Impact of estrogen population pharmacokinetics on a QSP model of mammary stem cell differentiation into myoepithelial cells

Justin Le Sauteur-Robitaille; Zhe Si Yu; Morgan Craig; Justin Le Sauteur-Robitaille; Zhe Si Yu; Morgan Craig

doi:10.3934/math.2021631

AIMS Mathematics

2021, Volume 6, Issue 10: 10861-10880. doi: 10.3934/math.2021631

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Research article Special Issues

Impact of estrogen population pharmacokinetics on a QSP model of mammary stem cell differentiation into myoepithelial cells

1.
Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, Canada
2.
Sainte-Justine University Hospital Research Centre

Received: 17 January 2021 Accepted: 16 July 2021 Published: 28 July 2021
MSC : 92B05, 92C32

Stem cell differentiation cascades are critical components of healthy tissue maintenance. Dysregulation in these systems can lead to serious diseases, including cancer. Myoepithelial mammary cells are produced from differentiated mammary stem cells in processes regulated, in part, by estrogen signalling and concentrations. To quantify and predict the production of mammary myoepithelial cell production by estrogen, we developed a mechanistic, quantitative systems pharmacology (QSP) model that includes the explicit characterization of free and unbound estrogen concentrations in circulation. Linking this model to a previously developed population pharmacokinetics model for ethinyl estradiol, a synthetic form of estrogen included in oral contraceptives, we predicted the effects of estrogen on myoepithelial cell development. Interestingly, pharmacokinetic intraindividual variability alone did not significantly impact on our modelos predictions, suggesting that combinations of physiological and pharmacokinetic variability drive heterogeneity in mechanistic QSP models. Our model is one component of an improved understanding of mammary myoepithelial cell production and development, and our results support the call for mechanistically constructed systems models for disease and pharmaceutical modelling.

Keywords:

Citation: Justin Le Sauteur-Robitaille, Zhe Si Yu, Morgan Craig. Impact of estrogen population pharmacokinetics on a QSP model of mammary stem cell differentiation into myoepithelial cells[J]. AIMS Mathematics, 2021, 6(10): 10861-10880. doi: 10.3934/math.2021631

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Abstract

1. Introduction

In large scale surveys, it is a usual practice to prefer multi-stage sampling to estimate the population characteristics over single-stage sampling. The main purpose to use multi-stage sampling is the clear reduction in the cost of survey operations even if estimates derived from multi-stage sampling are likely to be less efficient than those of the single-stage sampling. Sukhatme et al. ^[1] advised some ratio and regression type estimators in two-stage sampling using single auxiliary variable when first stage units are of unequal or equal sizes. The suitable use of auxiliary information in estimation stage results as a considerable reduction in the mean square error of the estimator. By making use of auxiliary information, Srivastava and Garg ^[2] proposed separate-type estimator for the estimation of population mean in two-stage sampling design. Taking inspiration from Srivastava and Garg ^[2], Koyuncu and Kadilar ^[3] and Jabeen et al. ^[4] proposed separate type estimator under two-stage sampling. In the literature, the use of two-phase sampling under two-stage sampling design is not well documented. Saini and Bahl ^[5] proposed estimator under two-stage sampling design using double sampling for stratification and multi-auxiliary information. A generalized ratio cum product estimator for population mean in simple random sampling was developed by Singh et al. ^[6]. Shabbir ^[7] produces the estimators of population mean under stratified two phase sampling.

In the literature (^[8,9,10,11]), the use of two phase sampling under two stage sampling design is not well documented. Also the shrinkage estimators have been discussed several times in literature by considering the unbiasedness of the estimators but no one has discussed the situation when the property of unbiasedness is not fulfill that is very common in real life applications. In order to fill this gap, we are motivated to produce two-exponential estimators under two stage two phase sampling that is discussed in section 2. Also we will discuss general shrinkage estimator in section 3. We will compare both estimators mathematically and by using real population data in section 4 and 5. Finally the conclusion will be discussed in section 6.

Let a population consists of N first stage units, each containing ${M_i}$ second-stage units where i = 1, 2, …, N. Let a first-stage sample of size n ( $\subset$ N) is selected and subsequently a second-stage sample of ${m_{i(1)}}$ ( $\subset$ ${M_i}$ ) units is selected and information on some auxiliary variables say ${x_{ij(1)}}$ is taken. Here it is assumed that each first stage unit/cluster is of different size so each cluster is assigned a weight ${\eta _i} = \frac{{{M_i}}}{{\bar M}}$ to it. Let a sub-sample (second-phase sample) of ${m_{i(2)}}$ units is selected from ${m_{i(1)}}$ (first-phase sample) such that ${m_{i(2)}} \subset {M_i}$ . Let ${m_{i(2)}}$ units are observed so as to collect information regarding study variable ${y_{ij(2)}}$ and auxiliary variables ${x_{ij(2)}}$ . Let ${\bar X_{ts}} = \frac{1}{N}\sum\limits_{i = 1}^N {\frac{1}{{\bar M}}{M_i}{{\bar X}_{i.}}}$ be the mean in the population and ${\bar X_i} = \frac{1}{{{M_i}}}\sum\limits_{j = 1}^{{M_i}} {{x_{ij}}}$ be the mean of i^th first stage unit in the population. Let ${\bar x_{ts(1)}} = \frac{1}{n}\sum\limits_{i = 1}^n {{\eta _i}{{\bar x}_{i(1)}}}$ and ${\bar x_{i(2)}} = \frac{1}{{{m_{i(2)}}}}\sum\limits_{j = 1}^{{m_{i(2)}}} {{x_{ij(2)}}}$ respectively are the means of first-phase and second-phase sample in two-stage sampling where ${\bar x_{i(1)}}$ and ${\bar x_{i(2)}}$ be the sample means of first phase and second phase in ith stage. Let $S_{_{xb}}^2 = \frac{1}{{N - 1}}\sum\limits_{i = 1}^N {{{({\eta _i}{{\bar X}_i} - {{\bar X}_{ts}})}^2}}$ be the population variance between fsu’s and $S_{_{xwi}}^2 = \frac{1}{{{M_i} - 1}}\sum\limits_{i = 1}^n {{{({x_{ij}} - {{\bar X}_{i.}})}^2}}$ be the population variance within first stage units. Similarly these notations can be defined for other variables. Further we consider that the selection of units at each stage (or phase) has been made by simple random sampling without replacement.

2. Materials and method

2.1. Proposed generalized estimator

We propose a generalized estimator by considering the exponential relationship as:

${t^G} = {\bar y_{ts}}{\mkern 1mu} {\rm{exp}}\left( {\alpha \left( {1 - \frac{{a{\mkern 1mu} {{\bar x}_{ts(1)}}}}{{\left( {\bar X + (a - 1){\mkern 1mu} {{\bar x}_{ts(1)}}\;\;} \right)}}\;} \right)} \right){\rm{exp}}\left( {\beta \left( {1 - \frac{{b{\mkern 1mu} {{\bar z}_{ts(2)}}}}{{\left( {\bar Z + (b - 1){\mkern 1mu} {{\bar z}_{ts(2)}}\;\;} \right)}}\;} \right)\;} \right),{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu}$

(1)

Where ( $a$ , $b$ ) are constants to be determined such that the mean square error is minimum, ( $\alpha$ , $\beta$ ) are known constants takes the value (0, 1, -1) to produce different ratio-type and product-type estimators as presented in Table B1 (Appendix B).

