Moments of Dual Generalized Order Statistics from Inverted Kumaraswamy Distribution and Related Inference
- https://doi.org/10.2991/jsta.d.210219.001How to use a DOI?
- Moments, Dual generalized order statistics, Maximum likelihood estimation, Uniformly minimum variance unbiased estimation
In this paper, the exact and explicit expressions for single, product, and conditional moments for inverted Kumaraswamy distribution using dual generalized order statistics (dgos) are derived. Further, the expression for maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for the parameters of inverted Kumaraswamy distribution based on dgos are deduced. Also, we obtained the results for order statistics and lower record values by putting some specific values of the parameters of dgos. Finally, a simulation study is carried out for illustrative purpose.
- © 2021 The Authors. Published by Atlantis Press B.V.
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Generalized order statistics (gos) was introduced by Kamps  as a unified model of ordered random variables arranged in increasing order of magnitude which contains almost all models related to ordered random variables such as order statistics, records, Pfeifer records, progressive type-II censored order statistics, and sequential order statistics. But those models of ordered random variables which are in decreasing order of magnitude cannot be studied in this framework. To resolve such types of problems, Pawlas and Szynal  proposed the concept of dual (lower) generalized order statistics (dgos) which contains several models of ordered random variables those are arranged in decreasing order of magnitude, like reversed order statistics, lower records, and lower Pfeifer records. Burkschat et al.  established the direct relationship between gos and dgos in such a way that if are gos with cumulative distribution function (cdf) and are dgos with (cdf) , thenwhere denotes the convergence in distribution.
Therefore, to avoid the complications in calculations, one can use dgos when is available in compact and closed form and when a closed and compact form of is available, gos can be preferred.
Let be a sequence of independent and identically distributed (iid) random variables with absolutely continuous distribution function (df) and the probability density function (pdf) , . Further, let , , , , , such that for all . Then , are said to be dgos if their joint pdf is given by
Note that reduces to the order statistics from a sample of size if (i.e. and ) and when , reduces to lower records.
Here, we shall consider two cases:
Case I: , i.e. 's are pairwise different.
The pdf of is given by (Kamps )
And the joint of and is:
Therefore, the conditional of given , for is:
It may be noted that for , (Khan and Khan )
Case II: When :
The of is given by
Further, the conditional pdf of given , is
And the conditional pdf of given , is
Here, the results are derived for case-I as it includes case-II as a particular case of it.
In the last two decades, several researchers have analyzed statistical properties of continuous distributions based on gos, dgos and hence a wide literature is available on the various developments on dgos and gos. Pawlas and Szynal  discussed the relation for single and product moments of gos from Pareto, generalized Pareto, and Burr distributions. Cramer et al.  proved the existence of moments of gos. Ahsanullah  characterized the uniform distribution based on dgos. Athar et al.  obtained the explicit expressions for ratio and inverse moments of gos from Weibull distribution. Explicit expressions for single and product moments of gos from linear exponential distribution have been derived by Ahmad . Based on dgos, Khan et al.  established the recurrence relations for single and product moments of exponentiated Weibull distribution. Khan et al.  have characterized continuous distributions conditioned on a pair of nonadjacent dgos. Burkschat  has discussed the linear estimators and predictors from generalized Pareto distribution based on gos. Barakat and Adll  studied the asymptotic theory of extreme dgos. Athar and Faizan  deduced the explicit expression for moments of power function distribution based on dgos. Khan and Khan  obtained the ratio and inverse moments of gos from Burr distribution using hypergeometric functions. Domma and Hamdeni  discussed the truncated moments of dgos. Recurrence relations for moments of dgos from Weibull gamma distribution have been discussed by Mahmoud et al. . Recurrence relation for single and product moments of dgos from a general class of distribution have been derived by Saran et al. . Based on lower records, Khan and Arshad  have derived the uniformly minimum variance unbiased estimator(UMVUE) of reliability function and stress–strength reliability from proportional reversed hazard rate model. An exact and explicit expression of moments of Topp–Leone distribution based dgos have been deduced by Khan and Iqrar . They have also derived the expressions for maximum likelihood estimator(MLE) and UMVUE for the parameter of Topp–Leone distribution based on dgos. Interval estimation for Topp–Leone generated family of distributions based on dgos have been considered by Arshad and Jamal . Khatoon et al.  have considered the single, product, and conditional moments of gos as well as parameter estimation based on gos from Kumaraswamy power function distribution.
