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Estimation of the shape and scale parameters of Pareto distribution using ranked set sampling
Abstract
In the current paper, the estimation of the shape and location parameters α and c, respectively, of the Pareto distribution will be considered in cases when c is known and when both are unknown. Simple random sampling (SRS) and ranked set sampling (RSS) will be used, and several traditional and ad hoc estimators will be considered. In addition, the estimators of α, when c is known using an RSS version based on the order statistic that maximizes the Fisher information for a fixed set size, will be considered. These estimators will be compared in terms of their biases and mean square errors. The estimators based on RSS can be real competitors against those based on SRS.
... If the i 0 -th order statistic has the largest Fisher information, it is selected in this set for actual measurement. Thus we have w (i0) = 1 and the other w i = 0 (for more details, see [8,9]). After obtaining the optimal weight, a superior method of statistical inference under this RSS is looked for. ...
... Researches on parametric estimation in existing literatures were mainly focused on the maximum likelihood estimators (MLEs) [10,11] . For example, Abu-Dayyeh, et al. [8] studied the MLE of the shape parameter for Pareto distribution under the balanced RSS and the RSS based on maximizing the Fisher information, respectively. However, their results do not directly apply other population besides Pareto distribution. ...
... for odd n in Section 2 are iid and their pdfs are (8). Then the main part of likelihood function for these samples can be written as ...
This paper studies a maximum likelihood estimator (MLE) of the parameter for a continuous one-parameter exponential family under ranked set sampling (RSS). The authors first find the optimal RSS according to the character of the family, viz, arrange the RSS based on quasi complete and sufficient statistic of independent and identically distributed (iid) samples. Then under this RSS, some sufficient conditions for the existence and uniqueness of the MLE, which are easily used in practice, are obtained. Using these conditions, the existence and uniqueness of the MLEs of the parameters for some usual distributions in this family are proved. Numerical simulations for these distributions fully support the result from the above two step optimizations of the sampling and the estimation method. © 2017 Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg
... Reference [12] used RSS to derive maximum likelihood (ML) and Bayesian estimators for generalized exponential model. Reference [13] handled with parameter estimators of Pareto distribution using RSS. The parameter estimator of the Rayleigh distribution was regarded in [14] using different methods of estimations and ranking designs. ...
... The logarithm of Equation (13), denoted by ...
The ranked set sampling (RSS) methodology is an effective technique of acquiring data when measuring the units in a population is costly, while ranking them is easy according to the variable of interest. In this article, we deal with an RSS-based estimation of the inverted Kumaraswamy distribution parameters, which is extensively applied in life testing and reliability studies. Some estimation techniques are regarded, including the maximum likelihood, the maximum product of spacing’s, ordinary least squares, weighted least squares, Cramer–von Mises, and Anderson–Darling. We demonstrate a simulation investigation to assess the performance of the suggested RSS-based estimators via accuracy measures relative to simple random sampling. On the basis of actual data regarding the waiting times between 65 consecutive eruptions of Kiama Blowhole, additional conclusions have been drawn. The outcomes of simulation and real data application demonstrated that RSS-based estimators outperformed their simple random sampling counterparts significantly based on the same number of measured units.
... Nikulin and Highlight (2009) have studied some of the statistical properties of PGW distribution. The cumulative density function (cdf), probability density function (pdf) and quantile function of the power generalized Weibull (PGW(í µí¼, í µí»¼, í µí»½)) distribution are í µí°¹(í µí±¥; í µí¼, í µí»¼, í µí»½) = 1 − í µí± 567589: ; < = (1) í µí±(í µí±¥; í µí¼, í µí»¼, í µí»½) = í µí¼í µí»¼í µí»½í µí±¥ @65 71 + í µí¼í µí±¥ @ < B65 7í µí±í µí±¥í µí± D1 − 71 + í µí¼í µí±¥ @ < B E< (2) andí µí±¥ G , í µí±¢ = í µí°¹(í µí±¥) . ...
... Besides these studies, several authors have considered the estimation of the parameters of well-known distributions using RSS or modifications of it. For example, estimation of í µí±(í µí± < í µí±) using Some Modifications of RSS for Weibull Distribution was proposed by Akgül and Senoglu (2017) , The estimation of unknown parameters of exponential, extreme-value, logistic, Weibull and Pareto distributions was studied by Lam et al. (1994), and maximum likelihood estimators of the parameters of a modified Weibull distribution using extreme ranked set sampling was introduced by AlOmari and Al-Hadrami (2011), Bhoj (1997), AbuDayyeh et al. (2004), Helu et al. (2010) and Abu-Dayyeh et al. (2013).and a í µí±-stage RSS design uses í µí± [85 sample units from the target population to produce a sample of size í µí± after í µí± stages of ranking developed by Taconeli & Cabral (2019). ...
Parameter estimation Based on Double Ranked Set Sampling (DRSS) designs was recently developed by Sabry et al., (2019) and shows high efficiency and precision of the likelihood estimators when applied to the two-parameter Weibull distribution. In this paper the likelihood function of the General Double Ranked Set Sampling (GDRSS) design discussed by Taconeli & Cabral (2019) is derived and the two double ranked set sampling designs DRSS and GDRSS are compared along with the usual Ranked Set Sampling (RSS) and Extreme Ranked Set Sampling (ERSS) designs for the estimation of the parameters of the Power Generalized Weibull (PGW) distribution which is an extension of the two parameter Weibull distribution. An intensive simulation has been made to compare the one-and the two-stages designs. The results show that likelihood estimation based on DRSS and GDRSS designs provide more efficient estimators than the usual RSS and ERSS designs. Moreover, the GDRSS is slightly more efficient that the DRSS designs in the case of estimating the PGW distribution.
