Log-logistic parameters estimation using moving extremes ranked set sampling design

In statistical parameter estimation problems, how well the parameters are estimated largely depends on the sampling design used. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the scale and shape parameters for the log-logistic distribution. Several traditional estimators and ad hoc estimators will be studied under MERSS. The estimators under MERSS are compared to the corresponding ones under SRS. The simulation results show that the estimators under MERSS are significantly more efficient than the ones under SRS.


§1 Introduction
Cost effective sampling is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. The method of ranked set sampling (RSS) provides an effective way to achieve observational economy in terms of precision achieved per unit of sampling. Initially the concept of RSS was introduced by McIntyre (1952) as a process of increasing the precision of the sample mean as an estimator of population mean. Ranking can be performed based on expert judgment, visual inspection or any means that does not involve actually quantifying the observations. In RSS one first draws n 2 units at random from the population and partitions them into n sets of n units. The n units in each set are ranked without making actual measurements. From the first set of n units the unit ranked lowest is chosen for actual quantification. From the second set of n units the unit ranked second lowest is measured. The process is continued until the unit ranked largest is measured from the n−th set of n units. If a larger sample size is required then the procedure can be repeated k times.
Takahasi and Wakimoto (1968) established a very important statistical foundation for the theory of RSS. Later estimation of parameters of various commonly used distributions has been carried out using RSS (for details see Stokes (1995) The procedure of MERSS is described as follows: 1. Select n simple random samples of sizes 1, 2, 3, · · · , n, respectively. 2. Order the elements of each set by visual inspection or by some other cheap method, without actual measurement of the characteristic of interest.
3. Measure accurately the maximum ordered observation from the first set, then the second set, . . . , the last set. 4.
Step (3) is repeated on another n sets of size 1, 2, 3, · · · , n, respectively, however the minimum ordered observations are measured instead of the maximum ordered observations. 5. If needed, this process can be replicated k times (cycles). Clearly, only the two extreme values are used in MERSS, maximum or minimum of sets of varied size, whereas the ranks of all the elements of each set are needed in RSS. Since it is not difficult to identify maximum or minimum units, MERSS is a very useful modification of RSS. It allows for an increase of set size without introducing too many ranking errors.
AL A random variable X is said to have a log-logistic distribution with the scale parameter α and the shape parameter β if its distribution function is given by where x > 0, α > 0 and β > 0. The probability density function (pdf) corresponding to the distribution function in (1) is then given by We write LLD(α, β) to denote the distribution as defined in (1). The applications of log-logistic distribution are well known in wealth or income (see Fisk (1961)), hydrology for modelling stream flow rates and precipitation (see Shoukri et al. (1988)) and engineer of survival analysis (see Ashkar et al. (2003)). For further details on the importance and applications of a loglogistic distribution one may refer to Bennett (1983), Ahmad et al. (1988), Robson et al. (1999) and Geskus (2001).
Parameter estimation problems for the log-logistic distribution have been discussed by many authors. Among recent literature, Balakrishnan et al. (1987) studied the best linear unbiased estimator (BLUE) of the scale parameter of a log-logistic distribution under SRS. Chen (2006) discussed about the interval estimation for the shape parameter of the log-logistic distribution under SRS. Lesitha et al. (2013) provided an unbiased estimator and BLUE of the scale parameter of a log-logistic distribution under RSS. Further, inference on the parameters of the log-logistic distribution has been studied by many authors using SRS including Tiku et al. In this paper, we consider several traditional estimators and ad hoc estimators of the scale and shape parameters α and β from LLD (α, β) based on MERSS. In Sect. 2, we study an unbiased estimator or modified unbiased estimator and BLUE or modified BLUE of α and β in case when one parameter is known and ad hoc estimators in case when both parameters are unknown. In Sect. 3, we consider the MLEs of the parameters of this distribution. The relative efficiencies of all estimators are simulated and the conclusions will be presented in Sect. 4

. §2 Several types of estimators
In this section, we deal with several types of estimators of α and β of the LLD (α, β) under MERSS: (i) An unbiased estimator and BLUE of α defined from LLD (α, β) in which β is known, (ii) A modified unbiased estimator and modified BLUE of β when α is known and (iii) Ad hoc estimators of α and β when α and β are both unknown.

