Optic neuropathy related to Onodi cell mucocele: a systematic review and meta-analysis of randomized controlled trials

Turki M. Bin Mahfoz; Ahmad K. Alnemare; Turki M. Bin Mahfoz; Ahmad K. Alnemare

doi:10.3934/medsci.2021018

AIMS Medical Science

2021, Volume 8, Issue 3: 203-223. doi: 10.3934/medsci.2021018

Previous Article Next Article

Research article

Optic neuropathy related to Onodi cell mucocele: a systematic review and meta-analysis of randomized controlled trials

Turki M. Bin Mahfoz ^{1
,
,},
Ahmad K. Alnemare ²

1.
Otolaryngology Department, College of Medicine, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2.
Otolaryngology Department, College of Medicine, Majmaah University, Al-Majmaah 11952, Saudi Arabia

Received: 01 December 2020 Accepted: 10 March 2021 Published: 04 August 2021

Background

Onodi cells (OC) are important for rhinologists because they contain the optic canal, and are close to the optic nerve and internal carotid artery. Therefore, any pathologic processes within OCs, including infectious or inflammatory sinusitis, fungus ball, inverted papilloma, mucocele, or sino-nasal malignancy can cause devastating ophthalmologic complications. We aimed to define the different optic neuropathy conditions related to isolated OC mucoceles, determine the different symptoms of each condition and its risk factors, and explore the efficacy of the relevant diagnostic tools and treatment strategies.

Methodology

A comprehensive electronic search with time and language restrictions was conducted. Several known databases were included: PubMed, The Cochrane Library, and Web of Science from 1990 to 2020. We combined the search terms and limited the study to the English language. We removed duplicates, and the articles were screened based on title, abstract, and full text according to the PRISMA checklist.

Results

The electronic search strategy conducted in this review resulted in 409 hits. After removing duplicate studies and studies with inadequate information, 20 case studies were finally included in this analysis, 65% of which presented men (n = 13), and seven presented women (35%). The mean age reported in these studies was 54.75 with a standard deviation of 14.62 years. We found that visual disturbances that can lead to visual loss were present in most cases (75% of cases). Other symptoms included headache (35%) and eye pain (30%). The risk factors for developing optic neuropathy conditions related to OC mucoceles include being Asian and elderly with a history of either nasal or eye conditions. Our findings showed that the mean time between the onset of symptoms and intervention was 16.8 ± 21.8 days. Most patients underwent endoscopic sinus surgery to remove the OC mucocele (18/20; 90%) with a success rate of 77%. Pharmacologic intervention as a solo treatment (IV corticosteroids or /and antibiotic) was used in only six patients, with a success rate of only 33%. Computed tomography (CT) and magnetic resonance imaging (MRI) is the most commonly used diagnostic tools, with diagnostic success rates of 40% and 82.3%, respectively.

Conclusion

Optic neuropathy conditions related to OC mucoceles are very rare. However, a higher incidence of these conditions was observed in elderly Asian patients with a history of nasal or optic conditions. Visual disturbances are the most common symptoms accompanying any type of eye condition. Endoscopic sinus surgery is considered an effective and safe intervention for these patients, and the period between the onset of symptoms and surgery does not affect the outcomes of the surgery. Furthermore, treatment with corticosteroids and/or antibiotics cannot replace surgery and it can also worsen the condition. Moreover, MRI is superior to CT scans for demonstrating this abnormality, and both are superior to other diagnostic tools. Finally, further investigations should be conducted to study the causes of the low incidence of these conditions in the eastern region.

Keywords:

Citation: Turki M. Bin Mahfoz, Ahmad K. Alnemare. Optic neuropathy related to Onodi cell mucocele: a systematic review and meta-analysis of randomized controlled trials[J]. AIMS Medical Science, 2021, 8(3): 203-223. doi: 10.3934/medsci.2021018

Related Papers:

[1]	Ruizhi Yang, Dan Jin, Wenlong Wang . A diffusive predator-prey model with generalist predator and time delay. AIMS Mathematics, 2022, 7(3): 4574-4591. doi: 10.3934/math.2022255
[2]	Yingyan Zhao, Changjin Xu, Yiya Xu, Jinting Lin, Yicheng Pang, Zixin Liu, Jianwei Shen . Mathematical exploration on control of bifurcation for a 3D predator-prey model with delay. AIMS Mathematics, 2024, 9(11): 29883-29915. doi: 10.3934/math.20241445
[3]	Sahabuddin Sarwardi, Hasanur Mollah, Aeshah A. Raezah, Fahad Al Basir . Direction and stability of Hopf bifurcation in an eco-epidemic model with disease in prey and predator gestation delay using Crowley-Martin functional response. AIMS Mathematics, 2024, 9(10): 27930-27954. doi: 10.3934/math.20241356
[4]	Xin-You Meng, Fan-Li Meng . Bifurcation analysis of a special delayed predator-prey model with herd behavior and prey harvesting. AIMS Mathematics, 2021, 6(6): 5695-5719. doi: 10.3934/math.2021336
[5]	Heping Jiang . Complex dynamics induced by harvesting rate and delay in a diffusive Leslie-Gower predator-prey model. AIMS Mathematics, 2023, 8(9): 20718-20730. doi: 10.3934/math.20231056
[6]	Qinghui Liu, Xin Zhang . Chaos detection in predator-prey dynamics with delayed interactions and Ivlev-type functional response. AIMS Mathematics, 2024, 9(9): 24555-24575. doi: 10.3934/math.20241196
[7]	Hairong Li, Yanling Tian, Ting Huang, Pinghua Yang . Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model. AIMS Mathematics, 2024, 9(10): 29608-29632. doi: 10.3934/math.20241434
[8]	Fatao Wang, Ruizhi Yang, Yining Xie, Jing Zhao . Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator. AIMS Mathematics, 2023, 8(8): 17719-17743. doi: 10.3934/math.2023905
[9]	Eric M. Takyi, Charles Ohanian, Margaret Cathcart, Nihal Kumar . Sex-biased predation and predator intraspecific competition effects in a prey mating system. AIMS Mathematics, 2024, 9(1): 2435-2453. doi: 10.3934/math.2024120
[10]	Liye Wang, Wenlong Wang, Ruizhi Yang . Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect. AIMS Mathematics, 2023, 8(4): 9716-9739. doi: 10.3934/math.2023490

Abstract

Background

Methodology

Results

Conclusion

1. Introduction

In survey sampling, it is well known fact that suitable use of the auxiliary information may improves the precision of an estimator for the unknown population parameters. The auxiliary information can be used either at the design stage or at estimation stage to increase the accuracy of the population parameter estimators. Several authors presented modified different type of estimators for estimating the finite population mean including ^{[4,9,21,22,23,24,25,26,27]}.

The problem of estimation of finite population mean or total in two-stage sampling scheme using the auxiliary information has been well established. The two stage sampling scheme is an improvement over the cluster sampling, when it is not possible or easy to calculate all the units from the selected clusters. One of the main characteristic could be the budget, and it becomes too difficult to collect information from all the units within the selected clusters. To overcome this, one way is to select clusters, called first stage unit (fsus) and from the given population of interest, select a subsample from the selected clusters called the second stage units (ssu). This also benefits to increase the size of the first stage samples which consist of clusters, and assume to be heterogeneous groups. If there is no variation within clusters then might not be possible to collect information from all the units within selected clusters. In many situations, it is not possible to obtain the complete list of ultimate sampling units in large scale sample surveys, while a list of primary units of clusters may be available. In such situations, we select a random sample of first stage units or primary units using certain probability sampling schemes i.e simple random sampling (with or without replacement), systematic sampling and probability proportional to size (PPS), and then we can perform sub-sampling in selected clusters (first stage units). This approach is called two-stage sampling scheme.

Two-stage has a great varaity of applications, which go far beyond the immediate scop of sample survey. Whenever any process involves in chemical, physical, or biological tests that can be performed on a small amount of materail, it is likely to be drawn as a subsample from a larger amount that is itself a sample.

In large scale survey sampling, it is usual to adopt multistage sampling to estimate the population mean or total of the study variable $y$ . ^[13] proposed a general class of estimators of a finite population mean using multi-auxiliary information under two stage sampling scheme. ^[1] proposed an alternative class of estimators in two stage sampling with two auxiliary variables. ^[10] proposed estimators for finite population mean under two-stage sampling using multivariate auxiliary information. ^[12] suggested a detailed note on ratio estimates in multi-stage sampling. ^[6] given some stratagies in two stage sampling using auxiliary information. ^[3] suggested a class of predictive estimaotrs in two stage sampling using auxiliary information. ^[8] gave a generalized method of estimation for two stage sampling using two auxiliary variables. ^[5] suggested chain ratio estimators in two stage sampling. For certain related work, we refer some latest articles, i.e., ^{[14,15,16,17,18,19,20]}.

In this article, we propose an improved generalized class of estimators using two auxiliary variables under two-stage sampling scheme. The biases and mean sqaure errors of the proposed generalized class of estimators are derived up to first order of approximation. Based on the numerical results, the proposed class of estimators are more efficient than their existing counterparts.

2. Symbols and notation

Consider a finite population $U$ = $\{{U}_{1}, {U}_{2}, ..., {U}_{N}\}$ is divided into $N$ first-stage units (fsus) clusters in the population. Let N be the total number of first stage unit in population, n be the number of first stage units selected in the sample, ${M}_{i}$ be the number of second stage units (ssus) belongs to the ${i}^{th}$ first stage units (fsus), (i = 1, 2, …, N), and mi be the number of fsus selected from the ${i}^{th}$ fsu in the sample of n fsus, (i = 1, 2, …, n).