... Hassan (2013) obtained a Bayesian estimator for the shape and scale parameters of the EE distribution using RSS. Abu-Dayyeh et al. (2013) used RSS to estimate the shape and scale parameters of the Pareto distribution. Samuh and Qtait (2015) used median RSS (MRSS) to estimate the shape and scale parameters of the EE distribution. ...
Partial ranked set sampling (PRSS) is a cost-effective sampling method. It is a combination of simple random sample (SRS) and ranked set sampling (RSS) designs. The PRSS method allows flexibility for the experimenter in selecting the sample when it is either difficult to rank the units within each set with full confidence or when experimental units are not available. In this article, we introduce and define the likelihood function of any probability distribution under the PRSS scheme. The performance of the maximum likelihood estimators is examined when the available data are assumed to have an exponentiated exponential (EE) distribution via some selective RSS schemes as well as SRS. The suggested ranked schemes include the PRSS, RSS, neoteric RSS (NRSS), and extreme RSS (ERSS). An intensive simulation study was conducted to compare and explore the behaviour of the proposed estimators. The study demonstrated that the maximum likelihood estimators via PRSS, NRSS, ERSS, and RSS schemes are more efficient than the corresponding estimators under SRS. A real data set is presented for illustrative purposes.
... The exponentiated Pareto (EP) distribution was introduced by Gupta et al. (1998). The probability density function (pdf) of EP distribution is given by (1) Where and are shape parameters. The corresponding cumulative distribution function (cdf) is (2) and the quantile function is given by (3) Ranked set sampling (RSS) is a technique of data collection based on the rank of the units of the drawn sample. ...
... The performance of ML estimation under the RSS scheme has been developed and used in numerous studies. For example, Abu-Dayyeh et al. [11] for the Pareto; Aljohani et al. [12] for the modified Kies exponential; Bantan et al. [13] for the half logistic inverted Topp-Leone; Chen et al. [14] for the Pareto; Esemen and Gürler [15] for the generalized Rayleigh; He et al. [16] for the log-logistic; Sabry and Almetwally [17] for the exponential Pareto; Sabry et al. [18] for the Weibull; Singh and Mehta [19] for the log-logistic and Taconeli and Giolo [20] for the power Lindley and weighted Lindley distributions. Different results regarding parametric estimation based on the RSS scheme, including other estimation methods, are presented by Pedroso et al. [21] and Taconeli and Bonat [22]. ...
The generalized Bilal (GB) distribution can be defined as the distribution of the median of three independent random variables drawn from the Weibull distribution. Its failure rate function can be monotonic (decreasing or increasing) or upside-down bathtub-shaped. In this study, we aim to reveal some important properties of the GB distribution that have not been considered before. The findings are both theoretical and practical. From the theoretical viewpoint, we present explicit expressions for both single and product moments of order statistics from the GB distribution. The L-moments are derived as well. From the practical viewpoint, the parameter estimations are accomplished using the maximum likelihood (ML) method, which is based on two different sampling schemes: simple random sampling (SRS) and ranked set sampling (RSS) schemes. Furthermore, the asymptotic confidence intervals for the SRS and RSS estimators are discussed. For the sake of comparison and illustration, a simulation study and a real data example are presented. Concluding remarks are given at the end.
... When the phasors are characterized by PDFs, the summation of independent random variables is calculated using convolution or related techniques (see section IV of Stanton et al. [32]). It has been shown that the summation of Burr distributed random variables leads to series solutions [51][52][53][54][55][56][57]. However, for our case we need to add phasors The dependence of the ratio of the first moment (mean) squared divided by the non-centered second moment for the Burr distribution as a function of the Burr parameter b is shown in the blue curve. ...
Speckle statistics in ultrasound and optical coherence tomography have been studied using various distributions, including the Rayleigh, the K, and the more recently proposed Burr distribution. In this paper, we expand on the utility of the Burr distribution by first validating its theoretical framework with numerical simulations and then introducing a new local estimator to characterize sample tissues of liver, brain, and skin using optical coherence tomography. The spatially local estimates of the Burr distribution's power-law or exponent parameter enable a new type of parametric image. The simulation and experimental results confirm the potential for various applications of the Burr distribution in both basic science and clinical realms.
... Later, he used it for the study on wealth distribution in Italy in which more than 80% of the country's wealth was owned by 20% of its population. Estimation of the shape and scale parameters of the Pareto distribution using RSS and ERSS has been proposed by Abu-Dayyeh, Assrhani, and Ibrahim (2013) and Omar and Ibrahim (2013), respectively. Pareto distribution is a highly positively skew distribution. ...
Ranked set sampling (RSS) is a method of sampling that can be advantageous when quantification of all sampling units is costly but when small sets of units can be ranked according to the character under investigation by means of visual inspection or other methods not requiring actual measurements. RSS performs better than simple random sampling (SRS) to estimate the population mean. In original RSS procedure, the units corresponding to each rank are used. In this article, we propose to use RSS method with lowest order statistics from each sample to estimate the population mean of Pareto distribution which is highly positively skew. The Pareto distribution is chosen due to its application in social and scientific phenomenon including the distribution of wealth in a society. The estimator based on lowest order statistics with bias correction term has been proposed. Two cases, known and unknown scale parameter, have been considered. The simulation-based methods have also been included. It is shown that the gains in the relative precisions of population mean based on our proposed method are uniformly higher than those based upon the RSS and extreme RSS procedures. The proposed method with bias correction term is recommended for real applications.
