E-5
|
3.9E-5
|
3.8E-5
|
3.4E-5
|
MPS
|
|
0.5
|
0.866
|
0.886
|
0.8719
|
0.8777
|
0.8786
|
0.0022
|
0.0015
|
0.0016
|
0.0015
|
|
1.5
|
0.7071
|
0.7436
|
0.7241
|
0.7317
|
0.732
|
0.0067
|
0.0039
|
0.0047
|
0.0044
|
|
3.5
|
0.5477
|
0.5861
|
0.573
|
0.5772
|
0.5766
|
0.009
|
0.0052
|
0.0063
|
0.0059
|
|
5
|
0.4804
|
0.5144
|
0.5074
|
0.5087
|
0.5077
|
0.0089
|
0.0053
|
0.0063
|
0.006
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0054
|
0.0032
|
0.0038
|
0.0036
|
|
6
|
6
|
0.05
|
0.9837
|
0.9829
|
0.9804
|
0.9808
|
0.9815
|
5E-05
|
7E-05
|
7E-05
|
6E-05
|
MPS
|
|
0.5
|
0.866
|
0.8646
|
0.8514
|
0.8542
|
0.8573
|
0.002
|
0.0022
|
0.0019
|
0.0019
|
|
1.5
|
0.7071
|
0.7107
|
0.6955
|
0.6999
|
0.7023
|
0.0053
|
0.0045
|
0.0042
|
0.0043
|
|
3.5
|
0.5477
|
0.5547
|
0.5466
|
0.5508
|
0.5506
|
0.007
|
0.0051
|
0.0054
|
0.0055
|
|
5
|
0.4804
|
0.4873
|
0.4841
|
0.488
|
0.4863
|
0.0069
|
0.005
|
0.0056
|
0.0056
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0042
|
0.0034
|
0.0034
|
0.0034
|
Table (11). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 0.5 and a sample size of n = 54
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
9
|
6
|
0.05
|
0.9535
|
0.9526
|
0.9482
|
0.9495
|
0.951
|
0.0004
|
0.0003
|
0.0003
|
0.00029
|
MPS
|
|
0.5
|
0.7071
|
0.7112
|
0.6993
|
0.7024
|
0.704
|
0.0038
|
0.0033
|
0.0032
|
0.00314
|
|
1.5
|
0.5
|
0.5031
|
0.5005
|
0.4995
|
0.5
|
0.0036
|
0.003
|
0.003
|
0.00309
|
|
3.5
|
0.3536
|
0.3507
|
0.3591
|
0.3541
|
0.354
|
0.0026
|
0.0021
|
0.0024
|
0.00234
|
|
5
|
0.3015
|
0.2962
|
0.3083
|
0.3022
|
0.301
|
0.0023
|
0.0018
|
0.0023
|
0.00209
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0025
|
0.0021
|
0.0022
|
0.00219
|
|
3
|
18
|
0.05
|
0.9535
|
0.9498
|
0.946
|
0.9472
|
0.948
|
0.0007
|
0.0005
|
0.0004
|
0.00039
|
MPS
|
|
0.5
|
0.7071
|
0.7067
|
0.6947
|
0.6968
|
0.7
|
0.0051
|
0.0038
|
0.0036
|
0.0037
|
|
1.5
|
0.5
|
0.5011
|
0.4965
|
0.496
|
0.497
|
0.0055
|
0.0035
|
0.0036
|
0.00365
|
|
3.5
|
0.3536
|
0.3516
|
0.3565
|
0.3533
|
0.352
|
0.0048
|
0.0027
|
0.0032
|
0.00301
|
|
5
|
0.3015
|
0.2984
|
0.3065
|
0.3024
|
0.301
|
0.0045
|
0.0025
|
0.0031
|
0.00278
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0041
|
0.0026
|
0.0028
|
0.00271
|
Table (12). Real and estimated reliability values and their associated MSE values when θ = 0.8, β = 0.5 and a sample size of n = 54
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
9
|
6
|
0.05
|
0.9266
|
0.9269
|
0.9229
|
0.9237
|
0.925
|
0.0005
|
0.0004
|
0.0005
|
0.00039
|
MPS
|
|
0.5
|
0.5743
|
0.5793
|
0.5756
|
0.5777
|
0.578
|
0.0043
|
0.0028
|
0.0032
|
0.0031
|
|
1.5
|
0.3299
|
0.3274
|
0.3393
|
0.3379
|
0.336
|
0.0036
|
0.0024
|
0.0028
|
0.0027
|
|
3.5
|
0.1895
|
0.1808
|
0.2014
|
0.1977
|
0.194
|
0.0026
|
0.0018
|
0.002
|
0.00183
|
|
5
|
0.1469
|
0.1376
|
0.1589
|
0.155
|
0.152
|