Let ${y}_{ij}$ , ${x}_{ij}$ and ${z}_{ij}$ be values of the study variable $y$ and the auxiliary variables $\left(x\mathrm{a}\mathrm{n}\mathrm{d}z\right)$ respectively, for the ${j}^{th}$ ssus ${U}_{i} = (j = \mathrm{1, 2}, ..., {M}_{i})$ , in the ${i}^{th}$ fsus. The population mean of the study variable $y$ and the auxiliary variables $(x, z)$ are given by:

$\overline {Y} = \frac{1}{N}\sum _{i = 1}^{N}\;{u}_{i}{\overline {Y}}_{i}, \;\;\;\;\overline {X} = \frac{1}{N}\sum _{i = 1}^{N}\;{u}_{i}{\overline {X}}_{i}, \;\;\;\;\overline {Z} = \frac{1}{N}\sum _{i = 1}^{N}\;{u}_{i}{\overline {Z}}_{i} ,$

where

${\overline {Y}}_{i} = \frac{1}{{M}_{i}}\sum\limits_{j = 1}^{{M}_{i}}\;{y}_{ij}, {\overline {X}}_{i} = \frac{1}{{M}_{i}}\sum\limits_{j = 1}^{{M}_{i}}\;{x}_{ij}, {\overline {Z}}_{i} = \frac{1}{{M}_{i}}\sum\limits_{j = 1}^{{M}_{i}}\;{z}_{ij}, \;\;(i = \mathrm{1, 2}, ..., N).$

${u}_{i} = \frac{{M}_{i}}{\overline {M}} , \;{\rm{and}}\; \overline {M} = \frac{M}{N} , \; M = \sum _{i = 1}^{N}\;{M}_{i} ,$

$R = \frac{\overline {Y}}{\overline {X}} , \; {\rm{and}} \; {R}_{i} = \frac{{\overline {Y}}_{i}}{{\overline {X}}_{i}} ,$

${S}_{by}^{2} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Y}}_{i}-\overline {Y}{)}^{2} ,$

${S}_{bx}^{2} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {X}}_{i}-\overline {X}{)}^{2} ,$

${S}_{bz}^{2} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Z}}_{i}-\overline {Z}{)}^{2} ,$

${S}_{byx} = \frac{1}{N-1}\sum _{i = 1}^{N}\;\left({u}_{i}{\overline {Y}}_{i}-\overline {Y}\right)\left({u}_{i}{\overline {X}}_{i}-\overline {X}\right), \;\;\;\;\;\;\;\;{S}_{byz} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Z}}_{i}-\overline {Z})({u}_{i}{\overline {Y}}_{i}-\overline {Y}) ,$

${S}_{bxz} = \frac{1}{N-1}\sum _{i = 1}^{N}\;({u}_{i}{\overline {Z}}_{i}-\overline {Z})({u}_{i}{\overline {X}}_{i}-\overline {X}), \;\;\;\;\;\;\;\;{S}_{iy}^{2} = \frac{1}{{M}_{i}-1}\sum _{j = 1}^{{M}_{i}}\;({y}_{ij}-{\overline {Y}}_{i}{)}^{2} ,$

${S}_{ix}^{2} = \frac{1}{{M}_{i}-1}\sum _{j = 1}^{{M}_{i}}\;({x}_{ij}-{\overline {X}}_{i}{)}^{2}, \;\;\;\;\;\;\;\;{S}_{iz}^{2} = \frac{1}{{M}_{i}-1}\sum _{j = 1}^{{M}_{i}}\;({z}_{ij}-{\overline {Z}}_{i}{)}^{2} ,$

${S}_{iyx} = \frac{1}{{M}_{i}-1}({y}_{ij}-{\overline {Y}}_{i})({x}_{ij}-{\overline {X}}_{i}), \;\;\;\;\;\;\;\;{S}_{iyz} = \frac{1}{{M}_{i}-1}({y}_{ij}-{\overline {Y}}_{i})({z}_{ij}-{\overline {Z}}_{i}) ,$

${S}_{ixz} = \frac{1}{{M}_{i}-1}\sum\limits_{j = 1}^{{M}_{i}}\;({x}_{ij}-{\overline {X}}_{i})({z}_{ij}-{\overline {Z}}_{i}), \;\;\;\;\;\;\;\;(i = \mathrm{1, 2}, ...N).$

Similarly for sample data:

$\overline {y} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {y}}_{i} = {\overline {y}}^{\mathrm{*}}\left(say\right), \;\overline {x} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {x}}_{i} = {\overline {x}}^{\mathrm{*}}\left(say\right), \;\overline {z} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {z}}_{i} = {\overline {z}}^{\mathrm{*}}\left(say\right),$

where

${\overline {y}}_{i} = \frac{1}{{m}_{i}}\sum\limits_{j = 1}^{{m}_{i}}\;{y}_{ij}, \;\;\;\;{\overline {x}}_{i} = \frac{1}{{m}_{i}}\sum\limits_{j = 1}^{{m}_{i}}\;{x}_{ij}, \;\;\;\;{\overline {z}}_{i} = \frac{1}{{m}_{i}}\sum\limits_{j = 1}^{{m}_{i}}\;{z}_{ij},$

${s}_{by}^{2} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {y}}_{i}-\overline {y}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{bx}^{2} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {x}}_{i}-\overline {x}{)}^{2} ,$

${s}_{bz}^{2} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {z}}_{i}-\overline {z}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{byx} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {y}}_{i}-\overline {y}\left)\right({u}_{i}{\overline {x}}_{i}-\overline {x}) ,$

${s}_{byz} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {y}}_{i}-\overline {y})({u}_{i}{\overline {z}}_{i}-\overline {z}), \;\;\;\;\;\;\;\;{s}_{bxz} = \frac{1}{n-1}\sum _{i = 1}^{n}\;({u}_{i}{\overline {x}}_{i}-\overline {x})({u}_{i}{\overline {z}}_{i}-\overline {z}) ,$

${s}_{iy}^{2} = \frac{1}{{m}_{i}-1}({y}_{ij}-{\overline {y}}_{i}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{ix}^{2} = \frac{1}{{m}_{i}-1}({x}_{ij}-{\overline {x}}_{i}{)}^{2} ,$

${s}_{iz}^{2} = \frac{1}{{m}_{i}-1}({z}_{ij}-{\overline {z}}_{i}{)}^{2}, \;\;\;\;\;\;\;\;{s}_{iyx} = \frac{1}{{m}_{i}-1}\sum _{j = 1}^{{m}_{i}}\;({y}_{ij}-{\overline {y}}_{i}\left)\right({x}_{ij}-{\overline {x}}_{i}) ,$

${s}_{iyz} = \frac{1}{{m}_{i}-1}\sum _{j = 1}^{{m}_{i}}\;\left({y}_{ij}-{\overline {y}}_{i}\right)\left({z}_{ij}-{\overline {z}}_{i}\right), \;\;\;\;\;\;\;\;{s}_{ixz} = \frac{1}{{m}_{i}-1}\sum _{j = 1}^{{m}_{i}}\;({x}_{ij}-{\overline {x}}_{i})({z}_{ij}-{\overline {z}}_{i}) .$

In order to obtain the biases and mean sqaured errors, we consider the following relative error terms:

${e}_{0} = \frac{{\overline {y}}^{\mathrm{*}}-\overline {Y}}{\overline {Y}}, \;{e}_{1} = \frac{{\overline {x}}^{\mathrm{*}}-\overline {X}}{\overline {X}}, \;{e}_{2} = \frac{{\overline {z}}^{\mathrm{*}}-\overline {Z}}{\overline {Z}} ,$

$E\left({e}_{0}^{2}\right) = \lambda {C}_{by}^{2}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iy}^{2} = {V}_{y} ,$

$E\left({e}_{1}^{2}\right) = \lambda {C}_{bx}^{2}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{ix}^{2} = {V}_{x} ,$

$E\left({e}_{2}^{2}\right) = \lambda {C}_{bz}^{2}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iz}^{2} = {V}_{z} ,$

$E\left({e}_{0}{e}_{1}\right) = \lambda {C}_{byx}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iyx} = {V}_{yx},$

$E\left({e}_{0}{e}_{2}\right) = \lambda {C}_{byz}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{iyz} = {V}_{yz},$

$E\left({e}_{1}{e}_{2}\right) = \lambda {C}_{bxz}+\frac{1}{nN}\sum _{i = 1}^{n}\;{u}_{i}^{2}{\theta }_{i}{C}_{ixz} = {V}_{xz} ,$

${C}_{by} = \frac{{S}_{by}}{\overline {Y}}, \;\;\;\;\;\;\;\;{C}_{bx} = \frac{{S}_{bx}}{\overline {X}}, \;\;\;\;\;\;\;\;{C}_{bz} = \frac{{S}_{bz}}{\overline {Z}} ,$

${C}_{byx} = \frac{{S}_{byx}}{\overline {Y}\overline {X}}, \;\;\;\;\;\;\;\;{C}_{byz} = \frac{{S}_{byz}}{\overline {Y}\overline {Z}}, \;\;\;\;\;\;\;\;{C}_{bxz} = \frac{{S}_{bxz}}{\overline {X}\overline {Z}} ,$

${C}_{iyx} = \frac{{S}_{iyx}}{\overline {Y}\overline {X}}, \;\;\;\;\;\;\;\;{C}_{iyz} = \frac{{S}_{byz}}{\overline {Y}\overline {Z}}, \;\;\;\;\;\;\;\;{C}_{ixz} = \frac{{S}_{ixz}}{\overline {X}\overline {Z}} ,$

${C}_{iy} = \frac{{S}_{iy}}{\overline {Y}}, \;\;\;\;\;\;\;\;{C}_{ix} = \frac{{S}_{ix}}{\overline {X}}, \;\;\;\;\;\;\;\;{C}_{iz} = \frac{{S}_{iz}}{\overline {Z}} ,$

where,

${\theta }_{i} = \left(\frac{1}{{m}_{i}}-\frac{1}{{M}_{i}}\right), \lambda = (\frac{1}{n}-\frac{1}{N}) .$

3. Existing estimators

In this section, we consider several estimators of the finite population mean under two-stage sampling that are available in the sampling literature, the properties of all estimators considered here are obtained up-to the first order of approximation.