... Step 2: Divide the 3 elements randomly into sets each of size 2 elements and use the usual RSS procedure on each set to obtain RSS each of size of the form = { { (1) , (2) , … . ( ) }, = 1,2, … , } Step 3: Case I: For even set sizes ( = 2 ), select from the first sets the minimum ranked measurement and from the last r sets select the maximum ranked measurement. ...
In this paper, the derivation of the likelihood function for parameter estimation based on double ranked set sampling (DRSS) designs used by Sabry et al. (2019) for the estimation of the parameters of the power generalized Weibull distribution is considered. The developed likelihood function is then used for the estimation of the exponential Pareto distribution parameters. The maximum likelihood estimators (MLEs) are then investigated and compared to the corresponding ones based on simple random sampling (SRS) and ranked set sampling (RSS) designs. A Monte Carlo simulation is conducted and the absolute relative biases, mean square errors, and efficiencies are compared for the different designs. The relative efficiency of the DRSS estimates with respect to other designs was found to be higher in case of the exponential Pareto distribution (EP).
... some situations , the whole procedure to generate an RSS of size n is repeated r times throughout in this paper we consider the case r=1. A Monto -carlo simulation is applied for different sample sizes n= (6,10,14,20,30) with four sets values of parameters were selected set 1( = 0.75 , = 0,5 , = 1.5 ) and set 2 ( = ,75 , = 2 , = 1.5 ) set 3 ( = 3 , = 2 , = 1.5 ) , set 4 ( = 3 , = 2 , = 3 ). The estimators , , based on SRS and RSS are obtained by solving equations (4,5,6) and (16,17,18) by the R package. ...
The aim of this paper is to estimate the parameters of exponentiated Burr type XII distribution (EBXII) based on ranked set sampling (RSS) technique, and also simple random sampling(SRS) is provided by the method of maximum likelihood. Fisher information matrix for both (SRS) and (RSS) for the unknown parameters are derived. Simulation study compared between the estimators of both methods in terms of their biases, mean square errors, and efficiencies. It is shown that the estimators based on RSS are more efficient than those of SRS.
... The statistics literature has derived the sum of random variables of these distributions but the solutions generally involve complicated generalized functions or series. 47,48,[53][54][55][56][57] To simplify this, we examine a complex Burr summation. ...
After 100 years of theoretical treatment of speckle patterns from coherent illumination, there remain some open questions about the nature of ultrasound speckle from soft vascularized tissues. A recent hypothesis is that the fractal branching vasculature is responsible for the dominant echo pattern from organs such as the liver. In that case an analysis of cylindrical scattering structures arranged across a power law distribution of sizes is warranted. Using a simple model of echo strength and basic transformation rules from probability, we derive the first order statistics of speckle considering the amplitude, the intensity, and the natural log of amplitude. The results are given by long tailed distributions that have been studied in the statistics literature for other fields. Examples are given from simulations and animal studies, and the theoretical fit to these preliminary data support the overall framework as a plausible model for characterizing ultrasound speckle statistics.
... Second, we obtain the closed form estimator of R by using the modified maximum likelihood (MML) methodology. This methodology was first initiated by Mehrotra and Nanda (1974) and was adopted to RSS by Zheng and Al-Saleh (2002), Al-Saleh and Al-Hadhrami (2003) and Abu-Dayyeh, Assrhani, and Ibrahim (2013). In view of interval estimation, the asymptotic confidence interval (ACI) of R is constructed. ...
In this study, we consider the point and interval estimation of the stress–strength reliability R=PX<Y based on ranked set sampling when the stress X and the strength Y are both independent Burr Type X random variables. In the context of point estimation, we obtain the maximum likelihood (ML) estimator of R using iterative methods. We also use Mehrotra and Nanda’s modified maximum likelihood methodology, which gives explicit estimator of R as an alternative to the ML methodology. In view of interval estimation, we construct the asymptotic confidence interval of R. In addition, the bootstrap confidence intervals of R are constructed based on two different resampling methods. The performance of the proposed estimators (both point and interval) is compared with their simple random sampling counterparts. A real data set from an agricultural experiment is analyzed to show the implementation of the proposed methodologies.
... In most cases, estimators based on the RSS exhibit better qualities in comparison with estimators based on the SRS. The advantage of RSS estimators is shown for more theoretical distributions, see for example, [1], [4], [5], [22], [23]. The popularity of RSS increased during the last few decades, and many variations of the original model have been developed, see among others, [2], [3], [12], [21]. ...
The ranked set sampling (RSS) is a cost-effective method of sampling that can be used in a wide range of statistical problems. In this paper, the shape and the scale parameters of Nadarajah-Haghighi extension of the exponential distribution are estimated based on a simple random sample (SRS) and RSS. Three cases are considered: 1) the scale parameter is known; 2) the shape parameter is known; 3) both shape and scale parameters are unknown. Observations are done when the ranking mechanism in the ranked set sample is perfect and when it is not. Method of moments, the maximum likelihood method, and a modification of the maximum likelihood method are used. The obtained estimators are compared in terms of their biases and mean square errors (MSE). The results revealed that estimators based on RSS tend to show better properties (smaller bias and MSE) relative to their SRS counterparts, regardless of the quality of the ranking.
