A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations

Zahra Pirouzeh; Mohammad Hadi Noori Skandari; Kamele Nassiri Pirbazari; Stanford Shateyi; Zahra Pirouzeh; Mohammad Hadi Noori Skandari; Kamele Nassiri Pirbazari; Stanford Shateyi

doi:10.3934/math.20241151

AIMS Mathematics

2024, Volume 9, Issue 9: 23692-23710. doi: 10.3934/math.20241151

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Research article Special Issues

A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations

1.
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2.
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
3.
Department of Applied Mathematics, University of Venda, P. Bag X5050, Thohoyandu, Limpopo 950, South Africa

Received: 16 February 2024 Revised: 12 June 2024 Accepted: 14 June 2024 Published: 08 August 2024
MSC : 34K37, 49M25, 49M37

Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy.

Keywords:

Citation: Zahra Pirouzeh, Mohammad Hadi Noori Skandari, Kamele Nassiri Pirbazari, Stanford Shateyi. A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations[J]. AIMS Mathematics, 2024, 9(9): 23692-23710. doi: 10.3934/math.20241151

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Abstract

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	${\overline {y}}_{G2}^{\mathrm{*}}$

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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

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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

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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

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

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

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

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

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

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

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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

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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

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

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

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

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

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

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

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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

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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.

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