(ⅰ) The usual mean estimator ${\overline {y}}^{\mathrm{*}} = {\overline {y}}_{0}^{\mathrm{*}}$ and its variance under two-stage sampling are given by:

${\overline {y}}_{0}^{\mathrm{*}} = \frac{1}{n}\sum _{i = 1}^{n}\;{u}_{i}{\overline {y}}_{i} ,$

(1)

and

$V\left({\overline {y}}_{0}^{\mathrm{*}}\right) = {\overline {Y}}^{2}{V}_{y} = MSE\left({\overline {y}}_{0}^{\mathrm{*}}\right) .$

(2)

(ⅱ) The usual ratio estimator under two-stage sampling, is given by:

${\overline {y}}_{R}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right) ,$

(3)

where $\overline {X}$ is the known population mean of $x$ .

The bias and $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{R}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{R}^{\mathrm{*}}\right) = \overline {Y}\left[{V}_{x}-{V}_{yx}\right] ,$

(4)

and

$MSE\left({\overline {y}}_{R}^{\mathrm{*}}\right) = {\overline {Y}}^{2}\left[{V}_{y}+{V}_{x}-2{V}_{yx}\right] .$

(5)

(ⅲ) ^[2] Exponential ratio type estimator under two-stage sampling, is given by:

${\overline {y}}_{E}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}exp\left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}+{\overline {x}}^{\mathrm{*}}}\right) .$

(6)

The bias and $MSE$ of ${\overline {y}}_{E}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{E}^{\mathrm{*}}\right) = \overline {Y}\left[\frac{3}{8}{V}_{x}-\frac{1}{2}{V}_{yx}\right] ,$

(7)

and

$MSE\left({\overline {y}}_{E}^{\mathrm{*}}\right) = {\overline {Y}}^{2}\left[{V}_{y}+\frac{1}{4}{V}_{x}-{V}_{yx}\right] .$

(8)

(ⅳ) The traditional difference estimator under two-stage sampling is given by:

${\overline {y}}_{D}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}+d\left(\overline {X}-{\overline {x}}^{\mathrm{*}}\right) ,$

(9)

where $d$ is the constant.

The minimum variance of ${\overline {y}}_{D}^{\mathrm{*}}$ , is given by:

$V\left({\overline {y}}_{Dmin}^{\mathrm{*}}\right) = {\overline {Y}}^{2}{V}_{y}\left(1-{\rho }^{\mathrm{*}2}\right) = MSE\left({\overline {y}}_{D}^{\mathrm{*}}\right) ,$

(10)

where ${\rho }^{\mathrm{*}} = \frac{{V}_{yx}}{\sqrt{{V}_{y}}\sqrt{{V}_{x}}}$ .

The optimum value of $d$ is ${d}_{opt} = \frac{\overline {Y}{V}_{yx}}{\overline {X}{V}_{x}}$ .

(ⅴ) ^[7] Difference type estimator under two-stage sampling, is given by:

${\overline {y}}_{Rao}^{\mathrm{*}} = {d}_{0}{\overline {y}}^{\mathrm{*}}+{d}_{1}\left(\overline {X}-{\overline {x}}^{\mathrm{*}}\right) ,$

(11)

where ${d}_{0}$ and ${d}_{1}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{Rao}^{\mathrm{*}}$ to first order of approximation, is given by:

$Bias\left({\overline {y}}_{Rao}^{\mathrm{*}}\right) = ({d}_{0}-1)\overline {Y} ,$

(12)

and

$MSE({\overline {y}}_{Rao}^{\mathrm{*}}{)}_{min}\cong \frac{{\overline {Y}}^{2}({V}_{x}{V}_{y}-{V}_{yx}^{2})}{{V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x}} = \frac{{\overline {Y}}^{2}{V}_{y}(1-{\rho }^{\mathrm{*}2})}{1+{V}_{y}(1-{\rho }^{\mathrm{*}2})} .$

(13)

The optimum values of ${d}_{0}$ and ${d}_{1}$ are:

${d}_{0opt} = \frac{{V}_{x}}{{V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x}}$ and ${d}_{1opt} = \frac{\overline {Y}{V}_{yx}}{\overline {X}({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})}$ .

(ⅵ) The difference-in-ratio type estimator under two-stage sampling, is given by:

${\overline {y}}_{DR}^{\mathrm{*}} = \left[{d}_{2}{\overline {y}}^{\mathrm{*}}+{d}_{3}(\overline {X}-{\overline {x}}^{\mathrm{*}})\right]\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right) ,$

(14)

where ${d}_{2}$ and ${d}_{3}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DR}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{DR}^{\mathrm{*}}\right)\cong \overline {Y}({d}_{2}-1)-{d}_{2}\overline {Y}{V}_{yx}+{d}_{3}\overline {X}{V}_{x}+{d}_{2}{V}_{x}\overline {Y} ,$

(15)

and

$MSE({\overline {y}}_{DR}^{\mathrm{*}}{)}_{min}\cong \frac{{\overline {Y}}^{2}({V}_{x}^{2}{V}_{y}-{V}_{x}{V}_{yx}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2})}{({V}_{x}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2}-{V}_{x})} .$

(16)

The optimum values of ${d}_{2}$ and ${d}_{3}$ are:

${d}_{2opt} = \frac{{V}_{x}({V}_{x}-1)}{{V}_{x}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2}-{V}_{x}} ,$

${d}_{3opt} = \frac{-{\overline {Y}}^{2}({V}_{x}^{2}+{V}_{x}{V}_{y}-{V}_{x}{V}_{yx}-{V}_{yx}^{2}-{V}_{x}+{V}_{yx})}{\overline {X}({V}_{x}^{2}-{V}_{x}{V}_{y}+{V}_{yx}^{2}-{V}_{x})} .$

(ⅶ) The difference-in-exponential ratio type estimator under two-stage sampling, is given by:

${\overline {y}}_{DE}^{\mathrm{*}} = \left[{d}_{4}{\overline {y}}^{\mathrm{*}}+{d}_{5}(\overline {X}-{\overline {x}}^{\mathrm{*}})\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}-{\overline {x}}^{\mathrm{*}}}\right) ,$

(17)

where ${d}_{4}$ and ${d}_{5}$ are constants.

The bias and minimum $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DE}^{\mathrm{*}}$ to first order of approximation, are given by:

$Bias\left({\overline {y}}_{DE}^{\mathrm{*}}\right) = ({d}_{4}-1)\overline {Y}-\frac{1}{2}{d}_{4}\overline {Y}{V}_{yx}+\frac{3}{8}{d}_{4}\overline {Y}{V}_{x}+\frac{1}{2}{d}_{5}{V}_{x} ,$

(18)

and

$MSE({\overline {y}}_{DE}^{\mathrm{*}}{)}_{min}\cong \frac{-{\overline {Y}}^{2}\left({V}_{x}^{3}+16{V}_{x}^{2}{V}_{y}-16{V}_{x}{V}_{yx}^{2}-64{V}_{x}{V}_{y}+64{V}_{yx}^{2}\right)}{64{V}_{x}{V}_{y}-64{V}_{yx}^{2}+64{V}_{x}} .$

(19)

The optimum values of ${d}_{4}$ and ${d}_{5}$ are:

${d}_{4opt} = -\frac{1}{8}\frac{{V}_{x}({V}_{x}-8)}{({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})} ,$

${d}_{5opt} = \frac{\overline {Y}\left({V}_{x}^{2}+4{V}_{x}{V}_{y}-{V}_{x}{V}_{yx}-4{V}_{yx}^{2}-4{V}_{x}+8{V}_{yx}\right)}{\frac{1}{8}\overline {X}({V}_{x}{V}_{y}-{V}_{yx}^{2}+{V}_{x})} .$

(ⅷ) The difference-difference type estimator under two stage sampling, is given by:

${\overline {y}}_{DD}^{\mathrm{*}} = {\overline {y}}^{\mathrm{*}}+{d}_{6}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{d}_{7}(\overline {Z}-{\overline {z}}^{\mathrm{*}}) ,$

(20)

where ${d}_{6}$ and ${d}_{7}$ are constants.

The minimum variance or M $SE$ of ${\overline {y}}_{DD}^{\mathrm{*}}$ to first order of approximation, is given by:

$MSE({\overline {y}}_{DD}^{\mathrm{*}}{)}_{min}\cong \frac{{\overline {Y}}^{2}({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z})}{{V}_{x}{V}_{z}-{V}_{xz}^{2}} .$

(21)

The optimum values of ${d}_{6}$ and ${d}_{7}$ are:

${d}_{6} = \frac{-\overline {Y}({V}_{xz}{V}_{yz}-{V}_{yx}{V}_{z})}{\overline {X}({V}_{x}{V}_{z}-{V}_{xz}^{2})} ,$

${d}_{7} = \frac{\overline {Y}({V}_{x}{V}_{yz}-{V}_{xz}{V}_{yx})}{\overline {Z}({V}_{x}{V}_{z}-{V}_{xz}^{2})}.$

(ix) The difference-difference type estimator under two stage sampling, is given by:

${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}} = {d}_{8}{\overline {y}}^{\mathrm{*}}+{d}_{9}\left(\overline {X}-{\overline {x}}^{\mathrm{*}}\right)+{d}_{10}\left(\overline {Z}-{\overline {z}}^{\mathrm{*}}\right) ,$

(22)

where ${d}_{8}$ , ${d}_{9}$ and ${d}_{10}$ are constants.