2.2. Bias and mean square error of generalized estimator

To derive the bias and mean square error, we proceed as follows:

$\frac{{\bar y_{ts(2)}^{} - {{\bar Y}_{ts}}}}{{{{\bar Y}_{ts}}}} = {e_{0(2)}}\, , \, \, \frac{{\bar x_{ts(1)}^{} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}}}} = {e_{1(1)}}, \, \, \, \, \frac{{\bar z_{ts(2)}^{} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}} = {e_{2(2)}}, \, \, \frac{{\bar x_{ts(2)}^{} - {{\bar X}^{}}_{ts}}}{{{{\bar X}^{}}_{ts}}} = {e_{1(2)}}, \, \, \, \frac{{\, \bar z_{ts(1)}^{} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}} = {e_{2(1)}}$

Further we assume that $E\left({{e_{0(2)}}} \right) = E\left({{e_{1(2)}}} \right) = E\left({{e_{2(2)}}} \right) = E\left({{e_{2(1)}}} \right) = E\left({{e_{1(1)}}} \right) = 0$ , and some expectations under two-stage sampling design are obtained in order to obtain the bias and mean square error as,

$\left.\begin{array}{l} E({e_{0(2)}}^2) = {C_{020(2)}}, E({e_{1(2)}}^2) = {C_{200(2)}}, \\ E({e_{2(2)}}^2) = {C_{002(2)}}, E({e_{1(1)}}^2) = {C_{200(1)}}, \\ E({e_{2(1)}}^2) = {C_{002(1)}}, E({e_{1(2)}}{e_{1(1)}}) = {C_{200(1)}}\, \\ E({e_{0(2)}}{e_{1(2)}}) = {C_{110(2)}}, \, E({e_{1(2)}}{e_{2(2)}}) = {C_{101(2)}}\, , \\ E({e_{0(2)}}{e_{2(2)}}) = {C_{011(2)}}, E({e_{0(2)}}{e_{1(1)}}) = {C_{110(1)}}, \, \\ E({e_{1(2)}}{e_{2(1)}}) = {C_{101(1)}}\, , E({e_{0(2)}}{e_{2(1)}}) = {C_{011(1)}}\, \, \, \, \, \\ {\rm{Where}} \\ {C_{020(2)}} = \frac{1}{{{{\bar Y}^2}_{ts}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, S_{yb}^2 + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\, \, \left( {\frac{1}{{{m_{i(2)}}}} - \frac{1}{{{M_i}}}} \right)} \, S_{ywi}^2, } \right\}\, , \\ {C_{200(1)}} = \frac{1}{{{{\bar X}^2}_{ts}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, \, S_{xb}^2 + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\, \left( {\frac{1}{{{m_{i(1)}}}} - \frac{1}{{{M_i}}}} \right)} \, S_{xwi}^2} \right\} \\ {C_{200(2)}} = \frac{1}{{{{\bar X}^2}_{ts}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, S_{xb}^2 + \frac{1}{{nN}}\sum\limits_{i = 1}^N {\eta _i^2\, \left( {\frac{1}{{{m_{i(2)}}}} - \frac{1}{{{M_i}}}} \right)} \, S_{xwi}^2} \right\}, \\ {C_{002(1)}} = \frac{1}{{{{\bar Z}^2}_{ts}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, \, S_{zb}^2 + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\left( {\frac{1}{{{m_{i(1)}}}} - \frac{1}{{{M_i}}}} \right)} \, \, S_{zwi}^2} \right\}, \\ {C_{002(2)}} = \frac{1}{{{{\bar Z}^2}_{ts}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)S_{zb}^2 + \frac{1}{{nN}}\sum\limits_{i = 1}^N {\eta _i^2\left( {\frac{1}{{{m_{i(2)}}}} - \frac{1}{{{M_i}}}} \right)} S_{zwi}^2} \right\}, \\ {C_{110(1)}} = \frac{1}{{{{\bar Y}_{ts}}\, {{\bar X}_{ts}}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right){\rho _{xyb}}\, {S_{yb}}\, {S_{xb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\left( {\frac{1}{{{m_{i(1)}}}} - \frac{1}{{{M_i}}}} \right)} \, {\rho _{xywi}}\, {S_{ywi}}\, {S_{xwi}}} \right\}, \\ {C_{110(2)}} = \frac{1}{{{{\bar Y}_s}\, {{\bar X}_s}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right){\rho _{xyb}}\, {S_{yb}}\, {S_{xb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\, \eta _i^2\, \left( {\frac{1}{{{m_{i(2)}}}} - \frac{1}{{{M_i}}}} \right)} \, {\rho _{xywi}}\, {S_{ywi}}\, {S_{xwi}}} \right\}, \, \\ {C_{101(1)}} = \frac{1}{{{{\bar Z}_{ts}}\, {{\bar X}_{ts}}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right){\rho _{xzb}}\, {S_{zb}}\, {S_{xb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\, \eta _i^2\, \left( {\frac{1}{{{m_{i(1)}}}} - \frac{1}{{{M_i}}}} \right)} \, {\rho _{xzwi}}\, {S_{zwi}}\, {S_{xwi}}} \right\}, \\ {C_{101(2)}} = \frac{1}{{{{\bar Z}_{ts}}\, {{\bar X}_{ts}}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right){\rho _{xzb}}\, {S_{zb}}\, {S_{xb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\left( {\frac{1}{{{m_{i(2)}}}} - \frac{1}{{{M_i}}}} \right)} \, {\rho _{xzwi}}{S_{zwi}}{S_{xwi}}} \right\}, \,\\ {C_{200(1)}} = \frac{1}{{{{\bar Z}_{ts\, }}{{\bar X}_{ts}}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, {S_{xb}}\, {S_{xb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\, \left( {\frac{1}{{{m_{i(1)}}}} - \frac{1}{{{M_i}}}} \right)} \, {S_{xwi}}\, {S_{xwi}}} \right\} \\ {C_{011(2)}} = \frac{1}{{{{\bar Z}_{ts}}{{\bar Y}_{ts}}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, {\rho _{yzb}}\, {S_{zb}}\, {S_{yb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\left( {\frac{1}{{{m_{i(2)}}}} - \frac{1}{{{M_i}}}} \right)} \, {\rho _{yzwi}}{S_{zwi}}{S_{ywi}}} \right\}, \,\\ {C_{011(1)}} = \frac{1}{{{{\bar Z}_{ts}}\, {{\bar Y}_{ts}}}}\left\{ {\left( {\frac{1}{n} - \frac{1}{N}} \right)\, {\rho _{yzb}}\, {S_{zb}}\, {S_{yb}} + \frac{1}{{n\, N}}\sum\limits_{i = 1}^N {\eta _i^2\left( {\frac{1}{{{m_{i(1)}}}} - \frac{1}{{{M_i}}}} \right)} \, {\rho _{yzwi}}{S_{zwi}}{S_{ywi}}} \right\} \\ \\ \end{array} \right\}\, \,$

(2)

Using (2) we can express (1) to derive the bias and mean square error as,

${t^G} = {\bar Y_{ts}}\left( {1 + {e_{0(2)}}} \right){\rm{exp}}\left[ { - \frac{\alpha }{a}{e_{1(1)}}{{\left( {1 + \frac{{(a - 1)}}{a}{e_{1(1)}}} \right)}^{ - 1}}} \right]{\rm{exp}}\left[ { - \frac{\beta }{b}{e_{2(2)}}{{\left( {1 + \frac{{(b - 1)}}{b}{e_{2(2)}}} \right)}^{ - 1}}} \right],$

(3)

If $\left| {{e_{1(1)}}} \right| < 1$ , we expand the series, ${\left({1 + \frac{{(a - 1)}}{a}{e_{1(1)}}} \right)^{ - 1}}$ and ${\left({1 + \frac{{(b - 1)}}{b}{e_{2(2)}}} \right)^{ - 1}}$ up to the order n^-1, we get,

$\begin{gathered} {t^G} = {{\bar Y}_{ts}}\left( {1 + {e_{0(2)}}} \right){\rm{exp}}\left[ { - \frac{\alpha }{a}{e_{1(1)}}\left( {1 - \frac{{(a - 1)}}{a}{e_{1(1)}} + \frac{{{{(a - 1)}^2}}}{{{a^2}}}{e_{1(1)}}^2 + ...} \right)} \right]\, \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\rm{exp}}\left[ { - \frac{\beta }{b}{e_{2(2)}}\left( {1 - \frac{{(b - 1)}}{b}{e_{2(2)}} + \frac{{{{(b - 1)}^2}}}{{{b^2}}}{e_{2(2)}}^2 + ...} \right)} \right], \, \\ \end{gathered}$

(4)

If the contribution of terms involving powers in ${e_{0(2)}}$ , ${e_{1(1)}}$ and ${e_{2(2)}}$ higher than two is negligible, we have;

${t^G} = {\bar Y_{ts}}\left( {1 + {e_{0(2)}}} \right)\left[ {1 - \frac{\alpha }{a}{e_{1(1)}} + \frac{{\alpha {{(a - 1)}^2}}}{{{a^2}}}{e^2}_{1(1)} + \frac{1}{2}\frac{{{\alpha ^2}}}{{{a^2}}}{e^2}_{1(1)}} \right]\, \left[ {1 - \frac{\beta }{b}{e_{2(2)}} + \frac{{\beta {{(b - 1)}^2}}}{{{b^2}}}{e^2}_{2(2)} + \frac{1}{2}\frac{{{\beta ^2}}}{{{b^2}}}{e^2}_{2(2)}} \right], \,$

(5)