0.0022
|
0.0015
|
0.0017
|
0.0015
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0026
|
0.0018
|
0.002
|
0.0019
|
|
3
|
18
|
0.05
|
0.9266
|
0.9259
|
0.9183
|
0.919
|
0.92
|
0.0007
|
0.0006
|
0.0006
|
0.00054
|
WLS
|
|
0.5
|
0.5743
|
0.5813
|
0.565
|
0.5663
|
0.567
|
0.0058
|
0.0035
|
0.0037
|
0.00359
|
|
1.5
|
0.3299
|
0.3342
|
0.3341
|
0.3324
|
0.332
|
0.0059
|
0.0029
|
0.003
|
0.003
|
|
3.5
|
0.1895
|
0.1907
|
0.2008
|
0.1975
|
0.196
|
0.0045
|
0.0023
|
0.0022
|
0.00223
|
|
5
|
0.1469
|
0.1479
|
0.1597
|
0.1561
|
0.154
|
0.0037
|
0.002
|
0.0019
|
0.0019
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0041
|
0.00225
|
0.0023
|
0.002252
|
Table (13). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 1.5 and a sample size of n = 54
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
9
|
6
|
0.05
|
0.9837
|
0.9878
|
0.9856
|
0.9856
|
0.986
|
3E-05
|
2E-05
|
2E-05
|
2.4E-05
|
MPS
|
|
0.5
|
0.866
|
0.8955
|
0.8809
|
0.8812
|
0.883
|
0.0017
|
0.001
|
0.0011
|
0.00116
|
|
1.5
|
0.7071
|
0.7575
|
0.7352
|
0.7363
|
0.739
|
0.005
|
0.0029
|
0.0033
|
0.0034
|
|
3.5
|
0.5477
|
0.6005
|
0.5813
|
0.5826
|
0.584
|
0.0062
|
0.0038
|
0.0041
|
0.00425
|
|
5
|
0.4804
|
0.5284
|
0.5136
|
0.5146
|
0.515
|
0.0057
|
0.0037
|
0.0038
|
0.00397
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.0037
|
0.0023
|
0.0025
|
0.00256
|
|
3
|
18
|
0.05
|
0.9837
|
0.9839
|
0.9811
|
0.9815
|
0.982
|
4E-05
|
3E-05
|
3E-05
|
2.1E-05
|
MPS
|
|
0.5
|
0.866
|
0.8696
|
0.8521
|
0.8543
|
0.857
|
0.0017
|
0.001
|
0.0009
|
0.00085
|
|
1.5
|
0.7071
|
0.7155
|
0.6923
|
0.6942
|
0.697
|
0.0048
|
0.0022
|
0.0021
|
0.00208
|
|
3.5
|
0.5477
|
0.556
|
0.5401
|
0.5402
|
0.542
|
0.0066
|
0.0025
|
0.0026
|
0.00255
|
|
5
|
0.4804
|
0.4865
|
0.4769
|
0.476
|
0.478
|
0.0067
|
0.0024
|
0.0026
|
0.00255
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.004
|
0.0016
|
0.0017
|
0.00161
|
Table (14). Real and estimated reliability values and their associated MSE values when θ = 0.8, β = 0.5 and a sample size of n = 96
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
12
|
8
|
0.05
|
0.9535
|
0.952
|
0.9489
|
0.9498
|
0.9504
|
0.00018
|
0.00018
|
0.000159
|
0.00016
|
MPS
|
|
0.5
|
0.7071
|
0.7055
|
0.6982
|
0.6992
|
0.701
|
0.002
|
0.00177
|
0.001752
|
0.00178
|
|
1.5
|
0.5
|
0.4986
|
0.4978
|
0.4958
|
0.497
|
0.00216
|
0.00173
|
0.001966
|
0.00189
|
|
3.5
|
0.3536
|
0.3511
|
0.3571
|
0.3527
|
0.3529
|
0.00192
|
0.00148
|
0.001889
|
0.00166
|
|
5
|
0.3015
|
0.2987
|
0.3069
|
0.3019
|
0.3017
|
0.00181
|
0.0014
|
0.001842
|
0.00157
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.00161
|
0.00131
|
0.001521
|
0.00141
|
|
16
|
6
|
0.05
|
0.9535
|
0.9547
|
0.9515
|
0.9524
|
0.9531
|
0.00011
|
0.00013
|
0.000122
|
0.00012
|
MPS
|
|
0.5
|
0.7071
|
0.7157
|
0.7077
|
0.7094
|
0.7112
|
0.00164
|
0.00145
|
0.001465
|