... Although RSS actually is a nonparametric in its nature, a large number of publications for parametric estimation has been produced in the literature. From those, Abu-Dayyeh et al. [3] used different estimation methods for estimating the shape and scale parameters of Pareto distribution based on RSS and SRS. Sarikavanij et al. [4] made a comparison study between SRS and RSS for estimating the location and scale parameters of a two-parameter exponential distribution. ...
The problem of parameters estimation plays a significant role in various areas of academic researches. In this article, we propose three different methods of estimation for the parameters of location-scale family under ranked set sampling in the view of missing data mechanism. Through a series of Monte Carlo simulations, it is well investigated that the proposed methods are relatively robust from violating the perfect ranking condition and provide better performance over their competitors using bias and MSE (mean square error) criteria. An empirical data set is also used for illustrative purposes.
... It is appropriate for situations where quantification of sampling units is either costly or difficult, but ranking the units in a small set is easy and inexpensive. For further introduction of RSS, refer to Stokes (1995), Al-Saleh and Al-Hadhrami (2003), Abu-Dayyeh et al. (2013), Omar and Ibrahim (2013), Wangxue et al. (2013), Wangxue et al. (2016) and Wangxue et al. (2017). ...
In this article, maximum likelihood estimator(s) (MLE(s)) of the scale and shape parameters \(\alpha \) and \(\beta \) from log-logistic distribution will be respectively considered in cases when one parameter is known and when both are unknown under simple random sampling (SRS) and ranked set sampling (RSS). In addition, the MLE of one parameter, when another parameter is known using a RSS version based on the order statistic that maximizes the Fisher information for a fixed set size, will be considered. These MLEs will be compared in terms of asymptotic efficiencies. These MLEs based on RSS can be real competitors against those based on SRS. All efficiencies of these MLEs are simulated under imperfect ranking.
... Besides these studies, several authors have considered the estimation of the parameters of well-known distributions using RSS or modifications of it. For example, the estimation of unknown parameters of exponential, extreme-value, logistic, Weibull and Pareto distributions was studied by Lam et al. (1994), Bhoj (1997), Abu Dayyeh et al. (2004), Helu et al. (2010) and Abu Dayyeh et al. (2013). Also, Muttlak (2002, 2004) were based on RSS and its modifications. ...
In statistical literature, estimation of R=P(X < Y) is a commonly-investigated problem, and consequently, there have been considerable number of studies dealing with its estimation of it under simple random sampling (SRS). However, in recent years, the ranked set sampling (RSS) method have been widely-used in the estimation of R. In this study, we consider the estimation of R when the distribution of the both stress and strength are Weibull under the modification of RSS, which are extreme ranked set sampling (ERSS), median ranked set sampling (MRSS) and percentile ranked set sampling (PRSS). We obtain the estimators of R using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies under these modifications. Then the performances of proposed estimators are compared with the corresponding ML and MML estimators of R using SRS via a Monte-Carlo simulation study.
... Dey et al. [19] considered the estimation of Rayleigh distribution using ML and Bayesian methods and different sampling schemes these are SRS, RSS, MRSS and modified RSS. For some additional results and references, one can see [20][21][22][23][24][25][26]. The purpose of this paper is to provide results for estimation of the shape and the scale parameters of GR distribution using the ML method and to compare the results in the different sampling scheme such as SRS, RSS, ERSS and MRSS. ...
Ranked set sampling (RSS) is an efficient method for estimating parameters when exact measurement of observation is difficult and/or expensive. In this paper, we provide maximum likelihood estimation of the shape and scale parameters concerning generalized Rayleigh distribution based on RSS and its some modifications. We compare the biases, mean squared errors and relative efficiencies of estimators in simple random sampling, RSS, extreme RSS and median RSS with different set and cycle sizes. Comparison of the mean squared errors of estimators in RSS for the case of imperfect ranking are also given. Monte Carlo simulation study is performed by using Mathematica 11.0 with 10,000 repetitions.
... Methods have been developed for handling most basic statistical problems, including mean estimation (McIntyre 1952), variance estimation (MacEachern et al. 2002), estimation of the distribution function (Stokes and Sager 1988), parametric point estimation (Stokes 1995), and nonparametric interval estimation (Frey 2007). Methods have also been developed for more specialized statistical problems such as group sequential testing (Hussein et al. 2013), parametric location and scale parameter estimation for the Pareto family (Abu-Dayyeh et al. 2013), ratio estimation of the mean (Jafari Jozani et al. 2012;Kadilar et al. 2009), mean estimation using moving extreme RSS (Abu-Dayyeh and Al Sawi 2009), and maximum likelihood estimation using rounded data (Li et al. 2012). Research areas where RSS has been applied include forestry (Halls and Dell 1966), environmental monitoring (Kvam 2003), and entomology (Howard et al. 1982). ...