${t^G} - {\bar Y_{ts}} = \, \, {\bar Y_{ts}}\left[ \begin{array}{l} {e_{0(2)}} - \frac{\alpha }{a}{e_{1(1)}} + \frac{{\alpha (a - 1)}}{{{a^2}}}{e^2}_{1(1)} + \frac{{\alpha \, \beta }}{{a\, b}}\, {e_{1(1)}}\, {e_{2(2)}} + \frac{{{\alpha ^2}}}{{2\, {a^2}}}\, {e^2}_{1(1)} - \frac{\beta }{b}{e_{2(2)}} + \frac{{{\beta ^2}}}{{2\, {b^2}}}\, {e_{2(2)}}^2 + \frac{{\beta \, (b - 1)}}{{{b^2}}}\, {e_{2(2)}}^2 \\ - \frac{\beta }{b}\, {e_{2(2)}}\, {e_{0(2)}} - \frac{\alpha }{a}\, {e_{1(1)}}\, {e_{0(2)}} \\ \end{array} \right],$

(6)

In order to get the bias, we take expectation on (6) and get,

$Bias\, (\, {t^G}) = {\bar Y_{ts}}\, \, \left[ {\frac{{{\alpha ^2}}}{{2\, {a^2}}}{C_{200(1)}} + \frac{{\alpha (a - 1)}}{{{a^2}}}{C_{200(1)}} + \frac{{{\beta ^2}}}{{2\, {b^2}}}{C_{002(2)}} + \frac{{\beta \, (b - 1)}}{{{b^2}}}{C_{002(2)}} - \frac{\beta }{b}{C_{011(2)}} - \frac{\alpha }{a}{C_{110(1)}} + \frac{\alpha }{a}\frac{\beta }{b}{C_{101(1)}}} \right]\, ,$

(7)

To get the mean square error of the estimator, we take square and retain terms up to first order of e’s then we take expectation of (6) and we obtain,

${\rm{MSE}}\left( {\, {t^G}} \right) = {\bar Y^2}_{ts}\, \left( {{C_{020(2)}} + {{\left( {\frac{\alpha }{a}} \right)}^2}{C_{200(1)}} + {{\left( {\frac{\beta }{b}} \right)}^2}{C_{002(2)}} - 2\, \frac{\alpha }{a}\, {C_{110(1)}} - 2\frac{\beta }{b}{C_{011(2)}} + 2\frac{\alpha }{a}\frac{\beta }{b}{C_{101(1)}}} \right)\, , \,$

(8)

For the following optimal value of the constants $\, a$ and $\, b$ , we achieve the minimum Variance among the class of proposed generalized estimator,

$\begin{gathered} a = \frac{{\alpha \left( {{C_{200(1)}}{C_{002(2)}} - {C_{101(2)}}^2} \right)}}{{{C_{110(1)}}{C_{002(2)}} - {C_{011(1)}}{C_{101(2)}}}} , \, \, \, \, \, \, \, {\rm{and }}\, b = \frac{{\beta \left( {{C_{200(1)}}\, {C_{002(2)}} - {C_{101(2)}}^2} \right)}}{{{C_{200(1)}}\, {C_{011(1)}} - {C_{110(1)}}{C_{101(2)}}}}, \end{gathered}$

(9)

We obtain minimum mean square error as,

$\min MSE\left( {{t^G}} \right) = {\bar Y^2}_{ts}\, \left( {{C_{020(2)}} - \frac{{{C_{110(1)}}^2{C_{002(2)}} + {C_{011(1)}}^2{C_{200(1)}} - 2{C_{110(1)}}{C_{101(2)}}{C_{011(1)}}}}{{{C_{200(1)}}{C_{002(2)}} - {C_{101(2)}}^2}}} \right),$

(10)

We observe from (10) that proposed generalized estimator gives us more precise results under the optimal conditions, as compare to its class of the estimators.

From (7)–(10), we get expressions of the bias, mean square error, optimal values and minimum mean square error for the exponential-type estimators presented in Table B1 as class of estimators. Some of the results are discussed as,

ⅰ). For $\alpha = 1, \beta = 1$ , we get a class of exponential-type ratio-cum-ratio estimators given in Table B1 and expressions of the mean square error and bias for these estimators are given as,

$\left. \begin{array}{l} {\rm{MSE}}{\mkern 1mu} (t_{}^{j - err}) = \left\{ {\bar Y_{ts}^2\left( {{C_{020(2)}} + \frac{1}{{{a^2}}}{\mkern 1mu} {C_{200(1)}} + \frac{1}{{{b^2}}}{\mkern 1mu} {C_{002(2)}} - 2\frac{1}{a}{C_{110(1)}} - 2\frac{1}{b}{C_{011(2)}} + 2\frac{1}{{ab}}{C_{101(1)}}} \right)\;\;j( \in G) = 1,3,5} \right\},\\ {\rm{and}}\\ {\rm{Bias}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} (t_{}^{j - err}) = \left\{ {\bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2{\mkern 1mu} {a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2{\mkern 1mu} {b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} - \frac{1}{a}{C_{110(1)}}} \right)\;\;j( \in G) = 1,3,5} \right\}, \end{array} \right\}$

(11)

Substitution of the different values of a and b yield the mean square error and bias for the estimators belongs to the class of exponential-type ratio-cum-ratio estimators. The optimal values which lead to minimum mean square error for the class of exponential ratio-cum-ratio estimators is obtained as,

$a = \frac{{{C_{200(1)}}\;\;{C_{002(2)}}\; - \;\;{C_{101(2)}}^2}}{{{C_{110(1)}}\;\;{C_{002(2)}}\;\; - \;{C_{011(1)}}\;\;{C_{101(2)}}}}{\mkern 1mu} \;\;{\mkern 1mu} {\rm{and}}{\mkern 1mu} {\mkern 1mu} b = \frac{{{C_{200(1)}}\;\;{C_{002(2)}}\; - \;\;{C_{101(2)}}^2}}{{{C_{200(1)}}\;\;{C_{011(1)}}\;\; - \;{C_{110(1)}}\;\;{C_{101(2)}}}}\;\;\;.$

ⅱ). For $\alpha = - 1, \beta = - 1$ , we get exponential-type product-cum-product estimators given in Table B1.The mean square error of $t_{}^G$ are expressed using as,

$\left. \begin{array}{l} {\rm{MSE}}(t_{}^{j - epp}) = \left\{ {\bar Y_{ts}^2\left( {{C_{020(2)}} + \frac{1}{{{a^2}}}{C_{200(1)}} + \frac{1}{{{b^2}}}{C_{002(2)}} + 2\frac{1}{a}{C_{110(1)}} + 2\frac{1}{b}{C_{011(2)}} + 2\frac{1}{{ab}}{C_{101(1)}}} \right)\;,\;\;\;\;\;\;\;\;\;\;\;j( \in G) = 2,4,6} \right\}\\ {\rm{and}}\\ {\rm{Biast}}_{}^{j - epp}) = \left\{ {\bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2{a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2{b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} + \frac{1}{a}{C_{110(1)}}} \right),\;\;\;\;\;\;\;\;\;\;j( \in G) = 2,4,6} \right\} \end{array} \right)$

(12)

Substitution of the different values of “a” and “b” yield the Mean Square Error and bias for the estimators belongs to the class of exponential-type ratio-cum-ratio estimators. The optimal values which lead to minimum mean square error for the class of exponential product-cum-product estimators is obtained as,

$a = \frac{{{C_{101(2)}}^2\;\; - {C_{200(1)}}\;\;{C_{002(2)}}}}{{{C_{110(1)}}\;\;{C_{002(2)}} - {C_{011(1)}}\;\;{C_{101(2)}}}}{\mkern 1mu} \;\;{\mkern 1mu} {\rm{and}}{\mkern 1mu} {\mkern 1mu} b = \frac{{{C_{101(2)}}^2\;\; - {C_{200(1)}}\;\;{C_{002(2)}}}}{{{C_{200(1)}}\;\;{C_{011(1)}}\;\; - {C_{110(1)}}\;\;{C_{101(2)}}}}\;\;.$

ⅲ). For $\alpha = - 1, \beta = 1$ , we get exponential-type product-cum-ratio estimators given in Table B1.The mean square error of $t_{}^G$ is expressed as,

$\left. \begin{array}{l} {\rm{MSE}}\, (t_{}^{j - erp}) = \left\{ {\bar Y_{ts}^2\left( {{C_{020(2)}} + \frac{1}{a}{C_{200(1)}} + \frac{1}{b}{C_{002(2)}} + 2\frac{1}{a}{C_{110(1)}} - 2\frac{1}{b}{C_{011(2)}} - 2\frac{1}{{ab}}{C_{101(1)}}} \right), \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, j( \in G) = 7, 9, 11} \right\} \\ {\rm{and}} \\ {\rm{Bias}} \, (t_{}^{j - erp}) = \left\{ {\bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2\, {a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2\, {b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} + \frac{1}{a}{C_{110(1)}}} \right) \, , \, \, \, j( \in G) = 7, 9, 11} \right\} \\ \end{array} \right)\, \,$