0.00149
|
|
1.5
|
0.5
|
0.5094
|
0.5074
|
0.5067
|
0.5074
|
0.00186
|
0.0015
|
0.001552
|
0.00156
|
|
3.5
|
0.3536
|
0.3599
|
0.3646
|
0.3614
|
0.3609
|
0.00157
|
0.00129
|
0.001451
|
0.00134
|
|
5
|
0.3015
|
0.3063
|
0.3133
|
0.3094
|
0.3084
|
0.00147
|
0.00122
|
0.001447
|
0.00128
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.00133
|
0.00112
|
0.001207
|
0.00116
|
Table (15). Real and estimated reliability values and their associated MSE values when θ = 0.8, β = 0.5 and a sample size of n = 96
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
12
|
8
|
0.05
|
0.9266
|
0.9296
|
0.925
|
0.9273
|
0.9277
|
0.00028
|
0.00019
|
0.000143
|
0.00015
|
WLS
|
|
0.5
|
0.5743
|
0.5899
|
0.581
|
0.5831
|
0.5837
|
0.00182
|
0.00129
|
0.00131
|
0.00137
|
|
1.5
|
0.3299
|
0.342
|
0.3449
|
0.3415
|
0.3411
|
0.00126
|
0.00124
|
0.001162
|
0.00117
|
|
3.5
|
0.1895
|
0.1955
|
0.2063
|
0.1999
|
0.1988
|
0.001
|
0.00116
|
0.000937
|
0.00089
|
|
5
|
0.1469
|
0.1511
|
0.1633
|
0.1564
|
0.1552
|
0.00093
|
0.00109
|
0.000834
|
0.00077
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.00106
|
0.00099
|
0.000877
|
0.00087
|
|
16
|
6
|
0.05
|
0.9266
|
0.9258
|
0.9238
|
0.9249
|
0.9255
|
0.00028
|
0.0002
|
0.000232
|
0.00021
|
MPS
|
|
0.5
|
0.5743
|
0.5741
|
0.5717
|
0.5737
|
0.5744
|
0.00217
|
0.00128
|
0.001335
|
0.00138
|
|
1.5
|
0.3299
|
0.3256
|
0.331
|
0.3289
|
0.3289
|
0.00116
|
0.00072
|
0.000724
|
0.00072
|
|
3.5
|
0.1895
|
0.1819
|
0.1921
|
0.1876
|
0.1871
|
0.00075
|
0.00045
|
0.000614
|
0.00048
|
|
5
|
0.1469
|
0.139
|
0.1497
|
0.1449
|
0.1443
|
0.00069
|
0.00039
|
0.000607
|
0.00044
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.00101
|
0.00061
|
0.000702
|
0.00065
|
Table (16). Real and estimated reliability values and their associated MSE values when θ = 0.5, β = 1.5 and a sample size of n = 96
|
m
|
k
|
t
|
Values
|
MSE
|
Best
|
|
Real
|
MLE
|
MPS
|
OLS
|
WLS
|
MLE
|
MPS
|
OLS
|
WLS
|
|
12
|
8
|
0.05
|
0.9837
|
0.9825
|
0.9822
|
0.9827
|
0.983
|
1.72E-05
|
1.76E-05
|
1.94E-05
|
1.61E-05
|
MPS
|
|
0.5
|
0.866
|
0.8597
|
0.8583
|
0.8609
|
0.8627
|
0.00075
|
0.00074
|
0.000832
|
0.00073
|
|
1.5
|
0.7071
|
0.7003
|
0.6993
|
0.7024
|
0.7042
|
0.00206
|
0.00192
|
0.002197
|
0.00203
|
|
3.5
|
0.5477
|
0.5435
|
0.5443
|
0.5455
|
0.5461
|
0.00282
|
0.00247
|
0.002771
|
0.00267
|
|
5
|
0.4804
|
0.4775
|
0.4794
|
0.4792
|
0.4791
|
0.0029
|
0.00248
|
0.002724
|
0.00267
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.00171
|
0.00152
|
0.001709
|
0.00162
|
|
16
|
6
|
0.05
|
0.9837
|
0.9838
|
0.9833
|
0.9834
|
0.9838
|
1.76E-05
|
1.54E-05
|
1.70E-05
|
1.44E-05
|
MPS
|
|
0.5
|
0.866
|
0.8675
|
0.8647
|
0.8652
|
0.8674
|
0.0008
|
0.00069
|
0.000739
|
0.00067
|
|
1.5
|
0.7071
|
0.7107
|
0.7077
|
0.7078
|
0.7105
|
0.00219
|
0.00185
|
0.001951
|
0.00189
|
|
3.5
|
0.5477
|
0.5512
|
0.5512