Partially rank-ordered set sampling (PROSS) is a generalization of ranked-set sampling (RSS) in which the ranker is not required to give a full ranking in each set. In this paper, we study the efficiency of the PROSS sample mean under perfect rankings for various PROSS schemes. We obtain conditions under which one PROSS scheme is always more efficient than another, and we also obtain conditions under which how the efficiencies of two PROSS schemes compare depends on the particular distribution. We completely determine how PROSS schemes compare in the two-subset case, and we also prove a conjecture of Ozturk (Environ Ecol Stat 18:757–779, 2011) about how the efficiency of the PROSS sample mean compares to that of the RSS sample mean.
... It was found that the estimators under PRSS are more efficient than the estimators based on simple random sampling. Abu-Dayyeh et al. (2013) used RSS for studying the estimation of the shape and location parameters of the Pareto distribution. The estimators were compared with their counterpart in SRS in terms of their biases and mean square errors. ...
The method of maximum likelihood estimation based on Median Ranked Set Sampling (MRSS) was used to estimate the shape and scale parameters of the Exponentiated Exponential Distribution (EED). They were compared with the conventional estimators. The relative efficiency was used for comparison. The amount of information (in Fisher's sense) available from the MRSS about the parameters of the EED were be evaluated. Confidence intervals for the parameters were constructed using MRSS.
... Ozturk has developed two sampling designs to create artificially stratified samples using RSS [15]. Readers are encouraged to perusal at a historical perspective of the RSS approach, see [16][17][18][19][20][21][22][23][24][25]. ...
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, estimators for the parameters of the Kumaraswamy-inverse Rayleigh distribution based on record values are obtained. These estimators are derived using the maximum likelihood and Bayesian methods. The Bayesian estimators are derived under the well-known squared error (SE) loss function. Prediction of the future sth record value is derived using the maximum likelihood and Bayesian methods. Simulation study is conduct to illustrate the findings.
In the current paper, we considered the Fisher information matrix from generalized Rayleigh distribution (GR) distribution in moving extremes ranked set sampling (MERSS). The numerical results show that the ranked set sample carry more information about λ and α than a simple random sample of equivalent size. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), a modified unbiased estimator and a modified best linear unbiased estimator (BLUE) of scale and shape λ and α from GR distribution in SRS and MERSS are studied. The numerical results show that the modified unbiased estimator and the modified BLUE of λ and α in MERSS are significantly more efficient than the ones in SRS.
In this article, based on progressive type-II censored competing risks data with binomial removals, a constant-stress accelerated life testing is discussed when the lifetime of experiment units follows the Pareto distribution. Under the assumption that the failure time of each cause is statistically independent, the estimation of unknown parameters is obtained by maximum likelihood estimation and Bayesian estimators under symmetric and asymmetric loss functions. The corresponding confidence and credible intervals are constructed with approximation theory, bootstrap method, and MCMC technique respectively. Additionally, the log-linear model is established for the life test and two goodness-of-fit test statistics are constructed. Finally, a simulation study is carried out for the power analysis of different censoring ratio schemes and the numerical results verify the feasibility of the constructed test statistics for the censored data with competing risks.
In the current paper, we considered the Fisher information matrix from the generalized Rayleigh distribution (GR) distribution in ranked set sampling (RSS). The numerical results show that the ranked set sample carries more information about λ and α than a simple random sample of equivalent size. In order to give more insight into the performance of RSS with respect to (w.r.t.) simple random sampling (SRS), a modified unbiased estimator and a modified best linear unbiased estimator (BLUE) of scale and shape λ and α from GR distribution in SRS and RSS are studied. The numerical results show that the modified unbiased estimator and the modified BLUE of λ and α in RSS are significantly more efficient than the ones in SRS.
Purpose: The study of speckle from imaging systems has a rich history, and recently it was proposed that a fractal or power law distribution of scatterers in vascularized tissue will lead to a form of the Burr probability distribution functions for speckle amplitudes. This hypothesis is generalized and tested in theory, simulations, and experiments. Approach: We argue that two broadly applicable conjectures are sufficient to justify the applicability of the Burr distribution for speckle from a number of acoustical, optical, and other pulse-echo systems. The first requirement is a multiscale power law distribution of weak scatterers, and the second is a linear approximation for the increase in echo intensity with size over some range of applicability. Results: The Burr distribution for speckle emerges under a wide variety of conditions and system parameters, and from this one can estimate the governing power law parameter, commonly in the range of 2 to 6. However, system effects including the imaging point spread function and the degree of focusing will influence the Burr parameters. Conclusions: A generalized pair of conditions is sufficient for producing Burr distributions across a number of imaging systems. Simulations and some theoretical considerations indicate that the estimated Burr power law parameter will increase with increasing density of scatters. For studies of speckle from living tissue or multiscale natural structures, the Burr distribution should be considered as a long tail alternative to classical distributions.
Ranked set sampling (RSS) is an efficient technique for estimating parameters and is applicable whenever ranking on a set of sampling units can be done easily by a judgment method or based on an auxiliary variable. In this paper, we assume [Formula: see text]to have bivariate Lomax distribution where a study variable [Formula: see text]is difficult and/or expensive to measure and is correlated with an auxiliary variable [Formula: see text] which is readily measurable. The auxiliary variable is used to rank the sampling units. In this article, we propose an estimator for the scale parameter of bivariate Lomax distribution using some of the modified RSS schemes. Efficiency comparison of the proposed estimators is performed numerically as well as graphically. A simulation study is also performed to demonstrate the performance of the proposed estimators. Finally, we implement the results to real-life datasets.
AMS classification codes: 62D05, 62F07, 62G30
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.