(13)

Substitution of the different values of “a” and “b” yield the mean square error and bias for the estimators belongs to the class of exponential-type ratio-cum-ratio estimators. The optimal values which lead to minimum mean square error for the class of exponential product-cum-ratio estimators are obtained as,

$a = \frac{{{C_{101(2)}}^2\; - {C_{200(1)}}\;\;{C_{002(2)}}}}{{{C_{110(1)}}\;\;{C_{002(2)}}\;\; - \;{C_{011(1)}}\;\;{C_{101(2)}}}}{\mkern 1mu} {\mkern 1mu} \;\;{\rm{and}}{\mkern 1mu} {\mkern 1mu} b = \frac{{{C_{200(1)}}\;\;{C_{011(1)}}\;\; - \;{C_{110(1)}}{C_{101(2)}}}}{{{C_{200(1)}}\;\;{C_{002(2)}}\;\; - {C_{101(2)}}^2}}\;\;.$

ⅳ). For $\alpha = 1, \beta = - 1$ , we get exponential-type ratio-cum-product estimators given in Table A1. The mean square error of ${t^G}$ is expressed as,

$\left. \begin{array}{l} {\rm{MSE}}\, \, (t_{}^{j - epr}) = \left\{ {\bar Y_{ts}^2\left( {{C_{020(2)}} + \frac{1}{a}{C_{200(1)}} + \frac{1}{b}{C_{002(2)}} - 2\frac{1}{a}{C_{110(1)}} + 2\frac{1}{b}{C_{011(2)}} - 2\frac{1}{a}\frac{1}{b}{C_{101(1)}}} \right),\, \, \, \, \, \, \, j( \in G) = 8, 10, 12} \right\} \\ {\rm{and}} \\ {\rm{Bias}}\, \, (t_{}^{j - epr}) = \left\{ {\bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2\, {a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2\, {b^2}}}{C_{002(2)}} + \frac{1}{b}{C_{101(2)}} - \frac{1}{a}{C_{110(1)}}} \right) \, , \, \, \, \, \, \, \, \, \, \, j( \in G) = 8, 10, 12} \right\} \\ \end{array} \right)\, \,$

(14)

$a = \frac{{{C_{200(1)}}\;\;{C_{002(2)}}\;\; - {C_{101(2)}}^2}}{{{C_{110(1)}}\;\;{C_{002(2)}}\;\; - {C_{011(1)}}{\;\;C_{101(2)}}}}\;\;{\mkern 1mu} {\mkern 1mu} {\rm{and}}{\mkern 1mu} {\mkern 1mu} b = \frac{{{C_{101}}^2 - {C_{200(1)\;\;}}{C_{002(2)}}}}{{{C_{200(1)}}\;\;{C_{011(1)}} - {C_{110(1)}}\;\;{C_{101(2)}}}}\;\;.$

It is to mention that minimum mean square error of $t_{}^{j - epr}$ , $t_{}^{j - epr}$ , $t_{}^{j - epr}$ , and $t_{}^{j - epr}$ in (11)–(14) will be same but the conditions under which these attain minimum value are not unique.

3. Proposed two-exponential general Shrinkage estimator

In literature, the shrinkage estimators have been discussed several times considering the property that the estimators are unbiased. In many practical situations the property of unbiasedness is not met so consequently the existing literature will not be helpful. Such deviations from the said property have motivated us to consider the shrinkage estimators following the property of biasedness. Important results are presented in theorem and subsequently proof is provided in Appendix A.

3.1. Theorem for estimating any population characteristic

For any general estimator $t$ , let ${t_s}$ be a general shrinkage estimator for estimating any population characteristic T (e.g., population mean) which is unknown to us,

$t_s^{} = \lambda t$

(15)

A general form for the bias and mean square error of a general shrinkage estimator up to the first order of approximation may be given as,

$Bias\left( {t_s^{}} \right) = \left( {\lambda - 1} \right)T + Bias\left( t \right)$

(16)

and

$MSE\left( {{t_s}} \right) = {T^2}{\left( {\lambda - 1} \right)^2} + {\lambda ^2}MSE\left( t \right) + 2T\lambda (\lambda - 1)Bias(t)$

(17)

And form of the optimal shrinkage estimator in (15) along with minimum bias and minimum mean square error will be as,

${t_s} = \frac{{\left( {{T^2} + MSE(t) + 2TBias(t)} \right)}}{{T(T + Bias(t)}}\;\;\;t$

(18)

$Bias\left( {t_{s}^{opt}} \right) = \left( {\frac{{\left( {{T^2} + MSE(t) + 2TBias(t)} \right)}}{{T(T + Bias(t)}} - 1} \right)T + Bias\left( t \right)$

(19)

and

$\min \, MSE\left( {t_s^{}} \right) = {T^2} - \frac{{{T^2}(T + Bias(t)}}{{{T^2} + MSE(t) + 2TBias(t)}}$

(20)

Proof: For details see Appendix A.

By using (15), generalized form of a shrinkage class of estimator is,

$t_s^G = \lambda \, {t_G}$

where

${t_G} = {\bar y_{ts}}{\mkern 1mu} {\rm{exp}}\left( {\alpha \left( {1 - \frac{{a{\mkern 1mu} {{\bar x}_{ts(1)}}}}{{\left( {{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}\;\;} \right)}}} \right)} \right){\rm{exp}}\left( {\beta \left( {1 - \frac{{b{\mkern 1mu} {{\bar z}_{ts(2)}}}}{{\left( {{{\bar Z}_{ts}} + (b - 1){{\bar z}_{ts(2)}}\;\;} \right)}}} \right)} \right)$

(21)

For different choices of the values of the constants $\alpha, \beta, a, b$ , we get different family of shrinkage estimators as given in Table B2 (See Appendix).

Following the Theorem, we can directly write the mean square error and bias expressions of $t_s^G$ using (7) and (8) respectively as,

$Bias\, (t_s^G) = \lambda \, {\bar Y_{ts}}\left[ {\frac{{\alpha \, (a - 1)}}{{{a^2}}}{C_{200(1)}} + \frac{{{\alpha ^2}}}{{2{a^2}}}\, {C_{200(1)}} + \frac{{\beta \, (b - 1)}}{{{b^2}}}{C_{002(2)}} + \frac{{{\beta ^2}}}{{2{b^2}}}{C_{002(2)}} - \frac{\beta }{b}{C_{101(2)}} - \frac{\alpha }{a}{C_{110(1)}}} \right] + {\bar Y_{ts}}(\lambda - 1)\, , {\rm{or}}$

$Bias\, (t_s^G) = \lambda \, Bias(t_{}^{G - e}) + {\bar Y_{ts}}\, (\lambda - 1)\, ,$

(22)

and

$\begin{gathered} MSE\, \left( {t_s^G} \right) = {\lambda ^2}\, {{\bar Y}^2}_{ts}\left( {{C_{020(2)}} + {{\left( {\frac{\alpha }{a}} \right)}^2}{C_{200(1)}} + {{\left( {\frac{\beta }{b}} \right)}^2}{C_{002(2)}} - 2\, \frac{\alpha }{a}{C_{110(1)}} - 2\frac{\beta }{b}{C_{011(2)}} + 2\frac{\alpha }{a}\frac{\beta }{b}{C_{101(1)}}} \right) + {\left( {\lambda - 1} \right)^2}{{\bar Y}_{ts}}^2 \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, + 2\lambda \left( {\lambda - 1} \right){{\bar Y}_{ts}}\left[ {\frac{{\alpha \, (a - 1)}}{{{a^2}}}{C_{200(1)}} + \frac{{{\alpha ^2}}}{{2\, {a^2}}}{C_{200(1)}} + \frac{{\beta \, (b - 1)}}{{{b^2}}}{C_{002(2)}} + \frac{{{\beta ^2}}}{{2\, {b^2}}}{C_{002(2)}} - \frac{\beta }{b}{C_{101(2)}} - \frac{\alpha }{a}{C_{110(1)}}} \right] \\ \end{gathered}$

$MSE\left( {t_s^G} \right) = {\lambda ^2}MSE\left( {t_{}^{G - e}} \right) + {\left( {\lambda - 1} \right)^2}{\bar Y_{ts}}^2 + 2{\bar Y_{ts}}\, \lambda \left( {\lambda - 1} \right)MSE\left( {t_{}^{G - e}} \right)$

(23)

The optimal value of $\lambda$ , which minimize the expression (23) is obtained as,