The bias and $\mathrm{M}\mathrm{S}\mathrm{E}$ of ${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$ to first order of approximation is given by:

$Bias\left({\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}\right) = \overline {Y}\left({d}_{8}-1\right) ,$

(23)

and

$MSE\left({\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}\right)\cong \frac{{\overline {Y}}^{2}\left({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}\right)}{{V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2}} .$

(24)

The optimum values of ${d}_{8}$ , ${d}_{9}$ and ${d}_{10}$ are given by:

${d}_{8} = \frac{{V}_{x}{V}_{z}-{V}_{xz}^{2}}{{V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2}} ,$

${d}_{9} = \frac{-\overline {Y}({V}_{xz}{V}_{yz}-{V}_{yx}{V}_{z})}{\overline {X}({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2})} ,$

${d}_{10} = \frac{\overline {Y}({V}_{x}{V}_{yz}-{V}_{xz}{V}_{yx})}{\overline {Z}({V}_{x}{V}_{y}{V}_{z}-{V}_{x}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}+2{V}_{xz}{V}_{yx}{V}_{yz}-{V}_{yx}^{2}{V}_{z}+{V}_{x}{V}_{z}-{V}_{xz}^{2})}.$

4. Proposed estimator

The principal advantage of our proposed improved generalized class of estimators under two-stage sampling is that it is more flexible, efficient, than the existing estimators. The mean square errors based on two data sets are minimum and percentage relative efficiency is more than hundred as compared to the existing estimators considered here. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3)$ , $\delta$ and $\gamma$ . On the lines of ^[2,7], we propose the following generalized improved class of estimators under two stage sampling for estimation of finite population mean using two auxiliary varaible as given by:

${\overline {y}}_{G}^{\mathrm{*}} = \left[{w}_{1}{\overline {y}}^{\mathrm{*}}+{w}_{2}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{w}_{3}(\overline {Z}-{\overline {z}}^{\mathrm{*}})\right]\left[\left\{exp\delta \left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}+{\overline {x}}^{\mathrm{*}}}\right)\right\}{\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right)}^{\gamma }\right] ,$

(25)

where ${w}_{i}(i = \mathrm{1, 2}, 3)$ are constants, whose values are to be determined; $\delta$ and $\gamma$ are constants i.e., ( $0\le \delta$ , $\gamma \le 1$ ) and can be used to construct the different estimators.

Using (25), solving ${\overline {y}}_{G}^{\mathrm{*}}$ in terms of errors, we have

${\overline {y}}_{G}^{\mathrm{*}}-\overline {Y} = ({w}_{1}-1)\overline {Y}+{w}_{1}\overline {Y}\left\{{e}_{0}-\frac{1}{2}{\alpha }_{1}{e}_{1}+\frac{1}{8}{\alpha }_{2}{e}_{1}^{2}-\frac{1}{2}{\alpha }_{1}{e}_{0}{e}_{1}\right\}$

$-{w}_{2}\overline {X}\left\{{e}_{1}-\frac{1}{2}{\alpha }_{1}{e}_{1}^{2}\right\}-{w}_{3}\overline {Z}\left\{{e}_{2}-\frac{1}{2}{\alpha }_{1}{e}_{1}{e}_{2}\right\} ,$

where

${\alpha }_{1} = \delta +2\gamma$ and ${\alpha }_{2} = \delta (\delta +2)+4\gamma (\delta +\gamma +1)$ .

The bias and $MSE$ of ${\overline {y}}_{G}^{\mathrm{*}}$ are given by:

$Bias\left({\overline {y}}_{G}^{\mathrm{*}}\right)\cong ({w}_{1}-1)\overline {Y}+{w}_{1}\overline {Y}\left\{\frac{1}{8}{\alpha }_{2}{V}_{x}-\frac{1}{2}{\alpha }_{1}{V}_{yx}\right\}+{w}_{2}\overline {X}{\alpha }_{1}\frac{{V}_{x}}{2}+{w}_{3}\overline {Z}{\alpha }_{1}\frac{{V}_{xz}}{2} ,$

(26)

and

$MSE\left({\overline {y}}_{G}^{\mathrm{*}}\right)\cong ({w}_{1}-1{)}^{2}+{w}_{1}^{2}{\overline {Y}}^{2}A+{w}_{2}^{2}{\overline {X}}^{2}B+{w}_{3}^{2}{\overline {Z}}^{2}C-{w}_{1}{\overline {Y}}^{2}D-{w}_{2}\overline {Y}\overline {X}E$

$-{w}_{3}\overline {Y}\overline {Z}F+2{w}_{1}{w}_{2}\overline {Y}\overline {X}G+2{w}_{1}{w}_{3}\overline {Y}\overline {Z}H+2{w}_{2}{w}_{3}\overline {X}\overline {Z}I ,$

(27)

where

$A = {V}_{y}+\frac{1}{4}{V}_{x}({\alpha }_{1}^{2}+{\alpha }_{2})-2{\alpha }_{1}{V}_{yx}, \;\;\;\;B = {V}_{x}, \;\;\;\;C = {V}_{z},$

$D = \frac{1}{4}{\alpha }_{2}{V}_{x}-{\alpha }_{1}{V}_{yx}, \;\;\;\;\;\;\;\;E = {\alpha }_{1}{V}_{x}, \;\;\;\;F = {\alpha }_{1}{V}_{xz},$

$G = {\alpha }_{1}{V}_{x}-{V}_{yx}, \;\;\;\;\;\;\;\;H = {\alpha }_{1}{V}_{xz}-{V}_{xz}, \;\; I = {V}_{xz} .$

Solving (27), the minimum $MSE$ of ${\overline {y}}_{G}^{\mathrm{*}}$ to first order of approximation are given by:

$MSE({\overline {y}}_{G}^{\mathrm{*}}{)}_{min} = {\overline {Y}}^{2}[1-\frac{{\mathrm{\Omega }}_{2}}{4{\mathrm{\Omega }}_{1}}] ,$

(28)

where

${\mathrm{\Omega }}_{1} = ABC-A{I}^{2}-B{H}^{2}-C{G}^{2}+2GHI+BC-{I}^{2} ,$

and

${\mathrm{\Omega }}_{2} = AB{F}^{2}+AC{E}^{2}-2AEFI+BC{D}^{2}-2BDFH-2CDEG-{D}^{2}{I}^{2}+2DEHI$

$+2DFGI-{E}^{2}{H}^{2}+2EFGH-{F}^{2}{G}^{2}+4BCD+B{F}^{2}-4BFH+C{E}^{2}$

$-4CEG-4D{I}^{2}-2EFI+4EHI+4FGI+4BC+4{I}^{2}.$

The optimum values of ${w}_{i}(i = 1, \;2, \;3)$ are given by:

${w}_{1opt} = \frac{{\mathrm{\Omega }}_{3}}{2{\mathrm{\Omega }}_{1}}, {w}_{2opt} = \frac{\overline {Y}{\mathrm{\Omega }}_{4}}{2\overline {X}{\mathrm{\Omega }}_{1}},$ and ${w}_{3opt} = \frac{\overline {Y}{\mathrm{\Omega }}_{5}}{2\overline {Z}{\mathrm{\Omega }}_{1}}$ ,

where

${\mathrm{\Omega }}_{3} = BCD-BFH-CEG-D{I}^{2}+EHI+FGI+2GI+2BC-2{I}^{2},$

${\mathrm{\Omega }}_{4} = ACE-AFI-CDG+DHI-E{H}^{2}+FGH+CE-2CG-FI+2HI,$

${\mathrm{\Omega }}_{5} = ABF-AEI-BDH+DGI+EGH-F{G}^{2}+BF-2BH-EI+2GI .$

From (28), we produce the following two estimators called ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ . Put $(\delta = 0, \gamma = 1)$ and $(\delta = 1, \gamma = 0)$ in (25), we get the following two estimators respectively:

$\left(i\right)\;\;{\overline {y}}_{G1}^{\mathrm{*}} = \left[{w}_{4}{\overline {y}}^{\mathrm{*}}+{w}_{5}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{w}_{6}(\overline {Z}-{\overline {z}}^{\mathrm{*}})\right]\left(\frac{\overline {X}}{{\overline {x}}^{\mathrm{*}}}\right) ,$

$\left(ii\right)\;\;{\overline {y}}_{G2}^{\mathrm{*}} = \left[{w}_{7}{\overline {y}}^{\mathrm{*}}+{w}_{8}(\overline {X}-{\overline {x}}^{\mathrm{*}})+{w}_{9}(\overline {Z}-{\overline {z}}^{\mathrm{*}})\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{\overline {X}-{\overline {x}}^{\mathrm{*}}}{\overline {X}+{\overline {x}}^{\mathrm{*}}}\right) ,$

where ${w}_{i}(i = 4, \;5, \;6, \;7, \;8, \;9)$ are constants. Solving ${\overline {y}}_{G1}^{\mathrm{*}}$ , in terms of errors, we have:

$({\overline {y}}_{G1}^{\mathrm{*}}-\overline {Y}) = \left[-\overline {Y}+{w}_{4}\overline {Y}+{w}_{4}\overline {Y}{e}_{o}-{w}_{5}\overline {X}{e}_{1}-{w}_{6}\overline {Z}{e}_{2}\right].\left[1-\frac{1}{2}{e}_{1}+\frac{3}{8}{e}_{1}^{2}\right] ,$

$\left({\overline {y}}_{G1}^{\mathrm{*}}-\overline {Y}\right) = \left[\begin{array}{ll}& {w}_{4}\overline {Y}-{w}_{4}\overline {Y}{e}_{1}+\frac{3}{8}{w}_{4}\overline {Y}{e}_{1}^{2}+{w}_{4}\overline {Y}{e}_{0}-\frac{1}{2}{w}_{4}\overline {Y}{e}_{0}{e}_{1}-\overline {Y}+\frac{1}{2}\overline {Y}{e}_{1}-\frac{3}{8}\overline {Y}{e}_{1}^{2}\\ & -{w}_{5}\overline {X}{e}_{1}+\frac{1}{2}{w}_{5}\overline {X}{e}_{1}^{2}-{w}_{6}\overline {Z}{e}_{2}+\frac{1}{2}{w}_{6}\overline {Z}{e}_{1}{e}_{2}\end{array}\right] .$

(29)

The bias and MSE of ${\overline {y}}_{G1}^{\mathrm{*}}$ , to first order of approximation is given by:

${\rm{Bias}}( {\overline {y}}_{G1}^{\mathrm{*}} ) = \frac{3}{8}{w}_{4}\overline {Y}{V}_{x}^{2} - \frac{1}{2}{w}_{4}\overline {Y}{V}_{yx}-\frac{3}{8}\overline {Y}{V}_{x}^{2}+\frac{1}{2}{w}_{5}\overline {X}{V}_{x}^{2}+\frac{1}{2}{w}_{6}\overline {Z}{V}_{xz} ,$

By squaring and taking expectation of (29), we get the mean square error:

$MSE\left({\overline {y}}_{G1}^{\mathrm{*}}\right) = \left[\begin{array}{ll}& {w}_{6}^{2}{V}_{z}^{2}+{w}_{4}^{2}{V}_{y}^{2}+{w}_{5}^{2}{V}_{x}^{2}-2{w}_{4}^{2}R{V}_{yx}+2{w}_{4}R{V}_{yx}-2R{w}_{5}{V}_{x}^{2}\\ & -2{w}_{4}{R}^{2}{V}_{x}^{2}+{w}_{4}^{2}R{V}_{x}^{2}+2{w}_{4}R{w}_{5}{V}_{x}^{2}-2R{w}_{6}{V}_{xz}+{w}_{4}^{2}{\overline {Y}}^{2}\\ & -2{w}_{4}{\overline {Y}}^{2}+{\overline {Y}}^{2}+{R}^{2}{V}_{x}^{2}-2{w}_{4}{w}_{5}{V}_{yx}-2{w}_{4}{w}_{6}{V}_{yz}+2{w}_{5}{w}_{6}{V}_{xz}\\ & +2{w}_{4}R{w}_{6}{V}_{xz}\end{array}\right] .$

(30)

Differentiate (30) with respect to ${w}_{4}$ , ${w}_{5}$ and ${w}_{6}$ , we get the optimum values of ${w}_{4}$ , ${w}_{5}$ and ${w}_{6}$ i.e.,

${w}_{4opt} = \frac{{\overline {Y}}^{2}({V}_{x}^{2}{V}_{z}^{2}-{V}_{xz}^{2})}{\left[\begin{array}{ll}& -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+R{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & -R{V}_{x}^{2}{V}_{xz}^{2}-{\overline {Y}}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}^{2}-{V}_{yx}^{2}{V}_{z}^{2}+2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} ,$

${w}_{5opt} = \frac{\left[\begin{array}{ll}& -{R}^{3}{V}_{x}^{4}{V}_{z}^{2}-{R}^{2}{V}_{x}^{4}{V}_{z}^{2}-{R}^{3}{V}_{x}^{2}{V}_{xz}^{2}-R{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}\\ & -{\overline {Y}}^{2}{V}_{yx}{V}_{z}^{2}+R{V}_{x}^{2}{V}_{yz}^{2}+R{V}_{xz}^{2}{V}_{y}^{2}+R{V}_{yx}^{2}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{xz}{V}_{yz}-2{R}_{Vxz}{V}_{yx}{V}_{yz}\end{array}\right]}{\left[\begin{array}{ll}& -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+R{V}_{x}^{2}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & -R{V}_{x}^{2}{V}_{xz}^{2}-{\overline {Y}}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}^{2}-{V}_{yx}^{2}{V}_{z}+2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} ,$

${w}_{6opt} = \frac{{\overline {Y}}^{2}({V}_{x}^{2}{V}_{yz}-{V}_{xz}{V}_{yx})}{\left[\begin{array}{ll}& -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+R{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & -R{V}_{x}^{2}{V}_{xz}^{2}-{\overline {Y}}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{yz}^{2}-{V}_{xz}^{2}{V}_{y}^{2}-{V}_{yx}^{2}{V}_{z}^{2}+2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} .$

Substituting the optimum values of ${w}_{4}$ , ${w}_{5}$ and ${w}_{6}$ in (30), we get the minimum mean square error of ${\overline {y}}_{G1}^{\mathrm{*}}$ , given by:

$MSE({\overline {y}}_{G1}^{\mathrm{*}}{)}_{min} = \frac{{\overline {Y}}^{2}\left[\begin{array}{ll}& {R}^{2}{V}_{x}^{4}{V}_{z}^{2}-R{V}_{x}^{4}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & +R{V}_{x}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{yz}^{2}+{V}_{xz}^{2}{V}_{y}^{2}+{V}_{yx}^{2}{V}_{z}^{2}-2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]}{\left[\begin{array}{ll}& {R}^{2}{V}_{x}^{4}{V}_{z}^{2}-R{V}_{x}^{4}{V}_{z}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}-{V}_{x}^{2}{V}_{y}^{2}{V}_{z}^{2}\\ & +R{V}_{x}^{2}{V}_{xz}^{2}+{\overline {Y}}^{2}{V}_{xz}^{2}+{V}_{x}^{2}{V}_{yz}^{2}+{V}_{xz}^{2}{V}_{y}^{2}+{V}_{yx}^{2}{V}_{z}^{2}-2{V}_{xz}{V}_{yx}{V}_{yz}\end{array}\right]} .$

(31)

Solving ${\overline {y}}_{G2}^{\mathrm{*}}$ , in terms of errors, we have

$({\overline {y}}_{G2}^{\mathrm{*}}-\overline {Y}) = \left[{w}_{7}\overline {Y}+{w}_{7}\overline {Y}{e}_{0}-\overline {Y}-{w}_{8}\overline {X}{e}_{1}-{w}_{9}\overline {Z}{e}_{2}\right]\left(1-{e}_{1}+{e}_{1}^{2}\right) ,$

$\left({\overline {y}}_{G2}^{\mathrm{*}}-\overline {Y}\right) = \left[\begin{array}{ll}& {w}_{7}\overline {Y}+{w}_{7}\overline {Y}{e}_{0}-\overline {Y}-{w}_{8}\overline {X}{e}_{1}-{w}_{9}\overline {z}{e}_{2}-{w}_{7}\overline {Y}{e}_{1}-{w}_{7}\overline {Y}{e}_{1}+\overline {Y}{e}_{1}\\ & +{w}_{8}\overline {X}{e}_{1}^{2}-{w}_{9}\overline {Z}{e}_{1}{e}_{2}+{w}_{7}\overline {Y}{e}_{1}^{2}-\overline {Y}{e}_{1}^{2}\end{array}\right] .$

(32)

The Bias and MSE ${\overline {y}}_{G2}^{\mathrm{*}}$ , to first order of approximation, is given by:

Bias( ${\overline {y}}_{G2}^{\mathrm{*}}$ ) = ${w}_{8}\overline {X}{V}_{x}^{2}-{w}_{9}\overline {Z}{V}_{xz}+{w}_{7}\overline {Y}{V}_{x}^{2}-\overline {Y}{V}_{x}^{2}.$

By squaring and taking expectation of (32), we get the mean square error:

$MSE\left({\overline {y}}_{G2}^{\mathrm{*}}\right) = 4R{w}_{7}{V}_{yx}-4R{V}_{x}^{2}{w}_{8}+{\overline {Y}}^{2}-2{w}_{7}{\overline {Y}}^{2}+3{R}^{2}{V}_{x}^{2}-6{w}_{7}{R}^{2}{V}_{x}^{2}+4{w}_{7}{w}_{9}R{V}_{xz}$

$-2{w}_{7}{w}_{8}{V}_{yx}-2{w}_{7}\overline {Y}{w}_{9}{V}_{yz}+2{w}_{8}{w}_{9}{V}_{xz}-4R{w}_{9}{V}_{xz}-4{w}_{7}^{2}R{V}_{yx}+3{w}_{7}^{3}{R}^{2}{V}_{x}^{2}$

$+{w}_{9}^{2}{V}_{z}^{2}+{w}_{8}^{2}{V}_{x}^{2}+4{w}_{7}{w}_{8}R{V}_{x}^{2}+{w}_{7}^{2}{\overline {Y}}^{2} .$

(33)

Differentiate (33) with respect to ${w}_{7}$ , ${w}_{8}$ and ${w}_{9}$ , we get the optimum values of ${w}_{7}$ , ${w}_{8}$ and ${w}_{9}$ i.e.,