|
0.5499
|
0.5518
|
0.00275
|
0.00238
|
0.002468
|
0.00248
|
|
5
|
0.4804
|
0.4829
|
0.4849
|
0.4829
|
0.4842
|
0.00265
|
0.00236
|
0.002435
|
0.00247
|
|
ALL
|
-
|
-
|
-
|
-
|
-
|
0.00168
|
0.00146
|
0.001522
|
0.00151
|
7- The Applied Side:
Bladder cancer is one of the common diseases that pose a threat to human life. It is also one of the diseases whose degree of reliability cannot be predicted or measured except after some time from the spread of the disease. In this paper, data were obtained for 96 bladder cancer patients, and the data represents. The period of remission of the disease in months (beginning of complete recovery). Data were taken from (Rady, et al, 2016).
7-1 Goodness-of-fit tests
In order to ensure that the data represented by the patient’s length of stay in the hospital follow a Lomax distribution, the Kolmogorov-Smirnov (K-S) test, the Anderson Darling (A-D) test, and the Chi-Squared test were used, as the null hypothesis states that the data have a Lomax distribution, as follows:
Distribution suitability graphs for the data under study were obtained using the statistical program EasyFit 5.6, and were as follows:
Figure (1) Probability density function curve for the Lomax distribution for real data
Figure (2) Cumulative distribution function curve for the Lomax distribution for real data
The results of goodness-of-fit tests also indicate that the data has a Lomax distribution, as follows:
Table (17). Chi-square test results for goodness of fit
|
Test
|
Calculated value
|
Critical value
|
Result
|
|
Kolmogorov-Smirnov
|
0.0967
|
0.12
|
follows Lomax distribution
|
|
Anderson Darling
|
1.3768
|
2.5018
|
follows Lomax distribution
|
|
Chi-Squared
|
6.1
|
12.592
|
follows Lomax distribution
|
7-2 Estimation using the (RSS) method
RSS technology was used to draw a sample size of 96 from the total sample according to the number of cycles (k=16) and the number of units in each group (m=6), and the drawn sample was as follows:
Table (18). Duration of hospital stay in months for the sample using (RSS)
|
3.64
|
9.02
|
7.66
|
5.85
|
7.87
|
26.31
|
|
2.46
|
3.52
|
3.31
|
9.74
|
10.75
|
11.79
|
|
1.26
|
3.7
|
3.36
|
7.28
|
5.06
|
18.1
|
|
2.83
|
7.26
|
2.64
|
13.11
|
5.41
|
4.33
|
|
0.81
|
3.36
|
9.74
|
7.63
|
13.29
|
79.05
|
|
1.35
|
2.09
|
6.97
|
6.76
|
3.64
|
17.14
|
|
7.39
|
0.81
|
7.28
|
23.63
|
12.03
|
21.73
|
|
3.64
|
1.35
|
7.59
|
6.25
|
14.77
|
5.34
|
|
0.81
|
1.46
|
5.49
|
11.64
|
7.62
|
18.1
|
|
0.9
|
1.05
|
9.02
|
7.59
|
17.36
|
20.28
|
|
5.17
|
9.22
|
0.81
|
10.75
|
10.66
|
16.62
|
|
2.87
|
0.9
|
5.09
|
8.65
|
7.09
|
79.05
|
|
0.9
|
4.23
|
12.07
|
7.66
|
9.74
|
19.13
|
|
3.82
|
5.32
|
4.26
|
14.77
|
16.62
|
8.37
|
|
1.4
|
6.54
|
4.34
|
13.8
|
14.24
|
21.73
|
|
0.81
|
5.32
|
5.41
|
5.85
|
17.36
|
34.26
|