$\lambda = \frac{{\left( {{T^2} + MSE{\mkern 1mu} ({t^{G - e}}\;\;) + 2{\mkern 1mu} T{\mkern 1mu} Bias({t^{G - e\;\;\;}})} \right)}}{{T{\mkern 1mu} (T + Bias({t^{G - e}}\;\;)}}$

(24)

Using (23), the optimal estimator for the shrinkage family of estimator ( $t_s^G$ ) is written as

$\left. {t_s^G = {{\bar y}_{ts}}\frac{{\left( {{{\bar Y}_{ts}}^2 + MSE({t^{G - e}}\;\;) + 2{{\bar Y}_{ts}}Bias({t^{G - e}}\;)} \right)}}{{{{\bar Y}_{ts}}({{\bar Y}_{ts}} + Bias({t^{G - e}}\;)}}{\mkern 1mu} {\rm{exp}}\left( {\alpha {\mkern 1mu} \left( {1 - \frac{{a{\mkern 1mu} {{\bar x}_{ts(1)}}}}{{\left( {{{\bar X}_{ts}} + (a - 1){\mkern 1mu} {{\bar x}_{ts(1)}}} \right)}}\;} \right)} \right){\rm{exp}}\left( {\beta \left( {1 - \frac{{b{\mkern 1mu} {{\bar z}_{ts(2)}}}}{{\left( {{{\bar Z}_{ts}} + (b - 1){\mkern 1mu} {{\bar z}_{ts(2)}}\;} \right)}}} \right)} \right)} \right\}$

(25)

The expression for minimum value of min $MSE\left({t_s^G} \right)$ is obtained as,

$\begin{gathered} \left. {\min MSE\left( {t_s^G} \right) = {{\bar Y}^2}_{ts} - \frac{{{{\bar Y}^2}_{ts}{{\left( {{{\bar Y}^{}}_{ts} + Bias\left( {{t^{G - e}}} \right)} \right)}^2}}}{{\left[ {{{\bar Y}^2}_{ts} + MSE\left( {{t^{G - e}}} \right) + 2{{\bar Y}^{}}_{ts}Bias\left( {{t^{G - e}}} \right)} \right]}}} \right\}\, \, \, \, \, \, \\ \min MSE\left( {t_s^G} \right) = {{\bar Y}^2}_{ts}\left[ {1 - \frac{{{{\left( {1 + \frac{{Bias\left( {{t_a}} \right)}}{{{{\bar Y}^{}}_{ts}}}} \right)}^2}}}{{\left[ {1 + \frac{{MSE\left( {{t_a}} \right)}}{{{{\bar Y}^2}_{ts}}} + 2\frac{{Bias\left( {{t_a}} \right)}}{{{{\bar Y}^{}}_{ts}}}} \right]}}} \right]\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \\ \end{gathered}$

(26)

From (22)–(26), we get expressions of the bias, mean square error, optimal values and minimum mean square error for the exponential-type estimators presented in Table B2 (See Appendix B) as class of estimators. Some of the results are discussed as,

ⅰ). For $\alpha = 1, \beta = 1$ , we get a class of exponential-type ratio-cum-ratio estimators given in Table B2 (See Appendix) and expressions of the mean square error and bias for these estimators are given as,

$\begin{array}{*{20}{l}} {{\rm{MSE}}{\mkern 1mu} (t_s^{j - err\;\;}) = \left\{ {{{\bar Y}^2}_{ts}\left[ {1 - \frac{{{{\left( {1 + \frac{{Bias\left( {t_{}^{i - err\;\;}} \right)}}{{{{\bar Y}_{ts}}}}\;\;} \right)}^2}}}{{\left[ {1 + \frac{{MSE{\mkern 1mu} \left( {t_{}^{i - err\;\;}\;\;} \right)}}{{{{\bar Y}^2}_{ts}}} + 2\frac{{Bias\left( {t_{}^{i - err\;\;}} \right)}}{{{{\bar Y}^2}_{ts}}}} \right]\;\;}}\;} \right]{\mkern 1mu} \;j( \in G) = 1,3,5} \right\}}\\ {{\rm{and}}}\\ {{\rm{Bias}}{\mkern 1mu} (t_s^{j - err\;\;}) = \left\{ {\bar Y_{ts}^{}\left( {\lambda \left( {\frac{{{\rm{Bias}}{\mkern 1mu} (t_{}^{j - err\;\;}\;\;)}}{{\bar Y_{ts}^{}}}} \right) + (\lambda - 1)} \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} j( \in G) = 1,3,5} \right\}{\mkern 1mu} {\mkern 1mu} } \end{array}$

(27)

Substitution of the different values of a and b produce the mean square error and bias for the estimators belongs to the class of exponential-type ratio-cum-ratio estimators. The optimal values which lead to minimum mean square error for the class of exponential product-cum-ratio estimators are obtained as,

$\begin{gathered} a = \frac{{{C_{200(1)}}{C_{002(2)}} - {C_{101(2)}}^2}}{{{C_{110(1)}}{C_{002(2)}} - {C_{011(1)}}{C_{101(2)}}}}\, \, and\, \, b = \frac{{{C_{200(1)}}{C_{002(2)}} - {C_{101(2)}}^2}}{{{C_{200(1)}}{C_{011(1)}} - {C_{110(1)}}{C_{101(2)}}}}, \, \, \lambda = \frac{{\left( {{T^2} + MSE\, ({t^{G - e}}) + 2\, T\, Bias\, ({t^{G - e}})} \right)}}{{T(T + Bias\, ({t^{G - e}})}}, \\ \, \, \, \, \, \, \, where\, \, MSE\left( {{t^G}} \right) = {{\bar Y}_{ts}}^2\left[ {{C_{020(2)}} + \frac{1}{{{a^2}}}{C_{200(1)}} + \frac{1}{{{b^2}}}{C_{002(2)}} - 2\frac{1}{a}{C_{110(1)}} - 2\frac{1}{b}{C_{011(2)}} + 2\frac{1}{a}\frac{1}{b}{C_{101(1)}}} \right]. \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\rm{Bias}}\, (t_{}^G) = \bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2\, {a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2\, {b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} - \frac{1}{a}{C_{110(1)}}} \right) \\ \end{gathered}$

ⅱ). For $\alpha = - 1, \beta = - 1$ , we get exponential-type product-cum-product estimators given in Table B2 (See Appendix B).The mean square error of $t_s^G$ is expressed using as,

$\begin{array}{*{20}{l}} {{\rm{MSE}}{\mkern 1mu} (t_s^{j - epp\;}) = \left\{ {{{\bar Y}^2}_{ts}\left[ {1 - \frac{{{{\left( {1 + \frac{{Bias\left( {t_{}^{i - err}\;\;} \right)}}{{{{\bar Y}_{ts}}}}\;} \right)}^2}}}{{\left[ {1 + \frac{{MSE\left( {t_{}^{i - err}} \right)}}{{{{\bar Y}^2}_{ts}}} + 2\frac{{Bias\left( {t_{}^{i - err}} \right)}}{{{{\bar Y}^2}_{ts}}}\;} \right]}}\;} \right]{\mkern 1mu} {\mkern 1mu} j( \in G) = 2,4,6} \right\}}\\ {{\rm{and}}}\\ {{\rm{Bias}}{\mkern 1mu} (t_s^{j - epp}\;) = \left\{ {\bar Y_{ts}^{}\left( {\lambda \frac{{{\rm{Bias}}{\mkern 1mu} (t_s^{j - epp}\;\;)}}{{\bar Y_{ts}^{}}} + (\lambda - 1)} \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} j( \in G) = 2,4,6} \right\}} \end{array}$

(28)

Substitution of the different values of a and b yield the mean square error and bias for the estimators belongs to the class of exponential-type product-cum-product estimators. The optimal values which lead to minimum mean square error for the class of exponential product-cum-ratio estimators are obtained as,