In this paper, estimation of parameters α and β for the log-extended exponential-geometric distribution will be respectively considered in cases when β is known and when both are unknown. Simple random sampling (SRS) and moving extremes ranked set sampling (MERSS) will be used, and several traditional estimators will be considered. The estimators using MERSS are compared to the corresponding ones using SRS. The numerical results show that the estimators using MERSS are significantly more efficient than the ones using SRS. A real data set is used for illustration.
After 100 years of theoretical treatment of speckle patterns from coherent illumination, there remain some open questions about the nature of ultrasound speckle from soft vascularized tissues. A recent hypothesis is that the fractal branching vasculature is responsible for the dominant echo pattern from organs such as the liver. In that case, an analysis of cylindrical scattering structures arranged across a power law distribution of sizes is warranted. Using a simple model of echo strength and basic transformation rules from probability, we derive the first order statistics of speckle considering the amplitude, the intensity, and the natural log of amplitude. The results are given by long tailed distributions that have been studied in the statistics literature for other fields. Examples are given from simulations and animal studies, and the theoretical fit to these preliminary data support the overall framework as a plausible model for characterizing ultrasound speckle statistics.
Pareto distributions are useful for modeling the loss data in many fields such as actuarial science, economics, insurance, hydrology and reliability theory. In this paper, we consider the simultaneous estimation of the risk parameters of Pareto distributions from the perspective of empirical Bayes, novel SURE-type shrinkage estimators are developed by employing the Stein’s unbiased estimate of risk (SURE). Specifically, due to the lacking of the analytic form for the risk function, we propose to estimate the hyperparameters by minimizing an unbiased estimate of an approximation of the risk function. Under mild conditions, we prove the optimality of the new shrinkage estimators. The performance of our estimators is illustrated with simulation studies and an analysis of a real auto insurance claim dataset.
Ranked Set Sampling is an efficient technique when it is difficult to measure sampling units in respect to cost or time. Although this technique can be used for every sample sizes, the small sample sizes are preferred for better ranking. However, when the sample sizes are small, it is very difficult to obtain distribution of the statistic for the statistical inference such as hypothesis test. In this case, resampling techniques like bootstrap can be used to construct pseudo distribution of the statistics. In this study, the bootstrap methods for hypothesis test about population mean under ranked set sampling is given. A simulation study is also performed to examine the performance of these methods.
Cost-effective sampling design 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. In the current paper, the Fisher information matrix of the log-extended exponential–geometric distribution LEEGD(α,β) with parameters α and β based on simple random sample, ranked set sample (RSS), median RSS (MRSS) and extreme RSS is discussed. We obtain the expressions for the Fisher information matrix in each case and use them to perform efficiency comparisons. It is found that MRSS is most efficient when one parameter is inferred at a time (with the other parameter known), while RSS is most efficient when both parameters are inferred simultaneously. A real data set is used for illustration.
Ranked set sampling (RSS) is an efficient method for estimating parameters when exact measurement of observation is difficult and/or expensive. In the current paper, several traditional and ad hoc estimators of the scale and shape parameters \(\theta \) and \(\alpha \) from the Pareto distribution \(p(\theta ,\alpha )\) will be respectively studied in cases when one parameter is known and when both are unknown under simple random sampling, RSS and some of its modifications such as extreme RSS(ERSS) and median RSS(MRSS). It is found for estimating of \(\theta \) from \(p(\theta ,\alpha )\) in which \(\alpha \) is known, the best linear unbiased estimator (BLUE) under ERSS is more efficient than the other estimators under the other sampling techniques. For estimating of \(\alpha \) from \(p(\theta ,\alpha )\) in which \(\theta \) is known, the modified BLUE under MRSS is more efficient than the other estimators under the other sampling techniques. For estimating of \(\theta \) and \(\alpha \) from \(p(\theta ,\alpha )\) in which both are unknown, the ad hoc estimators under ERSS are more efficient than the other estimators under the other sampling techniques. All efficiencies of these estimators are simulated under imperfect ranking. A real data set is used for illustration.
In this paper we use the maximum likelihood (ML) and the modified maximum likelihood (MML) methods to estimate the unknown parameters of the inverse Weibull (IW) distribution as well as the corresponding approximate confidence intervals. The estimates of the unknown parameters are obtained based on two sampling schemes, namely, simple random sampling (SRS) and ranked set sampling (RSS). Comparison between the different proposed estimators is made through simulation via their mean square errors (MSE), Pitman nearness probability (PN) and confidence length.
The minimum variance unbiased estimators (MVUEs) of the parameters for various distributions are extensively studied under ranked set sampling (RSS). However, the results in existing literatures are only locally MVUEs, i.e. the MVUE in a class of some unbiased estimators is obtained. In this paper, the global MVUE of the parameter in a truncated parameter family is obtained, that is to say, it is the MVUE in the class of all unbiased estimators. Firstly we find the optimal RSS according to the character of a truncated parameter family, i.e. arrange RSS based on complete and sufficient statistics of independent and identically distributed samples. Then under this RSS, the global MVUE of the parameter in a truncated parameter family is found. Numerical simulations for some usual distributions in this family fully support the result from the above two-step optimizations. A real data set is used for illustration.