Kumaraswamy distribution was introduced by Kumaraswamy . This distribution is applicable in many real-life situations and natural phenomenon that are restricted with lower and upper bounds. Over the last decades, there has been a great interest in studying the Kumaraswamy distribution among the researchers. Few of them are Nadarajah , Jones , Garg , Cordeiro and De-Castro , Mitnik , Safi and Ahmed , El-Deen et al. . El-Deen et al.  studied the statistical inference for this distribution based on gos. Abd Al-Fattah et al.  introduced the inverted Kumaraswamy distribution as the inverted distributions have wide range of applications in real-life situations. Usman and Ahsan-ul-Haque , Reyad et al.  have studied the generalization form of inverted Kumaraswamy distribution.
The pdf of inverted Kumaraswamy distribution is given by
And its cdf is given by
This paper comprises of 5 sections. In Section 2, the results for single, product, and conditional moments of inverted Kumaraswamy distribution based on dgos have been derived. Section 3 included the theoretical setup and expressions for the MLE of both parameters of inverted Kumaraswamy distribution based on dgos. In Section 4, we have deduced the UMVUE of one shape parameter of inverted Kumaraswamy distribution by assuming that the other shape parameter is known. Finally, a simulation study is being carried out to check the validity and applicability of the derived estimators based on order statistics and lower records which is given in Section 5.
2. MOMENTS OF dgos
In this section, we have derived the exact and explicit expressions for single, product, and conditional moments of inverted Kumaraswamy distribution based on dgos. Further, by putting the specific values of the parameters and , we have deduced single moment, product moment, and conditional moments of order statistics and lower records from inverted Kumaraswamy distribution Table 1-6.
Mean of order statistics from inverted Kumaraswamy distribution for , .
Variance of order statistics from inverted Kumaraswamy distribution for , .
Mean of lower records from inverted Kumaraswamy distribution for , .
Variance of lower records from inverted Kumaraswamy distribution for , .
Mean of dgos from inverted Kumaraswamy distribution for , .
Variance of dgos from inverted Kumaraswamy distribution for , .
2.1. Single Moment
Let be continuous iid non negative random variables and follow inverted Kumaraswamy distribution given in (1.12), then the single moment of dgos is given by
In case of dgos, for the inverted Kumaraswamy distribution given in (1.12), the single moment of dgos is given by
Putting and simplifying accordingly, we get
Now, using the relation , we obtained the result.
In case of lower records, the single moment from inverted Kumaraswamy distribution is given by
Assuming and simplifying appropriately, we get
Using the relation , (Prudnikov et al. , p. 502), we obtained the result.
It is pertinent to mention that the computation of can further be simplified by using the recurrence relationwhere is the derivative of diagamma function given by and is the gamma function (Khan et al. ).
for , the single moment of inverted Kumaraswamy distribution is given by
When , the single moment of order statistics from inverted Kumaraswamy distribution is given by
2.2. Product Moments
In this section, the product moment of and dgos is computed in terms of hypergeometric function from inverted Kumaraswamy distribution. Further, by choosing the different combination of the parameters of dgos (i.e., and ), we reduced the obtained result in terms of order statistics and lower record values. The obtained result is summarized in the form of theorem which is given below.
Let be n continuous iid non negative random variables that follow inverted kumaraswamy distribution given in (1.12) and be their corresponding dgos. For any and , the product moment of and dgos is given by
(Mathai and Saxena ).
Let and simplifying accordingly, we have
Again putting , the product moment of inverted Kumaraswamy distribution based on dgos is given byor
Now using the relation (Mathai and Saxena )we have,
Hence, we get the result.
However, in case of lower records, an exact expression for product moment from inverted Kumaraswamy distribution could not be obtained.
For , the product moment from inverted Kumaraswamy distribution is given as
In case of order statistics, i.e., , the product moment from inverted Kumaraswamy distribution is given by
2.3. Conditional Moments
In this section, we have obtained the conditional moments of given dgos () and given dgos () from inverted Kumaraswamy Distribution in terms of incomplete beta function.
The qth () conditional moment of sth given rth dgos (), , from inverted Kumaraswamy distribution is given by
The conditional moment of is given by
Putting , we get
Setting and then using the relation given in (2.9), we obtained the result.
The qth , conditional moment of , , from inverted Kumaraswamy distribution is given by
3. MLE FOR THE SHAPE PARAMETERS AND
In this section, we have deduced expressions for MLE of the shape parameters of the inverted Kumaraswamy distribution based on dgos.