$\begin{array}{*{20}{l}} {a = \frac{{{C_{200(1)}}\;\;{C_{002(2)}}\;\; - {C_{101(2)}}^2}}{{{C_{110(1)}}\;\;{C_{002(2)}}\;\; - {C_{011(1)\;\;}}{C_{101(2)}}}}{\mkern 1mu} {\mkern 1mu} \;\;and{\mkern 1mu} {\mkern 1mu} b = \frac{{{C_{200(1)}}\;\;{C_{002(2)}}\;\; - {C_{101(2)}}^2}}{{{C_{200(1)}}\;\;{C_{011(1)}}\;\; - {C_{110(1)}}\;\;{C_{101(2)}}}},\lambda = \frac{{\left( {{T^2} + MSE({t^{G - e}}\;\;) + 2TBias({t^{G - e}}\;\;)} \right)}}{{T(T + Bias({t^{G - e}}\;)}},}\\ {{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} where{\mkern 1mu} {\mkern 1mu} MSE\left( {{t^G}} \right) = {{\bar Y}_{ts}}^2\left[ {{C_{020(2)}} + \frac{1}{{{a^2}}}{C_{200(1)}} + \frac{1}{{{b^2}}}{C_{002(2)}} - 2\frac{1}{a}{C_{110(1)}} - 2\frac{1}{b}{C_{011(2)}} + 2\frac{1}{a}\frac{1}{b}{C_{101(1)}}} \right].}\\ {{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\rm{Bias}}t_{}^G) = \bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2{a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2{b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} - \frac{1}{a}{C_{110(1)}}} \right)} \end{array}$

ⅲ). For $\alpha = - 1, \beta = 1$ , we get exponential-type product-cum-ratio estimators given in Table B2. The mean square error of $t_s^G$ is expressed as,

$\begin{array}{*{20}{l}} {{\rm{MSE}}{\mkern 1mu} (t_s^{j - erp\;\;}) = \left\{ {{{\bar Y}^2}_{ts}\left[ {1 - \frac{{{{\left( {1 + \frac{{Bias\left( {t_{}^{i - err}} \right)}}{{{{\bar Y}_{ts}}}}\;\;} \right)}^2}}}{{\left[ {1 + \frac{{MSE\left( {t_{}^{i - err}\;} \right)}}{{{{\bar Y}^2}_{ts}}} + 2\frac{{Bias\left( {t_{}^{i - err}} \right)}}{{{{\bar Y}^2}_{ts}}}} \right]}}} \right]{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} j( \in G) = 7,9,11} \right\}}\\ {{\rm{and}}}\\ {{\rm{Bias}}{\mkern 1mu} (t_s^{j - erp\;\;}\;) = \left\{ {\bar Y_{ts}^{}\left( {\lambda \frac{{{\rm{Bias}}{\mkern 1mu} (t_s^{j - erp\;\;}\;\;)}}{{\bar Y_{ts}^{}}} + (\lambda - 1)} \right)j( \in G) = 7,9,11} \right\}} \end{array}$

(29)

Substitution of the different values of a and b produce the mean square error and bias for the estimators belongs to the class of exponential-type ratio-cum-product estimators. The optimal values which lead to minimum mean square error for the class of exponential ratio-cum-product estimators, are obtained as,

$\begin{array}{*{20}{l}} {a = \frac{{{C_{200(1)}}\;\;{C_{002(2)}} - {C_{101(2)}}^2}}{{{C_{110(1)}}{C_{002(2)}} - {C_{011(1)}}{C_{101(2)}}}}\;\;\;{\mkern 1mu} {\mkern 1mu} and{\mkern 1mu} {\mkern 1mu} b = \frac{{{C_{200(1)}}\;\;{C_{002(2)}} - {C_{101(2)}}^2}}{{{C_{200(1)}}\;\;{C_{011(1)}} - \;\;{C_{110(1)}}{C_{101(2)}}}}\;\;,\lambda = \frac{{\left( {{T^2} + MSE({t^{G - e}}\;) + 2TBias({t^{G - e}}\;\;)} \right)}}{{T(T + Bias({t^{G - e}})}},}\\ {{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} where{\mkern 1mu} {\mkern 1mu} MSE\left( {{t^{G - e}}} \right) = {{\bar Y}_{ts}}^2\left[ {{C_{020(2)}} + \frac{1}{{{a^2}}}{C_{200(1)}} + \frac{1}{{{b^2}}}{C_{002(2)}} - 2\frac{1}{a}{C_{110(1)}} - 2\frac{1}{b}{C_{011(2)}} + 2\frac{1}{a}\frac{1}{b}{C_{101(1)}}} \right].}\\ {{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} Biast_{}^{G - e}) = \bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2{a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2{b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} - \frac{1}{a}{C_{110(1)}}} \right)} \end{array}$

ⅳ). For $\alpha = 1, \beta = - 1$ , we get exponential-type ratio-cum-product estimators given in Table B2. The mean square error of $t_s^G$ is expressed as,

$\begin{array}{*{20}{l}} {{\rm{MSE}}{\mkern 1mu} (t_s^{j - epr}) = \left\{ {{{\bar Y}^2}_{ts}\left[ {1 - \frac{{{{\left( {1 + \frac{{Bias\left( {t_{}^{i - err}} \right)}}{{{{\bar Y}_{ts}}}}\;\;} \right)}^2}}}{{\left[ {1 + \frac{{MSE\left( {t_{}^{i - err}\;} \right)}}{{{{\bar Y}^2}_{ts}}} + 2\frac{{Bias\left( {t_{}^{i - err}} \right)}}{{{{\bar Y}^2}_{ts}}}\;\;\;} \right]}}} \right]{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} j( \in G) = 8,10,12} \right\}}\\ {{\rm{and}}}\\ {{\rm{Bias}}{\mkern 1mu} (t_s^{j - epr}) = \left\{ {\bar Y_{ts}^{}\left( {\lambda \frac{{{\rm{Bias}}{\mkern 1mu} (t_s^{j - epr\;})}}{{\bar Y_{ts}^{}}} + (\lambda - 1)} \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} j( \in G) = 8,10,12} \right\}} \end{array}$

(30)

Substitution of the different values of a and b yield the mean square error and bias for the estimators belongs to the class of exponential-type product-cum-ratio estimators. The optimal values which lead to minimum mean square error for the class of exponential product-cum-ratio estimators, are obtained as,

$\begin{gathered} a = \frac{{{C_{200(1)}}{C_{002(2)}} - {C_{101(2)}}^2}}{{{C_{110(1)}}{C_{002(2)}} - {C_{011(1)}}{C_{101(2)}}}}\, \, and\, \, b = \frac{{{C_{200(1)}}{C_{002(2)}} - {C_{101(2)}}^2}}{{{C_{200(1)}}{C_{011(1)}} - {C_{110(1)}}{C_{101(2)}}}}, \lambda = \frac{{\left( {{T^2} + MSE\, ({t^{G - e}}) + 2\, T\, Bias({t^{G - e}})} \right)}}{{T(T + Bias({t^{G - e}})}}, \\ \, \, \, \, \, \, \, where\, \, MSE\, \left( {{t^G}} \right) = {{\bar Y}_{ts}}^2\left[ {{C_{020(2)}} + \frac{1}{{{a^2}}}{C_{200(1)}} + \frac{1}{{{b^2}}}{C_{002(2)}} - 2\frac{1}{a}{C_{110(1)}} - 2\frac{1}{b}{C_{011(2)}} + 2\frac{1}{a}\frac{1}{b}{C_{101(1)}}} \right]. \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, {\rm{Bias}}\, \, (t_{}^G) = \bar Y_{ts}^{}\left( {\frac{{a - 1}}{{{a^2}}}{C_{200(1)}} + \frac{1}{{2\, {a^2}}}{C_{200(1)}} + \frac{{b - 1}}{{{b^2}}}{C_{002(2)}} + \frac{1}{{2\, {b^2}}}{C_{002(2)}} - \frac{1}{b}{C_{101(2)}} - \frac{1}{a}{C_{110(1)}}} \right) \\ \end{gathered}$

It is to mention that minimum mean square error of $t_s^{j - epr}$ , $t_s^{j - epr}$ , $t_s^{j - epr}$ , and $t_s^{j - epr}$ in (27)–(30) will be same but the conditions under which these attain minimum value are not unique.