This paper addresses the estimation of the parameter of Rayleigh distribution using different methods of frequentist and Bayesian estimation approaches based on different sampling schemes namely, simple random sample, ranked set sample, modified ranked set sample and median ranked set sample. Comparison between estimators is made through simulation via their biases, relative efficiency, and Pitman measures of closeness criteria under both perfect and imperfect ranking. The performance of the estimators based on ranked set sample and median ranked set sample are better than the other estimators based on simple random sample and also modified ranked set sample.
This paper discusses the generalized Pareto distribution (GPD) and its application to the statistical analysis of extreme wind speeds. Its main advantage is that it makes use of all relevant data on the high wind gusts produced by the storms of interest, not just the annual maxima, and it is not necessary to have a value for every year to carry out the analysis. The GPD is closely related to the generalized extreme value distribution (GEVD), and can be used to determine the appropriate value of shape factor, k, for use in the GEVD. However, a negative shape factor, corresponding to a Type II GEVD, is physically unrealistic, and should be avoided for long-term wind speed predictions.
The maximum livelihood estimator (MLE) using a ranked set sample (RSS) usually has no closed expression because the maximum likelihood equation involves both hazard and inverse hazard functions, and may no longer be efficient when the judgment ranking is imperfect. In this paper, we consider a modified MLE (MMLE) using RSS for general parameters, which has the same expression as the MLE using a simple random sample (SRS), except that the SRS in the MLE is replaced by the RSS. The results show that, for the location parameter, the MMLE is always more efficient than the MLE using SRS, and for the scale parameter, the MMLE is at least as efficient as the MLE using SRS, when the same sample size is used. Under the perfect judgment ranking, numerical examples also show that the MMLE has good efficiency relative to the MLE based on RSS. When the judgment error is present, we conduct simulations to show that the MMLE is more robust than the MLE using RSS.
Ranked set sampling employs judgment ordering to obtain an estimate of a population mean. The method is most useful when the measurement or quantification of an element is difficult but the elements of a set of given size are easily drawn and ranked with reasonable success by judgment. In each set all elements are ranked but only one is quantified. Sufficient sets are processed to yield a specified number of quantified elements and the mean for each rank. The average of these means is an unbiased estimate of the population mean regardless of the errors in the ranking. Precision relative to random sampling, with the same number of units quantified, depends upon properties of the population and success in ranking. The ranked set concept is reviewed with particular consideration of the errors in judgment ordering.
We assume throughout this paper that the population under consideration has the distribution function F(x) and the density function f(x) with finite mean ~ and finite variance a 2. It should be noted that we assume nothing about the distribution except the above existence assumption, that is, we shall consider a non-parametric problem. When we are concerned with estimation of the population mean: we often encounter the situations where the measurement of the quantity of each element drawn from the population is very laborious but several elements can easily be arranged in the order of magnitude, for example, the case where the elements can be arranged without the measurement of each quantity. In practice the number of elements which are easily arranged will possibly be two or three, but we shall consider the general case. The following three examples will give us a better understanding of the situations : Example 1. Let us suppose that the quantity under consideration is the length of a kind of bacterial cells and the length Of the cells in a microscopic field is measured by using a micrometer. While the operation for the measurement will be laborious, the order of magnitude of two or three cells in the same microscopic field may be found by a glance in most cases. Example 2. Let us suppose that the quantity under consideration is the height of trees. We can find by a glance the order of height of two or three trees standing nearly each other.
A queuing system is described in which service times are conditioned upon a random parameter μ, such that the conditional service distribution is exponential and μ has a gamma density. It is shown that the resultant unconditional distribution of service times is Paretian. Several measures of effectiveness are discussed and the question of statistical estimation of service parameters is also explored.
A modification of ranked set sampling (RSS) called moving extremes ranked set sampling (MERSS) is considered parametrically, for the location parameter of symmetric distributions. A maximum likelihood estimator (MLE) and a modified MLE are considered and their properties are studied. Their efficiency with respect to the corresponding estimators based on simple random sampling (SRS) are compared for the case of normal distribution. The method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be real competitors to the MLE using (SRS). The procedure is illustrated using tree data. Copyright © 2003 John Wiley & Sons, Ltd.
Examines the skewed distribution of firms by size. Discusses the adequacy of previous economic explanations regarding firm size that are based on the static cost curve. It is shown that the static cost curve for the firm may predict the minimum size of a firm in an industry with a known size value, but it will not predict the size distribution of firms. Therefore, an alternative theory of firm size based on a stochastic model of the growth process is proposed. It is postulated that size has no effect on the expected percentage of firm growth--i.e., that each firm in every size class has the same average chance of increasing or decreasing in size regardless of its current size. The empirical data from a transition matrix for the 500 largest U.S. industrial corporations from 1954 to 1956 show that the frequency distributions of percentage changes in size of small, medium, and large firms were similar. However, this data encompasses an entire economy rather than a single industry. In order to find the size distribution of firms by industry, Bain's estimates of minimum efficient plant size and the U.S. Census of Manufacturers data on the size distribution of plants are used to compare the minimum efficient scales suggested for several industries. The findings suggest that the stochastic model provides new ways of interpreting the data on firm size distribution, and any deviation from the results predicted by the model reflect some departure from the law of proportionate effect or from one of the other assumptions in the model. Concludes with implications for economic policy and further research. (SFL)
Moving Extremes Ranked Set Sampling (MERSS) is a useful modification of Ranked Set Sampling (RSS). Unlike RSS, MERSS allows
for an increase of set size without introducing too much ranking error. The method is considered parametrically under exponential
distribution. Maximum likelihood estimator (MLE), and a modified MLE are considered and their properties are studied. The
method is studied under both perfect and imperfect ranking (with error in ranking). It appears that these estimators can be
real competitors to the MLE using the usual simple random sampling (SRS).