Estimation of parameters through MLE based on gos and dgos has been discussed by many authors. Habibullah and Ahsanullah  have discussed the parameters of Pareto distribution based on gos. MLE and Bayes estimator of the parameters of Burr-XII distribution based on gos have been obtained by Jaheen  after that Malinowska et al.  estimated the location and scale parameter of Burr-XII distribution for gos. Bayesian and nonBayesian estimation for Weibull distribution based on gos have been given by Abo-eleneen . Abo-Elfotouh and Nassar  have considered the MLE and Bayesion estimation of parameters of Weibull extension model for gos. Safi and Ahmed  have discussed the MLE for the parameter of Kumaraswamy distribution based on gos. MLE and Bayesian estimator for the parameter of Kumaraswamy distribution based on gos have been obtained by El-Deen et al. . Kim  have discussed the parameter estimation of generalized exponential distribution under dgos. MLE, UMVUE, and Bayes estimates of stress–strength reliability based on gos for exponential distribution have been Khan and Khatoon .
The log likelihood function of (3.1) is given by
Differentiating both sides of (3.2) w.r.t. , and equating to zero, we get
In case of lower records , the likelihood equation is given by
And the MLE of and in case of lower records can be obtained by solving the normal equations
4. UMVUE AND MLE OF WHEN IS KNOWN
In this section, we have obtained UMVUE and MLE of parameter of the inverted Kumaraswamy distribution based on dgos when is known say .
Replacing in the equation (3.1) and re-write the joint pdf of dgos from inverted Kumaraswamy distribution as
From equation (4.1), it can be easily seen that is a complete sufficient statistics for . From the normalized spacing of dgos, one can easily prove that follows gamma distribution with shape parameter and scale parameter denoting (see Arshad and Jamal ). Therefore,and hence the UMVUE of is
Further, adopting the same procedure described in the previous section (3), the MLE of is given by
In case of lower records, UMVUE and MLE can be obtained as
5. SIMULATION STUDY
In this section, the MLE for both the shape parameters, MLE and UMVUE of one shape parameter by taking another one is known of inverted Kumaraswamy distribution based on order statistics and lower records are computed. Here, R software is used for data simulation and computation purpose.
When both the shape parameters are unknown, we considered a Monte Carlo method of simulation to obtain the MLEs for the parameters by considering that , sample size for order statistics and for lower records . For the entire simulation, we consider 1000 replication of the process and the average estimate, bias, and mean square error (MSE) are computed for order statistics and lower records which are reported in Tables 7 and 8 respectively.
MLE of parameters of inverted Kumaraswamy distribution based on order statistics.
MLE of parameters of inverted Kumaraswamy distribution based on lower records.
For another case, i.e., when one shape parameter is known, the data is generated from and the MLE and UMVUE are obtained for for fixed value of , for order statistics and for lower records. Based on 1000 repetition, the average MSE for MLE and UMVUE and average bias for MLE also are noted which are listed in Tables 9 and 10 for order statistics and lower records, respectively.
MLE and UMVUE of when is known for inverted Kumaraswamy distribution based on order statistics.
MLE and UMVUE of when is known for inverted Kumaraswamy distribution based on lower records.
From Tables 7 and 8, one can easily see that for various configuration of parameters, biases and MSEs are decreasing as the sample size is increasing for order statistics and lower record case. Also it is noted at , MSEs of are smaller than MSEs of and when , MSE () MSE () for all choices of sample sizes. Moreover, it is noted that order statistics performed better compared to lower records based on the simulated result.
From Tables 9 and 10, it is evident that for all the parameter values and sample sizes, , i.e., UMVUE are more adequate than MLE for both order statistics and lower records, and also UMVUE free from bias. Based on the simulated result, we can conclude that UMVUE perform better than MLE when one shape parameter is known and in this situation, use of UMVUE is recommended.
In this paper, we have obtained exact and explicit expression for single, product, and conditional moments of dgos from inverted Kumaraswamy distribution. The results are obtained in terms of Gauss hypergeometric and beta functions. Further, by adjusting the parameter values of dgos, we obtained the moments of order statistics and lower record values. Based on these results, mean and variances of order statistics, lower record values, and dgos are calculated. We have also obtained MLE for all shape parameters and UMVUE for one shape parameter by taking another one is known as inverted Kumaraswamy distribution based on dgos. Based on the simulation result, we conclude that UMVUE perform better than MLE when one shape parameter is known and it is also noted that MSEs of MLE and UMVUE are decreasing as the sample size increases.
The authors would like to thanks the reviewers for reading the manuscript carefully, for detailed comments, and helpful suggestions which greatly helped to improve the manuscript.
Cite this article
TY - JOUR AU - Bushra Khatoon AU - M. J. S. Khan AU - Zubdahe Noor PY - 2021 DA - 2021/03/03 TI - Moments of Dual Generalized Order Statistics from Inverted Kumaraswamy Distribution and Related Inference JO - Journal of Statistical Theory and Applications SP - 251 EP - 266 VL - 20 IS - 2 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.d.210219.001 DO - https://doi.org/10.2991/jsta.d.210219.001 ID - Khatoon2021 ER -