${w}_{7opt} = \frac{({V}_{x}^{2}{V}_{z}^{2}-{V}_{xz}^{2})(-{R}^{2}{V}_{x}^{2}+{\overline {Y}}^{2})}{{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{xz}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz}+{V}_{yx}^{2}{V}_{z}^{2}} ,$

${w}_{8opt} = \frac{\left[\begin{array}{ll}& 2{\overline {Y}}^{2}R{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{R}^{2}{V}_{x}^{2}{V}_{xz}{V}_{yz}+{R}^{2}{V}_{x}^{2}{V}_{yx}{V}_{z}^{2}\\ & +{\overline {Y}}^{3}{V}_{xz}{V}_{yz}-{\overline {Y}}^{2}{V}_{yz}{V}_{z}^{2}-4\overline {Y}R{V}_{xz}{V}_{yx}{V}_{yz}+2R{V}_{yx}^{2}{V}_{z}\end{array}\right]}{\left[\begin{array}{ll}& {R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}\\ & +{\overline {Y}}^{2}{V}_{xz}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz}+{V}_{yx}^{2}{V}_{z}^{2}\end{array}\right]} ,$

${w}_{9opt} = -\frac{-\overline {Y}{R}^{2}{V}_{x}^{4}{V}_{yz}+{\overline {Y}}^{3}{V}_{x}^{2}{V}_{yz}+{R}^{2}{V}_{x}^{2}{V}_{xz}{V}_{yx}-{\overline {Y}}^{2}{V}_{xz}{V}_{yx}}{{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}-{\overline {Y}}^{2}{V}_{x}^{2}{V}_{z}^{2}-{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}+{\overline {Y}}^{2}{V}_{xz}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz}+{V}_{yx}^{2}{V}_{z}^{2}} .$

Substituting the optimum values of ${w}_{7}$ , ${w}_{8}$ and ${w}_{9}$ in (33), we get the minimum mean square error of ${\overline {y}}_{G2}^{\mathrm{*}}$ , given by:

$MSE({\overline {y}}_{G2}^{\mathrm{*}}{)}_{min} = \frac{({R}^{2}{V}_{x}^{2}-{\overline {Y}}^{2})({\overline {Y}}^{2}{V}_{x}^{2}{V}_{yz}^{2}+{V}_{yx}^{2}{V}_{z}^{2}-2\overline {Y}{V}_{xz}{V}_{yx}{V}_{yz})}{\left[\begin{array}{ll}& -{\overline {Y}}^{2}({V}_{x}^{2}{V}_{yz}^{2}-{V}_{x}^{2}{V}_{z}^{2}+{V}_{xz}^{2})+2{\overline {Y}}^{2}{V}_{xz}{V}_{yx}{V}_{yz}\\ & -{R}^{2}{V}_{x}^{4}{V}_{z}^{2}+{R}^{2}{V}_{x}^{2}{V}_{xz}^{2}-{V}_{yx}^{2}{V}_{z}^{2}\end{array}\right]} .$

(34)

We can generate the considered and many more estimators from (25), by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3),$ $\delta$ and $\gamma$ , given in Table 1.

Table 1. Members of the proposed generalized family of estimators.

${w}_{1}$	${w}_{2}$	${w}_{3}$	$\sigma$	$\gamma$	Estimator
1	0	0	0	0	${\overline {y}}^{\mathrm{*}}$
1	0	0	0	1	${\overline {y}}_{R}^{\mathrm{*}}$
1	0	0	1	0	${\overline {y}}_{E}^{\mathrm{*}}$
1	$d$	0	0	0	${\overline {y}}_{D}^{\mathrm{*}}$
${d}_{0}$	${d}_{1}$	0	0	0	${\overline {y}}_{Rao}^{\mathrm{*}}$
${d}_{2}$	${d}_{3}$	0	0	1	${\overline {y}}_{DR}^{\mathrm{*}}$
${d}_{4}$	${d}_{5}$	0	1	0	${\overline {y}}_{DE}^{\mathrm{*}}$
0	${d}_{6}$	${d}_{7}$	0	0	${\overline {y}}_{DD}^{\mathrm{*}}$
${d}_{8}$	${d}_{9}$	${d}_{10}$	0	0	${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$
${w}_{4}$	${w}_{5}$	${w}_{6}$	0	1	${\overline {y}}_{G1}^{\mathrm{*}}$
${w}_{7}$	${w}_{8}$	${w}_{9}$	1	0	${\overline {y}}_{G2}^{\mathrm{*}}$

| Show Table

DownLoad: CSV

5. Numerical study

Population 1. [Source: ^[11], Model Assisted Survey Sampling]

There are 124 countries (second stage units) divided into 7 continents (first stage units) according to locations. Continent ${7}^{th}$ consists of only one country therefore, we placed ${7}^{th}$ continent in ${6}^{th}$ continent.

We considered:

$y$ = 1983 import (in millions U.S dollars),

$x$ = 1983 export (in millions U.S dollars),

$z$ = 1982 gross national product (in tens of millions of U.S dollars).

The data are divided into 6 clusters, having $N =$ 6, and $n =$ 3. Also ${\sum }_{i = 1}^{N}\;{M}_{i} =$ 124, $\overline {M} =$ 20.67. In , we show cluster sizes, and population means of the study variable $\left(y\right)$ and the auxiliary variables $(x, z)$ . Tables 3 and 4 give some results.

Table 2. Cluster sizes with population means.

No. of clusters	${M}_{i}$	${m}_{i}$	${u}_{i}$	${\overline {Y}}_{i}$	${\overline {X}}_{i}$	${\overline {Z}}_{i}$
1	38	15	1.8387	2254.6	1901.1	1029.158
2	14	6	0.6774	25533.14	22083.21	25671.57
3	11	4	0.5323	3602.82	5835.455	5028.818
4	33	13	1.5968	12156.79	12438.85	7533.939
5	24	10	1.1613	34226.79	33198	16314.42
6	4	2	0.1936	26392.5	29360.5	43967.75

| Show Table

DownLoad: CSV

Table 3. Statistical computation.

$({u}_{i}{\overline {Y}}_{i}-\overline {Y}{)}^{2}$	$({u}_{i}{\overline {X}}_{i}-\overline {X}{)}^{2}$	$({u}_{i}{\overline {Z}}_{i}-\overline {Z}{)}^{2}$	$\left({u}_{i}{\overline {Y}}_{i}-\overline {Y}\right)$ $({u}_{i}{\overline {X}}_{i}-\overline {X})$	$\left({u}_{i}{\overline {X}}_{i}-\overline {X}\right)$ $({u}_{i}{\overline {Z}}_{i}-\overline {Z})$	$\left({u}_{i}{\overline {Y}}_{i}-\overline {Y}\right)$ $({u}_{i}{\overline {Z}}_{i}-\overline {Z})$
109395354.7	116233397.3	69704363.4	112762554.6	90010971.3	87323155.9
7243544.5	465731.6	51103837.3	1836722	4878593.3	19239878.4
160959577.1	124780258.7	57219949.1	141720068	84498047.6	95969259.7
23109139.4	31199313.33	3200403.1	26851243.6	9992516.3	8599916.4
632160672.8	589329821.2	75771962.7	610369671.8	211316533.3	218860811.8
90158398.4	73831543.1	2989684.1	81587582.8	14857085.7	16417829.8

| Show Table

DownLoad: CSV

Table 4. Statistical computations of variances and covariances.

${S}_{iy}^{2}$	${S}_{ix}^{2}$	${S}_{iz}^{2}$	${S}_{ixy}$	${S}_{ixz}$	${S}_{iyz}$
14634002.89	13229390.42	3667896.461	12035361.66	5676138.848	7031654.09
5199331742	3024354709	6568461403	3920918987	42963119811	5785526585
17474303.56	67544530.07	63348742.76	3322301379	62246450.49	32019714.86
510689624	689903319	440717912.5	586829812.3	522773788	447429378.1
1530618991	1588803380	408376223	1544450491	757258674.4	7559765056
1361248223	1782024492	5663081987	1557362451	3157897870	2755900798

| Show Table

DownLoad: CSV

${S}_{by}^{2} = 204605337.7, \;\;\;\;\;\;\;\;{S}_{bx}^{2} = 187168013, \;\;\;\;\;\;\;\;{S}_{bz}^{2} = 51998039.99,$

${S}_{byx} = 195025568.6, \;\;\;\;\;\;\;\;{S}_{byz} = 89282170.45, \;\;\;\;\;\;\;\;{S}_{bxz} = 83110749.52$

${V}_{y} = 0.27028, \;\;\;\;\;\;\;\;{V}_{x} = 0.25137, \;\;\;\;\;\;\;\;{V}_{z} = 0.30933,$

${V}_{yx} = 0.25723, \;\;\;\;\;\;\;\;{V}_{yz} = 0.24573, \;\;\;\;\;\;\;\;{V}_{xz} = 0.22493.$

$\overline {Y} = 14604.76564, \;\;\;\;\;\;\;\;\overline {X} = 14276.72113, \;\;\;\;\;\;\;\;\overline {Z} = 10241.22672.$

Population 2. [Source: ^[11], Model Assisted Survey Sampling]

Similarly we considered the data as mentioned in Population 1,

$y$ = 1983 import (in millions U.S dollars),

$x$ = 1981 military expenditure (in tens of millions U.S dollars),

$z$ = 1980 population (in millions).

The data are divided into 6 clusters having $N = 6$ , $n = 3$ , ${\sum }_{i = 1}^{N}\;{M}_{i} = 124, \;\overline {M} = 20.67$ .