4. Efficiency comparison

In this section, some efficiency conditions have been derived in terms of mean square error and bias. The efficiency comparison of proposed two-exponential general shrinkage estimator (see section 3) and two-exponential estimator (see section 2) with unbiased estimator in two-stage sampling. In order to derive the efficiency conditions, consider the following notations,

${A_1} = {C_{020(2)}} + {\left( {\frac{\alpha }{a}} \right)^2}{C_{200(1)}} + {\left( {\frac{\beta }{b}} \right)^2}{C_{002(2)}} - 2\left( {\frac{\alpha }{a}} \right){C_{110(1)}} - 2\left( {\frac{\beta }{b}} \right){C_{011(2)}} + 2\left( {\frac{\alpha }{a}} \right)\left( {\frac{\beta }{b}} \right){C_{101(1)}}$

${A_2} = \left[ {\frac{{\alpha (a - 1)}}{{{a^2}}}{C_{200(1)}} + \frac{{{\alpha ^2}}}{{2{a^2}}}{C_{200(1)}} + \frac{{\beta (b - 1)}}{{{b^2}}}{C_{002(2)}} + \frac{{{\beta ^2}}}{{2{b^2}}}{C_{002(2)}} - \frac{\beta }{b}{C_{101(2)}} - \frac{\alpha }{a}{C_{110(1)}}} \right]$

${A_3} = {C_{020(2)}} + {C_{200(1)}} + {C_{002(2)}} - 2{C_{110(1)}} - 2{C_{011(2)}} + 2{C_{101(1)}}$

${A_4} = \left( {{C_{020(2)}} - \frac{{{C_{110(1)}}^2\;\;{C_{002(2)}} + {C_{011(1)}}^2\;\;{C_{200(1)}}\;\; - 2{C_{110(1)}}\;\;{C_{101(2)}}\;\;{C_{011(1)}}}}{{{C_{200(1)}}\;\;{C_{002(2)}}\;\; - {C_{101(2)}}^2}}} \right)$

${A_5} = \left[ {{\lambda ^2} + 2\lambda (\lambda - 1)\left( {\frac{1}{\alpha }{C_{200(1)}} + \frac{1}{2}{C_{200(1)}}} \right)} \right]$

${A_6} = \left[ {{\lambda ^2}\left( {{C_{110(1)}} - \frac{\beta }{b}{C_{101(!)}}} \right) + \lambda (\lambda - 1)\left( {{C_{200(1)}} - {C_{110(1)}}} \right)} \right]$

${A_7} = \left[ {{\lambda ^2}\left( {{{\left( {\frac{\beta }{b}} \right)}^2}{C_{002(2)}} - 2\frac{\beta }{b}{C_{011(2)}}} \right) + {{(\lambda - 1)}^2} + 2\lambda (\lambda - 1)\left( {\frac{{\beta (b - 1)}}{{{b^2}}}{C_{002(2)}} + {{\left( {\frac{\beta }{b}} \right)}^2}{C_{002(2)}} - \frac{\beta }{b}{C_{101(1)}}} \right)} \right]$

4.1. When is known and is unknown

The efficiency conditions may be written as:

ⅰ. ${MSE}\left(t_{s}^{G}\right)-{MSE}\left(\bar{y}_{t s}\right) \leq 0$ if

$\min \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{C_{020(2)}}} \right)} \right)} }}{{\left( {1 - {A_2}} \right)}}} \right) \leqslant \lambda \leqslant \max \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{C_{020(2)}}} \right)} \right)} }}{{\left( {1 - {A_2}} \;\right)}}} \;\right),$

(31)

ⅱ. $MSE\, (t_s^G) - MSE\, (t_r^{}) \leqslant 0$

$\min \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{A_3}} \right)} \right)} }}{{\left( {1 - {A_2}} \right)}}} \right) \leqslant \lambda \leqslant \max \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{A_3}} \right)} \right)} }}{{\left( {1 - {A_2}} \;\right)}}}\; \right)$

(32)

ⅲ. $MSE\, (t_s^G) - MSE\, (t_{}^G) \leqslant 0$

$\min \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{A_1}} \right)} \right)} }}{{\left( {1 - {A_2}} \right)}}} \right) \leqslant \lambda \leqslant \max \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{A_1}} \right)} \right)} }}{{\left( {1 - {A_2}}\; \right)}}} \right)$

(33)

ⅳ. $MSE\, (t_s^G) - \min MSE\, (t_{}^G) \leqslant 0$

$\min \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{A_4}} \right)} \right)} }}{{\left( {1 - {A_2}} \right)}}} \right) \leqslant \lambda \leqslant \max \left( {\frac{{\left( {1 - {A_2}} \right) \pm \sqrt {\left( {{{\left( {1 - {A_2}} \right)}^2} - \left( {1 + {A_1} + 2{A_2}} \right)\left( {{A_4}} \right)} \right)} }}{{\left( {1 - {A_2}} \right)}}} \;\right)$

(34)

4.2. If a is unknown λ, b is known

ⅰ. ${MSE}\left(t_{s}^{G}\right)-{MSE}\left(\bar{y}_{t s}\right) \leq 0$ if

$\min \left( {\frac{{{A_6} \pm \sqrt {{A_6}^2 - {A_5}\left( {{A_7} - {C_{020(2)}}} \right)} }}{{\alpha {A_5}}}} \right) \leqslant \frac{1}{a} \leqslant \max \left( {\frac{{{A_6} \pm \sqrt {{A_6}^2 - {A_5}\left( {{A_7} - {C_{020(2)}}} \right)} }}{{\alpha {A_5}}}} \right)$

(35)

ⅱ. $MSE\, (t_s^G) - MSE\, (t_r^{}) \leqslant 0$

$\min \left( {\frac{{{A_6} \pm \sqrt {{A_6}^2 - {A_5}\left( {{A_7} - {A_3}} \right)} }}{{\alpha {A_5}}}} \right) \leqslant \frac{1}{a} \leqslant \max \left( {\frac{{{A_6} \pm \sqrt {{A_6}^2 - {A_5}\left( {{A_7} - {A_3}} \right)} }}{{\alpha {A_5}}}} \right)$

(36)

Substituting different values of ( $\alpha, \beta, a, b$ ) in above equations, we get different mathematical comparisons for the estimators given in Table B2 (See Appendix B). Also similar comparison is obtained if we assume $b$ as unknown and $\lambda, a$ is known.

5. Numerical study

For the demonstration of the performance of proposed two-exponential general shrinkage estimator, we take real population consists of four clusters with unequal first stage units. The description about the populations is given in Table B1 in Appendix B. The mean square error and percentage relative efficiency values for each of the estimators are given in Table C2.For this population, correlations within the clusters are positive (see Table C1). It is therefore the population is applicable only for ratio/ratio-type estimators.

The performance of the proposed two-exponential general shrinkage estimator and two-exponential estimator under two stage two phase sampling has been expressed in form of their mean square error and percentage relative efficiency values in Table C2. From Table C2 we see that the minimum mean square error of two-exponential general shrinkage estimator $t_s^G$ that is equal to the mean square error of $t_s^5$ . The percentage relative efficiency demonstrate the same result as the percentage relative efficiency of the proposed two-exponential general shrinkage is high among the family of proposed two-exponential general shrinkage and $t_s^5$ is the same efficient as $t_s^G$ .

We also see that the two-exponential general shrinkage estimator (section 3) is better to be used as compare to two-exponential estimator (section 2). The performance of $t_s^1$ is better than ${t^1}$ , similarly $t_s^3$ performs better than $t_{}^3$ . So it is concluded that for this particular population the use of two-exponential general shrinkage estimator in two stage two phase sampling produce better results.

6. Conclusion

Finally from the above empirical results, it is concluded that the performance of the two-exponential general shrinkage estimator (section 3) is higher as compare to proposed two-exponential estimator in two stage two phase sampling. So the two-exponential general shrinkage estimator (section 3) is acceptable for the real life application in two stage two phase sampling design.

Acknowledgments

We are grateful to the anonymous reviewers for their valuable comments and suggestions in improving the manuscript.

Conflict of interest

The authors declare no conflict of interest.

Appendix A

Proof of Theorem. Let $t$ be any general estimator of some population parameter T in two stage sampling for which a general shrinkage estimator ${t_s}$ is defined as,

$t_s^{} = \lambda t,$

(37)

By definition the bias of $t_s^{}$ is

$Bias\left( {{t_s}} \right) = \lambda E\left( t \right) - T$

$= \lambda E\left( {t - T} \right) + \left( {\lambda - 1} \right)T$

$= \lambda Bias\left( t \right) + \left( {\lambda - 1} \right)T$

(38)

By definition the MSE of ${t_s}$ may be defined as,

$MSE\left( {t_s^{}} \right) = E{\left( {t_s^{} - T} \right)^2}$

$= E{\left( {\lambda t - T} \right)^2}$

consider $t = T(1 + {e_{_T}})$ , with $E({e_{_T}}) = 0$ and $E({e_{_T}}^2) = \frac{{Var(t)}}{{{T^2}}} = \frac{{MSE\left(t \right)}}{{{T^2}}}$ .