Unbiased estimator and BLUE of α when β is known
Let {x 1 , x 2 , x 3 , · · · , x 2n } be a simple random sample of size 2n from (1) in which β is known.
Thus the BLUE of α under SRS is given bŷ Let {x 11 , x 22 , x 33 , · · · , x nn , y 11 , y 12 , y 13 , · · · , y 1n } be a moving extremes ranked set sample of size 2n from (1) in which β is known. Then xii α has the same density as the ith order statistic of an SRS of size i from f 1 (x)(see David(1981)), i.e. the pdf of xii α is Also y1i α has the same density as the first order statistic of an SRS of size i from f 1 (x)(see David(1981)), i.e. the pdf of y1i α is and Thus an UE of α under MERSS is given bŷ According to the lemma (see Casella et al., 2002, p.338), combine (9)-(11) with (12), we havê 2.2 Modified unbiased estimator and modified BLUE of β when α is known = 0. Hence we consider the estimators of β based on order statistics βln then it can be seen that Thus the BLUE of 1 β under SRS is given by 1 2n Then we suggest the following estimator of β which will be called the modified unbiased estimator of β.
Let {x 22 , x 33 , x 44 , · · · , x n+1n+1 , y 12 , y 13 , y 14 , · · · , y 1n+1 } be a moving extremes ranked set sample of size 2n from LLD(α, β) with α is known. Then βln xii α has the same density as the ith order statistic of an SRS of size i from f 2 (x), i.e. the pdf of βln xii α is Also βln y1i α has the same density as the first order statistic of an SRS of size i from and Thus an unbiased estimator of 1 β under MERSS is given by 1 2n Then we suggest the following estimator of β Then we suggest the following estimator of β which will be called the modified BLUE of β.

Ad hoc estimators of α and β
If Z = lnX, then the pdf of Z where θ = lnα and λ = 1 β . Let {z 1 , z 2 , z 3 , · · · , z 2n } be a simple random sample of size 2n from (27) and . The most well known estimators of location parameter θ and scale parameter λ using the order statistics, are the BLUEs (Arnold et al., 1992 and Balakrishnan et al., 1992) which can be written aŝ which will be called the modified BLUE of α and β, respectively.
Then we can obtain ad hoc estimators of α and β andβ In this section, we consider MLEs of the parameters of LLD(α, β) under MERSS. Under some regularity conditions, the asymptotic efficiency of the MLEs can be obtained from the inverse of the Fisher information matrix.

The fisher information matrix for α and β under SRS
is given by Reath et al. (2018).
Let {x 11 , x 22 , x 33 , · · · , x nn , y 11 , y 12 , y 13 , · · · , y 1n } be a moving extremes ranked set sample of size 2n from LLD(α, β), then the pdfs of x ii and y 1i (i = 1, 2, ..., m) are respectively In order to get the MLEs, we start with the likelihood function The log-likelihood function is where C is a constant. Then we have and (34) The second-order derivative of α and β for the lnL M ERSS are computed as and respectively. Thus we have and In this section, we will compare the relative efficiencies of the above estimators in Sect.2 and Sect.3.
The efficiency ofα M ERSS,U E with respect to (w.r.t.)α SRS, BLU E is and It can be seen that aef f i (i=7, 8,9) are free of α and β and aef f 7 > 1 for n > 2.
From Tables 1-4, we conclude the following: (2) ef f 2 > 1, which meansα M ERSS, BLU E is more efficientα SRS, BLU E . (13) aef f 9 > 1, which means the MLEs of α and β under MERSS is more efficient than the MLE of α and β under SRS. (14) In conclusion, the MLEs of α and β under MERSS are more efficient than that of α and β under SRS in Sect.3. (15) In conclusion, the MERSS is more efficient than SRS in estimating the scale and shape parameters of the log-logistic distribution.