In , we show cluster sizes, and means of the study variable $\left(y\right)$ and the auxiliary variables $(x, z)$ . Tables 6 and 7 give some computation results.

Table 5. Cluster sizes with population means.

No. of clusters	${M}_{i}$	${m}_{i}$	${u}_{i}$	${\overline {Y}}_{i}$	${\overline {X}}_{i}$	${\overline {Z}}_{i}$
1	38	15	1.8387	13.03684	418.3421	11.88421
2	14	6	0.6774	27.35	10065.21	26.1857
3	11	4	0.5323	23.13636	484.45	21.8818
4	33	13	1.5968	79.65455	3377.75	75.2424
5	24	10	1.1613	20.28333	4929.41	20.9583
6	4	2	0.1936	74.15	30676.25	70.975

| Show Table

DownLoad: CSV

Table 6. Statistical computation.

$({u}_{i}{\overline {Y}}_{i}-\overline {Y}{)}^{2}$	$({u}_{i}{\overline {X}}_{i}-\overline {X}{)}^{2}$	$({u}_{i}{\overline {Z}}_{i}-\overline {Z}{)}^{2}$	$({u}_{i}{\overline {Y}}_{i}-\overline {Y})({u}_{i}{\overline {X}}_{i}-\overline {X})$	$({u}_{i}{\overline {X}}_{i}-\overline {X})({u}_{i}{\overline {Z}}_{i}-\overline {Z})$	$({u}_{i}{\overline {Y}}_{i}-\overline {Y})({u}_{i}{\overline {Z}}_{i}-\overline {Z})$
109395354.7	11504653.07	171.5489	35430628.04	44434.42674	136842.964
7243544.5	7165707.84	296.6622	−461113.24893	7355674.750	−47336.072
160959577.1	15247937.12	544.5471	49517362.59	91132.5364	295949.600
23109139.4	1556060.53	7312.6397	6074534.22	106667.8685	416415.347
632160672.8	2494694.19	112.4143	39914472.69	−16750.3943	−268005.273
90158398.4	3214861.18	451.071	−16997583.03	−38085.13867	201365.558

| Show Table

DownLoad: CSV

Table 7. Statistical computations of variances and covariances.

${S}_{iy}^{2}$	${S}_{ix}^{2}$	${S}_{iz}^{2}$	${S}_{iyx}$	${S}_{ixz}$	${S}_{iyz}$
270.9083357	594166.8257	222.4889331	6380.484353	5806.286629	245.4143812
3906.928847	1281691972	3683.070549	2135135.281	2082979.098	3792.853077
1339.404545	461472.2727	1174.031636	13075.32182	12298.74909	1253.851727
45082.17318	53848774.81	83850.37836	1082424.717	1476243.493	43109.42511
368.9423188	52672480.78	364.7838949	117010.6551	116939.2322	366.7860145
18401.07	3453923758	16855.5025	7970505.317	7628400.308	17611.33833

| Show Table

DownLoad: CSV

${S}_{by}^{2} = 2002.428957, \;\;\;\;\;\;\;\;{S}_{bx}^{2} = 8236782.79, \;\;\;\;\;\;\;\;{S}_{bz}^{2} = 1782.2076,$

${S}_{byx} = 28451.30273, \;\;\;\;\;\;\;\;{S}_{byz} = 1888.920758, \;\;\;\;\;\;\;\;{S}_{xz} = 27939.79,$

${V}_{y} = 0.48633, \;\;\;\;\;\;\;\;{V}_{x} = 0.39654, \;\;\;\;\;\;\;\;{V}_{z} = 0.72000,$

${V}_{yx} = 0.14250, \;\;\;\;\;\;\;\;{V}_{yz} = 0.48726, \;\;\;\;\;\;\;\;{V}_{xz} = 0.16552,$

$\overline {Y} = 36.7702, \;\;\;\;\;\;\;\;\overline {X} = 4163.56, \;\;\;\;\;\;\;\;\overline {Z} = 34.8552.$

The results based on Tables 2–7 are given in and having biasses, mean square errors, and percentage relative efficiencies of the poposed and exisitng estimators w.r.t ${\overline {y}}_{0}^{\mathrm{*}}.$ Tables 8 and 9 show that the proposed estimators perform well as compared to the existing estimators considered here.

Table 8. Biases of different estimators in both data sets.

Estimator	Population 1	Population 2
${\overline {y}}_{0}^{\mathrm{}}$ , ${\overline {y}}_{D}^{\mathrm{}}$ , ${\overline {y}}_{DD}^{\mathrm{*}}$	0	0
${\overline {y}}_{R}^{\mathrm{*}}$	−85.58393	9.341102
${\overline {y}}_{E}^{\mathrm{*}}$	−501.692	2.847944
${\overline {y}}_{Rao}^{\mathrm{*}}$	−102.2916	−11.14854
${\overline {y}}_{DR}^{\mathrm{*}}$	62517.58	−5.729291
${\overline {y}}_{DE}^{\mathrm{*}}$	−1040.227	−4.254352
${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$	−8687.674	0.8665627
${\overline {y}}_{G1}^{\mathrm{*}}$	−911.2082	42.56601
${\overline {y}}_{G2}^{\mathrm{*}}$	4097419	−660.0231

| Show Table

DownLoad: CSV

Table 9.

$\mathrm{M}\mathrm{S}\mathrm{E}$ and

$\mathrm{P}\mathrm{R}\mathrm{E}$ of different estimators w.r.t

${\overline {y}}_{0}^{\mathrm{*}}$ .

	Population 1		Population 2
Estimators	MSE	PRE	MSE	PRE
${\overline {y}}_{0}^{\mathrm{*}}$	57649726.19	100	657.541	100
${\overline {y}}_{R}^{\mathrm{*}}$	1532261.42	3762.39	808.353	81.3433
${\overline {y}}_{E}^{\mathrm{*}}$	16186983.44	356.149	598.912	109.789
${\overline {y}}_{D}^{\mathrm{*}}$	1503085.83	3835.42	588.307	111.768
${\overline {y}}_{Rao}^{\mathrm{*}}$	1492567.93	3862.45	409.935	160.401
${\overline {y}}_{DR}^{\mathrm{*}}$	1489069.32	3871.53	341.831	192.358
${\overline {y}}_{DE}^{\mathrm{*}}$	1189664.93	4845.88	366.982	179.176
${\overline {y}}_{DD}^{\mathrm{*}}$	1025752.55	5620.24	208.189	315.839
${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$	1020843.33	5647.26	180.409	364.472
${\overline {y}}_{G1}^{\mathrm{*}}$	1019205.50	5656.34	165.866	396.429
${\overline {y}}_{G2}^{\mathrm{*}}$	747118.42	7716.28	159.646	411.875

| Show Table

DownLoad: CSV

The following expression is used to obtain the Percent Relative Efficiency (PRE), i.e.,

$PRE = \frac{MSE\left({\overline {y}}_{0}^{\mathrm{*}}\right)}{MSE\left({\overline {y}}_{i}^{\mathrm{*}}\right)}\times 100 ,$

where $i = 0, R, E, D, Rao, DR, DE, DD, DD\left(R\right), {G}_{1}, {G}_{2}.$

6. Discussion

As mentioned above, we used two real data sets to obtain the biases, MSEs or variances and PREs of all estimators under two-stage sampling scheme when using two auxiliary variables. In Tables 2–4 and Tables 5–7, we present the summary statistic of both population. From and , we observed that the proposed class of estimators ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ are more precise than the existing estimators ${\overline {y}}_{0}^{\mathrm{*}}$ , ${\overline {y}}_{R}^{\mathrm{*}}$ , ${\overline {y}}_{E}^{\mathrm{*}}$ , ${\overline {y}}_{D}^{\mathrm{*}}$ , ${\overline {y}}_{Rao}^{\mathrm{*}}$ , ${\overline {y}}_{DR}^{\mathrm{*}}$ , ${\overline {y}}_{DE}^{\mathrm{*}}$ , ${\overline {y}}_{DD}^{\mathrm{*}}$ , ${\overline {y}}_{DD\left(R\right)}^{\mathrm{*}}$ in terms of MSEs and PREs. It is clear that the proposed improved generalized class of estimators, i.e., performs better than the estimators. As we increase the sample size the mean square error values decreases, and percentage relative efficiency give best results, which are the expected results.

7. Conclusions

In this manuscript, we proposed a generalized class of estimators using two auxiliary variables under two-stage sampling for estimating the finite population mean. In addition, some well-known estimators of population mean like traditional unbiased estimator, usual ratio, exponential ratio type, traditional difference type, Rao difference type, difference-in- ratio type, difference-in-exponential ratio type, difference-in-difference, difference-difference ratio type estimator are created to be members of our suggested improved generalized class of estimators. Expression for the biases and mean squared error have been generated up to the first order of approximation. We identified 11 estimators as members of the proposed class of estimators by substituting the different values of ${w}_{i}(i = \mathrm{1, 2}, 3)$ , $\delta$ and $\gamma$ . Both generalized class of estimators ${\overline {y}}_{G1}^{\mathrm{*}}$ and ${\overline {y}}_{G2}^{\mathrm{*}}$ perform better as compared to all other considered estimators, although ${\overline {y}}_{G2}^{\mathrm{*}}$ is the best. In Population 2, the performance of ratio estimator $\left({\overline {y}}_{R}^{\mathrm{*}}\right)$ is weak. The gain in Population 1 is more as compared to Population 2.

Acknowledgments

The authors are thankful to the Editor-in-Chief and two anonymous referees for their careful reading of the paper and valuable comments which leads to a significant improvement in article.