$= E{\left\{ {\lambda \left( {T\left( {1 + {e_T}} \right)} \right) - T} \right\}^2}$

$= {T^2}E\left\{ {{\lambda ^2}{{\left( {1 + {e_T}} \right)}^2} + 1 - 2\lambda \left( {1 + {e_T}} \right)} \right\}$

$MSE\left( {t_s^{}} \right) = {T^2}\left\{ {{\lambda ^2}\left( {1 + \frac{{MSE\left( t \right)}}{{{T^2}}}} \right) + 1 - 2\lambda } \right\}$

(39)

We may also write (B-3) as

$MSE\left( {t_s^{}} \right) = {\lambda ^2}MSE\left( t \right) + {\left( {\lambda - 1} \right)^2}{T^2}$

$MSE\left( {t_s^{}} \right) = {\lambda ^2}{T^2}C_t^2 + {\left( {\lambda - 1} \right)^2}{T^2} \;\;{\rm{Where}}\;\;C_t^2 = \frac{{MSE\left( t \right)}}{{{T^2}}}$

(40)

Where $C_t^{}$ is a coefficient of variation of an estimator t.

Partially differentiating (40) w.r.t $\lambda$ and simplifying, we get:

$\lambda = \frac{1}{{1 + {T^{ - 2}}\;MSE\left( t \right)}}\;,$

(41)

$\lambda = \frac{1}{{1 + C_t^2}},$

(42)

Now using (41) in (15), (16) and (17) respectively, we may get,

${t_s} = \frac{t}{{1 + {T^{ - 2}}\;MSE\left( t \right)}}$

(43)

$Bias\left( {t_s^{}} \right) = \frac{1}{{1 + {T^{ - 2}}\;MSE\left( t \right)}}\left[ {Bias\left( t \right) + {T^{ - 2}}MSE\left( t \right)T} \right]$

(44)

$MS{E_{\min }}\left( {t_s^{}} \right) = \frac{{MSE\left( t \right)}}{{1 + {T^{ - 2}}\;MSE\left( t \right)}}$

(45)

Alternatively we may also produce the expressions in (44) to (46) as,

${t_s} = \frac{t}{{1 + C_t^2}}$

(46)

$Bias\left( {t_s^{}} \right) = \frac{1}{{1 + C_t^2}}\left( {Bias\left( t \right) + C_t^2T} \right)$

(47)

$MS{E_{\min }}\left( {t_s^{}} \right) = \frac{{{T^2}C_t^2}}{{1 + C_t^2}}$

(48)

Appendix B

Table B1. Some members of the proposed two exponential estimator (

${t^G}$ ).

Ratio-cum-Ratio estimator $\alpha = 1, \beta = 1$	Product -cum-product estimator $\alpha = - 1, \beta = - 1$	$a$	$b$
${t^{1 - err}} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + {{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar Z}_{ts}} - {{\bar z}_{ts(2)}}}}{{{{\bar Z}_{ts}} + {{\bar z}_{ts(2)}}}}} \right)$	$t_{}^{2 - epp} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar x}_{ts(1)}} + {{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar z}_{ts(2)}} + {{\bar Z}_{ts}}}}} \right)$	2	2
$t_{}^{3 - err} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar Z}_{ts}} - {{\bar z}_{ts(2)}}}}{{{{\bar Z}_{ts}}}}} \right)$	$t_{}^{4 - epp} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}}} \right)$	1	1
$t_{}^{5 - err} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar Z}_{ts}} - {{\bar z}_{ts(2)}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$t_{}^{6 - epp} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$a$	$b$
Product -cum-ratio estimator $\alpha = - 1, \beta = 1$	Ratio-cum-product estimator $\alpha = 1, \beta = - 1$	$a$	$b$
$t_{}^{7 - epr} = {\bar y_{ts}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + {{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + {{\bar z}_{ts(2)}}}}} \right)$	$t_{}^{8 - erp} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar x}_{ts(1)}} + {{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar z}_{ts(2)}} + {{\bar Z}_{ts}}}}} \right)$	2	2
$t_{}^{9 - epr} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(1)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}}} \right)$	$t_{}^{10 - erp} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}}} \right)$	1	1
$t_{}^{11 - epr} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$t_{}^{12 - erp} = {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$a$	$b$

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Table B2. Some members of the generalized shrinkage estimator (

$t_s^G$ ).

Ratio-cum-product estimator $\alpha = 1, \beta = 1$	Product -cum-product estimator $\alpha = - 1, \beta = - 1$	$a$	$\lambda$	$b$
$t_s^{1 - err} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + {{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar Z}_{ts}} - {{\bar z}_{ts(2)}}}}{{{{\bar Z}_{ts}} + {{\bar z}_{ts(2)}}}}} \right)$	$t_s^{2 - epp} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}} + {{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + {{\bar z}_{ts(2)}}}}} \right)$	2	$\lambda$	2
$t_s^{3 - err} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar Z}_{ts}} - {{\bar z}_{ts(2)}}}}{{{{\bar Z}_{ts}}}}} \right)$	$t_s^{4 - epp} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}}} \right)$	1	$\lambda$	1
$t_s^{5 - erp} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts}}}}} \right)\exp \left({\frac{{{{\bar Z}_{ts}} - {{\bar z}_{ts(2)}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$t_s^{6 - epp} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$a$	$\lambda$	$b$
Product -cum-ratio estimator $\alpha = - 1, \beta = 1$	Ratio-cum-product estimator $\alpha = 1, \beta = - 1$	$a$	$\lambda$	$b$
$t_s^{7 - epr} = \lambda {\bar y_{ts}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + {{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + {{\bar z}_{ts(2)}}}}} \right)$	$t_s^{8 - erp} = \lambda {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar x}_{ts(1)}} + {{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar z}_{ts(2)}} + {{\bar Z}_{ts}}}}} \right)$	2	$\lambda$	2
$t_s^{9 - epr} = \lambda {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(1)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}}} \right)$	$t_s^{10 - erp} = \lambda {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}}}}} \right)$	1	$\lambda$	1
$t_s^{11 - epr} = \lambda {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar X}_{ts}} - {{\bar x}_{ts(1)}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$t_s^{12 - erp} = \lambda {\bar y_{ts(2)}}\exp \left({\frac{{{{\bar x}_{ts(1)}} - {{\bar X}_{ts}}}}{{{{\bar X}_{ts}} + (a - 1){{\bar x}_{ts(1)}}}}} \right)\exp \left({\frac{{{{\bar z}_{ts(2)}} - {{\bar Z}_{ts}}}}{{{{\bar Z}_{ts}} + \left({b - 1} \right){{\bar z}_{ts(2)}}}}} \right)$	$a$	$\lambda$	$b$

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Appendix C

Table C1. Data Statistics for Population-Ⅰ.

	Population-Ⅰ (unequal fsu’s)
Cluster	1	2	3	4
${M_i}$	18	14	12	20
${m_i}$	9	7	6	10
${\bar Y_i}$	25.77722	22.79286	28.43500	23.0905
${\bar X_i}$	51.06389	46.49700	67.00217	57.11855
${C_{{y_i}}}^2$	0.58025	0.39297	0.34783	0.31545
${C_{{x_i}}}^2$	0.43322	0.29984	0.41947	0.40689
${\rho _{i1}}$	0.88373	0.83895	0.82425	0.82113

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Table C2. MSEs and PREs of the adapted and suggested class of estimators.

Estimators	MSE	PRE With new fraction
${\overline y _s}$	24.44649	100
$t_{}^{1 - err}$	11.66857	209.51
$t_{}^{3 - err}$	20.27969	120.5467
$t_{}^{5 - err}$	11.46787	213.1738
$t_s^{1 - err}$	11.58499	211.0229
$t_s^{3 - err}$	18.92219	129.1974
$t_s^{5 - err}$	11.25906	217.1317
$t_s^{G - opt}$	11.25906	217.1317

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AIMS Mathematics

Impact of estrogen population pharmacokinetics on a QSP model of mammary stem cell differentiation into myoepithelial cells

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Abstract

1. Introduction

2. Materials and method

2.1. Proposed generalized estimator

2.2. Bias and mean square error of generalized estimator

3. Proposed two-exponential general Shrinkage estimator

3.1. Theorem for estimating any population characteristic

4. Efficiency comparison

4.1. When is known and is unknown

4.2. If a is unknown λ, b is known

5. Numerical study

6. Conclusion

Acknowledgments

Conflict of interest

Appendix A

Appendix B

Appendix C

References

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AIMS Mathematics

Impact of estrogen population pharmacokinetics on a QSP model of mammary stem cell differentiation into myoepithelial cells

Related Papers:

Abstract

1. Introduction

2. Materials and method

2.1. Proposed generalized estimator

2.2. Bias and mean square error of generalized estimator

3. Proposed two-exponential general Shrinkage estimator

3.1. Theorem for estimating any population characteristic

4. Efficiency comparison

4.1. When is known and is unknown

4.2. If a is unknown λ, b is known

5. Numerical study

6. Conclusion

Acknowledgments

Conflict of interest

Appendix A

Appendix B

Appendix C

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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