Motivated by hydrological problems, the exact distributions of the sum X + Y, the product X
Y and the ratio X/(X + Y) are derived when X and Y are independent Pareto random variables. A detailed application of the results is provided to extreme rainfall data from
Florida.
The truncated shifted Pareto (TSP) distribution, a variant of the two-parameter Pareto distribution, in which one parameter is added to shift the distribution right and left and the right-hand side is truncated, is used to model size distributions of oil and gas fields for resource assessment. Assumptions about limits to the left-hand and right-hand side reduce the number of parameters to two. The TSP distribution has advantages over the more customary lognormal distribution because it has a simple analytic expression, allowing exact computation of several statistics of interest, has a J-shape, and has more flexibility in the thickness of the right-hand tail. Oil field sizes from the Minnelusa play in the Powder River Basin, Wyoming and Montana, are used as a case study. Probability plotting procedures allow easy visualization of the fit and help the assessment.
This paper examines the Pareto and primacy measures of the size distribution of cities. The mean Pareto exponent for a sample of 44 countries is 1.136, somewhat greater than the exponent of one implied by the rank-size rule. We find that value of the Pareto exponent is quite sensitive to the definition of the city and the choice of city sample size. The significance of non-linear terms in variants of the Pareto distribution also indicate that the rank-size rule is only a first approximation to a complete characterization of the size distribution of cities within a country. The relatively low correlation between primacy and Pareto measures confirms the need for a variety of measures of city size distributions. This paper also suggests that large cities are growing faster than small cities in most of the countries in our sample. This is indicated by the positive coefficient on the first non-linear term introduced into the Pareto equation. Finally, variations in the Pareto exponent and measures of primacy are partly explained by economic, demographic, and geographic factors.
Approximate maximum likelihood estimators have been obtained for the normal and gamma distributions and their efficiencies
compared to those for the best linear unbiased estimators for these distributions.
The unique minimum variance unbiased (UMVU) estimate of the probability distribution function of the Pareto distribution is derived. It is shown that the distribution function and the r th moment associated with the UMVU estimate are also UMVU estimators. The p.d.f. and its estimator are compared graphically. An estimate of the 100p th percentile is given. It is seen that a function of this estimator has a chi-square distribution.
The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the
scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can
not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika
61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form
for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test
(MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis
against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified
tests are good competitors of the LRT using MERSS and simple random sampling (SRS).
A new method of sampling is described. Take the largest in the first of n sets, each of n random items, the second largest in the second set, and so on to the smallest in the nth set. The sample of n items selected in this way is an unbiased sample of the population. For typical unimodal distributions the mean of such a sample is slightly less than (n + 1)/2 times more efficient than the mean of n items taken at random. The application of the ranked sample method to pasture measurement is discussed.
Estimation in the Pareto distribution · doi:10.2143/AST.20.2
- Rytgaard
- Rytgaard
Loss distributions Application of the generalized Pareto distribution to extreme value anal-ysis in wind engineering Use of the Truncated Shifted Pareto distribution in assessing size distribution of oil and gas field
- Rv Klugman Sa Holmes Jd
- Moriarty
RV, Klugman SA (1984) Loss distributions. Wiley, New York Holmes JD, Moriarty WW (1999) Application of the generalized Pareto distribution to extreme value anal-ysis in wind engineering. J Wind Eng Ind Aerodyn 83:1–10 Houghton JC (1988) Use of the Truncated Shifted Pareto distribution in assessing size distribution of oil and gas field. Math Geol 20(8):907–937. ISBN 0-4718792-9-0
Distributions in statistics Theory of point estimation A method of unbiased selective sampling, using ranked sets
- Johnson Nl Kotz
Johnson NL, Kotz S (1970) Distributions in statistics. Continuous univariate distributions—2. Wiley, New York Lehmann EL (1983) Theory of point estimation. Wiley, New York McIntyre GA (1952) A method of unbiased selective sampling, using ranked sets. Aust J Agric Res 3:385– 390
Pareto random variables for hydrological modeling Estimating the parameters of a Pareto distribution Old and new methods of estimation and the Pareto distribution
- S Ali
- Mm
S, Ali MM (2008) Pareto random variables for hydrological modeling. Water Resour Manag 22:1381–1393 Petersen JL (2000) Estimating the parameters of a Pareto distribution. Technical Report. University of Montana Quandt RE (1966) Old and new methods of estimation and the Pareto distribution. Metrika 10(1):55–82
Pareto distributions. International Co-operative Publishing House Asymptotic theory of statistics and probability Ranked set sampling theory with order statistics background The Pareto distributions as a queue service discipline
- Arnold
- Bc
Arnold BC (1983) Pareto distributions. International Co-operative Publishing House, Fairland DasGupta A (1984) Asymptotic theory of statistics and probability. Springer, New York Dell DR, Clutter JL (1972) Ranked set sampling theory with order statistics background. Biometrics 28:545–555 Harris CM (1968) The Pareto distributions as a queue service discipline. Oper Res 16:307–313
Estimating the parameters of a Pareto distribution
- J L Petersen
- RV Hogg