The two parameters of the Lomax distribution were estimated using the (MPS) method because it is the best method in simulation experiments. The results were ( 6.563425; 55.962806), and based on these estimates the reliability function for this distribution was estimated, as follows:
based on this equation, the reliability function was estimated at the survival times of the studied data, as follows:
Table (19). Values of the reliability function estimated using the (RSS) method
|
t
|
|
t
|
|
t
|
|
|
0.81
|
0.9100
|
5.17
|
0.5599
|
10.66
|
0.3184
|
|
0.90
|
0.9006
|
5.32
|
0.5510
|
10.75
|
0.3156
|
|
1.05
|
0.8851
|
5.34
|
0.5498
|
11.64
|
0.2893
|
|
1.26
|
0.8640
|
5.41
|
0.5457
|
11.79
|
0.2851
|
|
1.35
|
0.8552
|
5.49
|
0.5411
|
12.03
|
0.2786
|
|
1.40
|
0.8503
|
5.85
|
0.5207
|
12.07
|
0.2775
|
|
1.46
|
0.8445
|
6.25
|
0.4991
|
13.11
|
0.2512
|
|
2.09
|
0.7861
|
6.54
|
0.4841
|
13.29
|
0.2470
|
|
2.46
|
0.7540
|
6.76
|
0.4731
|
13.80
|
0.2354
|
|
2.64
|
0.7389
|
6.97
|
0.4628
|
14.24
|
0.2258
|
|
2.83
|
0.7234
|
7.09
|
0.4571
|
14.77
|
0.2150
|
|
2.87
|
0.7202
|
7.26
|
0.4491
|
16.62
|
0.1815
|
|
3.31
|
0.6858
|
7.28
|
0.4481
|
17.14
|
0.1731
|
|
3.36
|
0.6820
|
7.39
|
0.4430
|
17.36
|
0.1698
|
|
3.52
|
0.6701
|
7.59
|
0.4340
|
18.10
|
0.1589
|
|
3.64
|
0.6613
|
7.62
|
0.4326
|
19.13
|
0.1452
|
|
3.70
|
0.6569
|
7.63
|
0.4322
|
20.28
|
0.1314
|
|
3.82
|
0.6483
|
7.66
|
0.4309
|
21.73
|
0.1161
|
|
4.23
|
0.6199
|
7.87
|
0.4216
|
23.63
|
0.0991
|
|
4.26
|
0.6178
|
8.37
|
0.4006
|
26.31
|
0.0797
|
|
4.33
|
0.6132
|
8.65
|
0.3893
|
34.26
|
0.0435
|
|
4.34
|
0.6125
|
9.02
|
0.3750
|
79.05
|
0.0031
|
|
5.06
|
0.5666
|
9.22
|
0.3675
|
|
|
|
5.09
|
0.5648
|
9.74
|
0.3488
|
|
|
The estimated reliability function has been plotted with the true reliability function as follows:
Figure (3). Drawing the true and estimated reliability functions based on the (RSS) method
Discussion
The observed and recorded data in the current work were consistent with Lozano et al. (17), who found that the onset of puberty and prevalence of estrus behavior started at 5.4 to 6.9 months. In addition to that, the bodyweight data agreed with Ehtesham and Vakili, Nieto et al. (18,19) that recorded an increase in body weight at the time of puberty 210-240 days, reaching 35 Kg. Body mass plays a crucial role via changes in growth hormone during aging, especially at the early stages of life till puberty, with the consequence to fat deposition in the body. Fat deposition plays a vital role in puberty and the onset of estrus via increased leptin formation and its release in the body (20). This is logically accepted and supports our finding, which records an increase in leptin level as age progresses and increases body weight (21,22). Therefore, body weight strongly correlates with reproductive health system function and activity in terms of interference with sexual hormones.