Conflict of interest

The authors declare no conflict of interest.

Conflict of interest

The authors declare no conflicts of interest in this paper.

References

[1]	Ónodi A (1904) III. Die Sehstörungen und Erblindung nasalen Ursprunges, bedingt durch Erkrankungen der hinteren Nebenhöhlen. Ophthalmologica 12: 23-46. doi: 10.1159/000290212
[2]	Kitagawa K, Hayasaka S, Shimizu K, et al. (2003) Optic neuropathy produced by a compressed mucocele in an Onodi cell. Am J Ophthalmol 135: 253-254. doi: 10.1016/S0002-9394(02)01941-4
[3]	Kennedy D, Bolger W, Zinreich S Diseases of the sinuses: diagnosis and management (2001) .
[4]	Fukuda Y, Chikamatsu K, Ninomiya H, et al. (2006) Mucocele in an Onodi cell with simultaneous bilateral visual disturbance. Auris Nasus Larynx 33: 199-202. doi: 10.1016/j.anl.2005.11.024
[5]	Nonaka M, Fukumoto A, Nonaka R, et al. (2007) A case of a mucocele in an Onodi cell. J Nippon Med Sch 74: 325-328. doi: 10.1272/jnms.74.325
[6]	Lim SA, Sitoh YY, Lim TC, et al. (2008) Clinics in diagnostic imaging (120). Right rhinogenic optic neuritis secondary to mucocoele of the Onodi cell. Singapore Med J 49: 84-87.
[7]	Moeller CW, Welch KC (2010) Prevention and management of complications in sphenoidotomy. Otolaryngol Clin North Am 43: 839-854. doi: 10.1016/j.otc.2010.04.009
[8]	Yoon KC, Park YG, Kim HD, et al. (2006) Optic neuropathy caused by a mucocele in an Onodi cell. Jpn J Ophthalmol 50: 296-298. doi: 10.1007/s10384-005-0299-4
[9]	Yamaguchi K, Ohnuma I, Takahashi S, et al. (1997) Magnetic resonance imaging in acute optic neuropathy by sphenoidal mucocele. Int Ophthalmol 21: 9-11. doi: 10.1023/A:1005889203677
[10]	Zukin LM, Hink EM, Liao S, et al. (2017) Endoscopic management of paranasal sinus mucoceles: meta-analysis of visual outcomes. Otolaryngol Head Neck Surg 157: 760-766. doi: 10.1177/0194599817717674
[11]	Victores A, Foroozan R, Takashima M (2012) Recurrent Onodi cell mucocele: rare cause of 2 different ophthalmic complications. Otolaryngol Head Neck Surg 146: 338-339. doi: 10.1177/0194599811415806
[12]	Evans C (1981) Aetiology and treatment of fronto-ethmoidal mucocele. J Laryngol Otol 95: 361-375. doi: 10.1017/S0022215100090836
[13]	Klink T, Pahnke J, Hoppe F, et al. (2000) Acute visual loss by an Onodi cell. Br J Ophthalmol 84: 801-802. doi: 10.1136/bjo.84.7.799d
[14]	Toh ST, Lee JCY (2007) Onodi cell mucocele: rare cause of optic compressive neuropathy. Arch Otolaryngol Head Neck Surg 133: 1153-1156. doi: 10.1001/archotol.133.11.1153
[15]	Chmielik LP, Chmielik A (2017) The prevalence of the Onodi cell - Most suitable method of CT evaluation in its detection. Int J Pediatr Otorhinolaryngol 97: 202-205. doi: 10.1016/j.ijporl.2017.04.001
[16]	Thanaviratananich S, Chaisiwamongkol K, Kraitrakul S, et al. (2003) The prevalence of an Onodi cell in adult Thai cadavers. Ear Nose Throat J 82: 200-204. doi: 10.1177/014556130308200314
[17]	Anniko M, Bernal-Sprekelsen M, Bonkowsky V, et al. (2010) Otorhinolaryngology, Head and Neck Surgery Berlin Heidelberg: Springer-Verlag New York. doi: 10.1007/978-3-540-68940-9
[18]	Lim CC, Dillon WP, McDermott MW (1999) Mucocele involving the anterior clinoid process: MR and CT findings. AJNR Am J Neuroradiol 20: 287-290.
[19]	Nickerson JP, Lane AP, Subramanian PS, et al. (2011) Onodi cell mucocele causing acute vision loss: radiological and surgical correlation. Clin Neuroradiol 21: 245-248. doi: 10.1007/s00062-011-0056-7
[20]	Ogata Y, Okinaka Y, Takahashi M (1998) Isolated mucocele in an Onodi cell. ORL J Otorhinolaryngol Relat Spec 60: 349-352. doi: 10.1159/000027623
[21]	Jones NS, Strobl A, Holland I (1997) A study of the CT findings in 100 patients with rhinosinusitis and 100 controls. Clin Otolaryngol Allied Sci 22: 47-51. doi: 10.1046/j.1365-2273.1997.00862.x
[22]	Chee E, Looi A (2009) Onodi sinusitis presenting with orbital apex syndrome. Orbit 28: 422-424. doi: 10.3109/01676830903177419
[23]	Bockmühl U, Kratzsch B, Benda K, et al. (2005) Paranasal sinus mucoceles: surgical management and long term results. Laryngorhinootologie 84: 892-898. (Article in German). doi: 10.1055/s-2005-870572
[24]	Josepphson JS, Herrera A Mucocele of the paranasal sinuese: endoscopic diagnosis and treatment in Advanced ‎Endoscopic Sinus Surgery (1995) .51-59.
[25]	Lee JM, Au M (2016) Onodi cell mucocele: case report and review of the literature. Ear Nose Throat J 95: E4-8. doi: 10.1177/014556131609500905
[26]	Nathe C, Shen E, Crow RW (2018) Complete orbital apex syndrome from an onodi cell mucocele: a case report. Clin Ophthalmol 2: 63-66. doi: 10.35841/clinical-ophthalmology.2.2.63-66
[27]	Chafale VA, Lahoti SA, Pandit A, et al. (2015) Retrobulbar optic neuropathy secondary to isolated sphenoid sinus disease. J Neurosci Rural Pract 6: 238-240. doi: 10.4103/0976-3147.153233
[28]	Tzamalis A, Diafas A, Riga P, et al. (2020) Onodi cell mucocele-associated optic neuropathy: a rare case report and review of the literature. J Curr Ophthalmol 32: 107-113.
[29]	Kwon KW, Oh JS, Kim JW, et al. (2019) Onodi cell mucocele causing isolated trochlear nerve palsy: a case report. Medicine (Baltimore) 98: e15475. doi: 10.1097/MD.0000000000015475
[30]	Cheon YI, Hong SL, Roh HJ, et al. (2014) Fungal ball within Onodi cell mucocele causing visual loss. J Craniofac Surg 25: 512-514. doi: 10.1097/SCS.0000000000000678
[31]	Wu W, Sun MT, Cannon PS, et al. (2010) Recovery of visual function in a patient with an onodi cell mucocele compressive optic neuropathy who had a 5-Week interval between onset and surgical intervention: a case report. J Ophthalmol 2010: 483056.
[32]	Fukuda H, Fukumitsu R, Andoh M, et al. (2010) Small Onodi cell mucocele causing chronic optic neuropathy: case report. Neurol Med Chir (Tokyo) 50: 953-955. doi: 10.2176/nmc.50.953
[33]	Eça TF, Rodrigues PC, Dinis PB (2018) Surgical drainage plus optic nerve decompression in acute optic neuropathy by an Onodi cell mucocele. BAOJ Surg 4: 33.
[34]	Fleissig E, Spierer O, Koren I, et al. (2014) Blinding orbital apex syndrome due to Onodi cell mucocele. Case Rep Ophthalmol Med 2014: 453789.
[35]	Kashii S, Arakawa N, Taguch HKN Acute visual loss caused by an Onodi cell mucocele‎ (2016) .14.
[36]	Yen Nee See W, Sumugam K, Subrayan V (2016) Compressive optic neuropathy due to a large Onodi air cell: a case report and literature review. Allergy Rhinol (Providence) 7: 223-226.
[37]	Loo JL, Looi ALG, Seah LL (2009) Visual outcomes in patients with paranasal mucoceles. Ophthal Plast Reconstr Surg 25: 126-129. doi: 10.1097/IOP.0b013e318198e78e
[38]	Kim YS, Kim K, Lee JG, et al. (2011) Paranasal sinus mucoceles with ophthalmologic manifestations: a 17-year review of 96 cases. Am J Rhinol Allergy 25: 272-275. doi: 10.2500/ajra.2011.25.3624
[39]	Kimakura M, Oishi A, Miyamoto K, et al. (2009) Sphenoethmoidal mucocele masquerading as trochlear palsy. J AAPOS 13: 598-599. doi: 10.1016/j.jaapos.2009.09.003

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Medical Science

0.4

Metrics

Article views(2649) PDF downloads(107) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(1) / Tables(4)

AIMS Medical Science

Optic neuropathy related to Onodi cell mucocele: a systematic review and meta-analysis of randomized controlled trials

Related Papers:

Abstract

1. Introduction

2. Symbols and notation

3. Existing estimators

4. Proposed estimator

5. Numerical study

6. Discussion

7. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Medical Science

Optic neuropathy related to Onodi cell mucocele: a systematic review and meta-analysis of randomized controlled trials

Related Papers:

Abstract

1. Introduction

2. Symbols and notation

3. Existing estimators

4. Proposed estimator

5. Numerical study

6. Discussion

7. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog