1 Introduction

1.1 Significance of non-response

In sampling theory, it is generally assumed that the true value of each unit in the population can be determined without error. In practice, this assumption may be violated for various reasons and because of practical constraints that exist at the time of the survey. The problem of nonresponse in sample surveys is usually due to a lack of interest on the part of respondents in answering, persons not present at home, lack of knowledge about the survey or the questions asked ethical issues, and refusal of respondents to answer the given questionnaire. In postal surveys, questionnaires are mailed to units selected in the sample with a request that they be returned within a specified period of time. In general, however, many respondents do not answer the questions or do not return the completed questionnaire within the specified time. In such cases, one can use the information obtained from the available questionnaires, but this may result in a loss of estimator efficiency due to the smaller sample size. In general, the postal questionnaire method is very often used in sample surveys to reduce the cost of the survey. Due to the high non-response rate that occurs for the reasons mentioned above. The component of bias creeps into the estimation procedure, leading to results as inaccurate as would not be expected without the presence of non-response. For mail surveys, Hansen and Hurwitz (Hansen and Hurwitz 1946) proposed a method of subsampling from nonresponding units and provided an estimate based on values obtained from responding units and subsamples obtained by the personal interview method from some nonresponding units. The literature on the various sources of non-response and methods of eliminating the problem of non-response can be found in Kish (Kish 1965), Kumar et al. (Kumar et al. 2019), Singh et al. (Singh and Singh 2022). Cochran (Cochran 1977) referred to the failure to measure some of the units in the selected sample, while Zarkovich (Zarkovich 1966) and Ford (Ford 1976) referred to the same problem as missing data. Sudman (Sudman 1976) addressed the problem of bias due to non-cooperation. Sukhatme and Sukhatme (Sukhatme and Sukhatme 1970) described the effects of incomplete samples. All of these approaches refer to the phenomenon of non-response. Sarndal et al. (Sarndal et al. 2003), Groves (Groves 2004), and Dillman et al. (Dillman et al. 2002) and Chaudhary et al. (Chaudhary and Kumar 2016) have also defined various aspects and reasons for non-response.

1.2 Model-based approach

To make inference about the finite population, on the basis of a probability sample, by measuring the sample elements, two types of problems may arise Sarndal et al. (1992):

  1. (a)

    Inference about the finite population itself.

  2. (b)

    Inference about a model or a super-population from which the given population thought to have generated.

Case (a) can be dealt with design-based inference as it is usual in most of the survey sampling problems, but case (b) creates a different situation where the population is considered to be a random sample from a super-population which can be designated by a model \(D\). Among others the approach was advocated by Brewer (Brewer 1963), Royall (Royall et al. 1971), Cassel et al. (Cassel et al. 1976), Singh et al. (Singh et al. 2009), Basu (Basu 1958), Singh et al. (Singh et al. 2017) and Singh et al. (Sarndal et al. 2003) Recently, Ahmed and Shabbir (Shakeel Ahmed and Javid Shabbir 2019) have discussed the utility of the estimator for paradigm of the model-based and in the presence of non-ignorable non-response.

Royall and Herson (Royall and Herson 1973a) (Royall and Herson 1973b) suggested a particular type of super population model, termed as PRM, which is described as

$$Y_{t} = \delta_{0} \beta_{0} + \delta_{1} \beta_{1} x_{t} + \delta_{2} \beta_{2} x_{t}^{2} + \cdots + \delta_{J} \beta_{J} x_{t}^{J} + \varepsilon_{t} \left[ {v\left( {x_{t} } \right)} \right]^{1/2}$$
$$= \sum\limits_{j = 0}^{J} {\delta_{j} \beta_{j} x_{t}^{j} } + \varepsilon_{t} \left[ {v\left( {x_{t} } \right)} \right]^{1/2} for\,t = 1,2,...,N$$
(1)

\({\text{With}}E_{D} \left( {Y_{t} } \right) = h(x_{t} ) = \sum\limits_{j = 0}^{J} {\delta_{j} \beta_{j} x_{t}^{j} }\);

$$Var\left[ {Y_{t} } \right] = \sigma^{2} v\left( {x_{t} } \right);Cov\left( {Y_{s,} Y_{t} } \right) = 0,s \ne t$$

where \(Y_{t}\) is the study variable associated with the tth unit of the universe of size N, \(x_{t}\) is the value of the tth unit on the auxiliary variable X of the universe and it is non-negative.\(\varepsilon_{t} ;t = 1,2,...,N\) are independent random error with mean zero and variance \(\sigma^{2}\), \(\delta_{j}\)(j = 0, 1,.., J) is zero or one according as the term \(x_{t}^{j}\) is absent or present respectively. In the model (1), \(v\left( {x_{t} } \right)\) are known function of x-values and \(\beta_{j} ;j = 1,2,...,J\) are unknown model parameters. It is denoted this model as \(D\left[ {\delta_{0} ,\delta_{1} ,\delta_{2} ,...,\delta_{J} :v\left( x \right)} \right]\). Chambers (Chambers 1986) has described that in both sample survey theory and practice, mean of \(Y_{t}\) is proportional to \(x_{t}\). The variance of \(Y_{t}\) is proportional to \(v\left( {x_{t} } \right)\).

2 Initiation of the problem

2.1 Let a finite population of size N, denoted by Ω, consists of \(N_{1}\) respondents and \(N_{2}\) non-respondents units. We selected a sample of size n from a universe that consists \(n_{1}\) respondents and \(n_{2}\) non-respondents. For efficient estimate, we collect some information the non-respondents. Therefore, we select a sub-sample \(h_{2}\) from the non-respondents \(n_{2}\) units. Let the sample of size n and \(h_{2}\) be denoted by s and \(s_{h_{2}}\) respectively. Further let \(\Omega = s \cup \overline{s}\), \(s\) and \(\overline{s}\) are two disjoint sets, such that \(s\) represent the observed part of the universe and \(\overline{s}\) represents the missing part of the universe. Further, consider \(s = s_{1} \cup s_{2}\)(\(s_{1}\) and \(s_{2}\) samples of disjoint sets) and \(s_{2} = s_{{h_{2} }} \cup \overline{s}_{{h_{2} }}\). The value of where \(\overline{s}_{{h_{2} }}\) is sub-portion of \(s_{2}\) sample. We have discuss the notations:

Z = Variable X or Y

and

\(\overline{Z} = N^{ - 1} \sum\limits_{\Omega } {z_{t} }\): population mean of \(Z^{th}\) variable.

\(S_{Z}^{2} = (N - 1)^{ - 1} \sum\limits_{\Omega } {(z_{t} } - \overline{Z})^{2}\): \(Z^{th}\) population mean square.

Also sample means:

$$\overline{z}_{{s_{i} }} = n_{i}^{ - 1} \sum\limits_{{s_{i} }} {z_{t} \, } (i = 1,2); \, \overline{z}_{{s_{{h_{2} }} }} = h_{2}^{ - 1} \sum\limits_{{s_{h2} }} {z_{t} } ;$$
$$\overline{z}_{{\overline{s}_{{h_{2} }} }} = (n_{2} - h_{2} )^{ - 1} \sum\limits_{{\overline{s}_{{h_{2} }} }} {z_{t} } .$$

Higher order moments:

We considered for X

$$\overline{X}^{\left( j \right)} = \frac{1}{N}\sum\limits_{\Omega }^{{}} {x_{t}^{j} } ;\overline{x}^{\left( j \right)} = \frac{1}{n}\sum\limits_{s}^{{}} {x_{t}^{j} } ;\overline{x}_{{\overline{s}}}^{\left( j \right)} = M^{ - 1} \sum\limits_{{\overline{s}}} {x_{t}^{j} } ;\overline{x}_{{s_{i} }}^{\left( j \right)} = n_{i}^{ - 1} \sum\limits_{{s_{i} }}^{{}} {x_{t}^{j} } ; \, (i = 1,2)$$
$$\bar x_{{s_{{h_2}}}{\rm{ }}}^{(j)} = {({h_2})^{ - 1}}\sum\limits_{{s_{{h_2}}}} {x_{{\rm{t }}}^j;{\mkern 1mu} } \bar x_{{{\bar s}_{{h_2}}}{\rm{ }}}^{(j)} = \frac{1}{{({n_2} - {h_2})}}\sum\limits_{{{\bar s}_{{h_2}}}} {x_t^j}\quad {\rm{for }}\quad j = 1,2,3 \ldots$$

Obviously, we have seen that

$$\overline{X}^{(1)} = \overline{X};{\overline{\text{x}}}^{(1)} = \overline{x};\quad {\overline{\text{x}}}_{{{\text{s}}_{{\text{i}}} }}^{(1)} = \overline{x}_{{s_{i} }} (i = 1,2);$$
$${\overline{\text{x}}}_{{{\text{s}}_{{{\text{h}}_{{2}} }} }}^{(1)} = \overline{x}_{{s_{{h_{2} }} }} ;\quad {\overline{\text{x}}}_{{{\overline{\text{s}}}_{{}} }}^{(1)} = \overline{x}_{{\overline{s}}} ;\quad {\overline{\text{x}}}_{{{\overline{\text{s}}}_{{_{{h_{2} }} }} }}^{{^{(1)} }} = \overline{x}_{{\overline{s}_{{h_{2} }} }} .$$

Hansen and Hurwitz (Hansen and Hurwitz 1946) suggested the problem of non-response of population mean \(\overline{Y}\) is given by.

$$\overline{y}_{w}^{*} = \frac{{n_{1} \overline{y}_{{s_{1} }} + n_{2} \overline{y}_{{s_{{h_{2} }} }} }}{n}.$$
(2)

They obtained the variance of estimator as

$$V(\overline{y}_{w}^{*} ) = \left( {\frac{1}{n} - \frac{1}{N}} \right)S_{Y}^{2} + \frac{{(f_{2} - 1)}}{n}\frac{{N_{2} }}{N}S_{2Y}^{2} ;$$
(3)

where \(f_{2} = \frac{{n_{2} }}{{h_{2} }}\) and \(\begin{gathered} {\text{ S}}_{{{\text{2Y}}}}^{{2}} = (N_{2} - 1)^{ - 1} \sum\limits_{t = 1}^{{N_{2} }} {(y_{t} } - \overline{Y}_{{N_{2} }} )^{2} \hfill \\ \, \hfill \\ \end{gathered}\); \({\overline{\text{Y}}}_{{{\text{N}}_{{2}} }}\) shows non-respondent population mean.

3 Proposed family of strategy

Family \(T_{1}^{*} \left( \alpha \right)\): Here, we first write the family of one parameter estimators for population mean in the presence of non-response

$$T_{1}^{*} \left( \alpha \right) = \overline{y}_{w}^{*} \psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)$$
(4)
$${\text{where}}\,\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) = \exp \left[ {\alpha \left( {\frac{{\overline{X} - \overline{x}_{d} }}{{\overline{X} + \overline{x}_{d} }}} \right)} \right]$$
$${\text{where }}\overline{x}_{d} = d\overline{x} + \left( {1 - d} \right)\overline{X};d = \frac{n}{{\left( {N + n} \right)}}.$$
(5)

Remark 1

Instances of estimator \(T_{1}^{*} \left( \alpha \right)\) when \(\alpha\) = 0.

$$T_{1}^{*} (0) = \overline{y}_{w}^{*}$$
(6)

It is extended form of estimator developed by Hansen and Hurwitz (Groves 2004). They have also developed some other cases for \(\alpha\) = 1 and -1, we obtained an exponential-type ratio and product estimators.

4 \(D\)-Bias and \(D\)-MSE of \(T_{1}^{*} \left( \alpha \right)\)

The simple form of the model \(D\left[ {\delta_{0} ,\delta_{1} ,...,\delta_{J} :v\left( x \right)} \right]\), D—based estimator bias and MSE. Theorem 1: Value of D-bias of estimator \(T_{1}^{*} \left( \alpha \right)\) is

$$B_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right] = \sum\limits_{j = 0}^{J} {\delta_{j} \beta_{j} } \left[ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }}^{\left( j \right)} + n_{2} \overline{x}_{{s_{{h_{2} }} }}^{\left( j \right)} }}{n}} \right) - \overline{X}^{\left( j \right)} } \right]$$
(7)

The appendix A, Section –I mentioned proof of this Eq. (7)

Theorem 2

The value of D-MSE of the estimator \(T_{1}^{*} \left( \alpha \right)\) is given by.

$$M_{D} \left( {T_{1}^{*} \left( \alpha \right)} \right) = \left[ {B_{D} \left( {T_{1}^{*} \left( \alpha \right)} \right)} \right]^{2} + \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} \sigma^{2} \sum\limits_{{s_{1} }} {v\left( {x_{t} } \right)}$$
$$+ \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} \sigma^{2} \sum\limits_{{s_{{h_{2} }} }} {v\left( {x_{k} } \right)} + \frac{{\sigma^{2} }}{{N^{2} }}\left[ {\sum\limits_{{\overline{s}_{{h_{2} }} }} {v\left( {x_{t} } \right)} + \sum\limits_{{\overline{s}}} {v\left( {x_{t} } \right)} } \right].$$
(8)

The appendix A, Section –I mentioned proof of this Eq. (8).

Remark 2

We have seen from Eq. (8). The D-MSE of \(T_{1}^{*} \left( \alpha \right)\) is consisting two parts one is D -Bias and second is variance of estimator. We know that variance of the estimator does not depend on the function \(\sum\nolimits_{j = 0}^{J} {\delta_{j} \beta_{j} x_{t}^{j} }\). However, it is depending on error term \(\varepsilon_{t}\) and variance \(v\left( {x_{t} } \right)\). In the case spoil selection of polynomial regression there is no change in the variance of estimator.

5 \(D\)-Bias and \(D\)-MSE of particular cases of \(T_{1}^{*} \left( \alpha \right)\).

5.1. The PRM instance cases assumed two different forms with a functions \(h\left( {x_{t} } \right) = \sum\nolimits_{j = 0}^{J} {\delta_{j} \beta_{j} x_{t}^{j} }\) and function \(v\left( {x_{t} } \right)\). The instances forms of the model is given by

$$Y_{t} = \beta_{1} x_{t} + \varepsilon_{t} \left[ {x_{t} } \right]^{1/2}$$
(9)
$$Y_{t} = \beta_{0} + \beta_{1} x_{t} + \varepsilon_{t} \left[ {x_{t}^{2} } \right]^{1/2}$$
(10)
$$Y_{t} = \beta_{0} + \beta_{1} x_{t} + \beta_{2} x_{t}^{2} + \varepsilon_{t} \left[ {x_{t}^{g} } \right]^{{2^{ - 1} }} .$$
(11)

Obviously models (9), (10) & (11) may be denoted respectively as \(D\left[ {0,\,\,1\,\,:x} \right]\), \(D\left[ {1,\,\,1:x^{2} } \right]\) and \(D\left[ {1,\,\,1,\,\,1\,\,:x^{g} } \right]\). The variance function of model based idea was developed Cochran (Chambers 1986) and Brewer (Brewer 1963) when \(v\left( {x_{t} } \right)\) assumed to be \(x_{t}^{g}\) with \(0 \le g \le 2\). We motive by their contribution, we have provided six different PRMs:

$${\text{Model I}}\,D\left[ {0,\,\,1:1} \right] \Rightarrow h\left( {x_{t} } \right) = \beta_{1} x_{t} ,v\left( {x_{t} } \right) = x_{t}^{0}$$
(12)
$${\text{Model II}}\,D\left[ {0,1:x} \right] \Rightarrow h\left( {x_{t} } \right) = \beta_{1} x_{t} ,v\left( {x_{t} } \right) = x_{t}^{{}}$$
(13)
$${\text{Model III}}\,D\left[ {0,1:x^{2} } \right] \Rightarrow h\left( {x_{t} } \right) = \beta_{1} x_{t} ,v\left( {x_{t} } \right) = x_{t}^{2}$$
(14)
$${\text{Model IV}}\,D\left[ {1,1:1} \right] \Rightarrow h\left( {x_{t} } \right) = \beta_{0} + \beta_{1} x_{t} ,v\left( {x_{t} } \right) = x_{t}^{0}$$
(15)
$${\text{Model V}}\,D\left[ {1,1:x} \right] \Rightarrow h\left( {x_{t} } \right) = \beta_{0} + \beta_{1} x_{t} ,v\left( {x_{t} } \right) = x_{t}^{{}}$$
(16)
$${\text{Model VI}}\,D\left[ {1,1:x^{2} } \right] \Rightarrow h\left( {x_{t} } \right) = \beta_{0} + \beta_{1} x_{t} ,v\left( {x_{t} } \right) = x_{t}^{2}$$
(17)

6 D-MSE of \(T_{1}^{*} \left( \alpha \right)\) under Models I-VI

The MSE of the family of estimators \(T_{1}^{*} \left( \delta \right)\) in the models I–VI can be obtained from expression (8). These are presented as follows:

$$M_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right]_{I} = \left[ {\beta_{1} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }} + n_{2} \overline{x}_{{s_{{h_{2} }} }} }}{n}} \right) - \overline{X}} \right\}} \right]^{2}$$
$$+ \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} n_{1} \sigma^{2} + \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} h_{2} \sigma^{2}$$
$$+ \frac{{\sigma^{2} }}{{N^{2} }}\left[ {M + \left( {n_{2} - h_{2} } \right)} \right].$$
(18)
$$M_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right]_{II} = \left[ {\beta_{1} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }} + n_{2} \overline{x}_{{s_{{h_{2} }} }} }}{n}} \right) - \overline{X}} \right\}} \right]^{2}$$
$$+ \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} n_{1} \sigma^{2} \overline{x}_{{s_{1} }} + \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} h_{2} \sigma^{2} \overline{x}_{{s_{{h_{2} }} }}$$
$$+ \frac{{\sigma^{2} }}{{N^{2} }}\left\{ {\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{1} } + \sum\limits_{{\overline{s}}} {x_{1} } } \right\}.$$
(19)
$$M_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right]_{III} = \left[ {\beta_{1} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }} + n_{2} \overline{x}_{{s_{{h_{2} }} }} }}{n}} \right) - \overline{X}} \right\}} \right]^{2}$$
$$+ \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} \sigma^{2} \sum\limits_{{s_{1} }} {x_{t}^{2} } + \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} \sigma^{2} \sum\limits_{{s_{{h_{2} }} }} {x_{t}^{2} }$$
$$+ \frac{{\sigma^{2} }}{{N^{2} }}\left[ {\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{t}^{2} } + \sum\limits_{{\overline{s}}} {x_{t}^{2} } } \right].$$
(20)
$$M_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right]_{IV} = \left[ {\beta_{0} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - 1} \right\} + \beta_{1} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }} + n_{2} \overline{x}_{{s_{{h_{2} }} }} }}{n}} \right) - \overline{X}} \right\}} \right]^{2}$$
$$+ \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} n_{1} \sigma^{2} + \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} h_{2} \sigma^{2}$$
$$+ \frac{{\sigma^{2} }}{{N^{2} }}\left[ {M + \left( {n_{2} - h_{2} } \right)} \right].$$
(21)
$$M_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right]_{V} = \left[ {\beta_{0} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - 1} \right\} + \beta_{1} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }} + n_{2} \overline{x}_{{s_{{h_{2} }} }} }}{n}} \right) - \overline{X}} \right\}} \right]^{2}$$
$$+ \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} n_{1} \sigma^{2} \overline{x}_{{s_{1} }} + \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} h_{2} \sigma^{2} \overline{x}_{{s_{{h_{2} }} }}$$
$$+ \frac{{\sigma^{2} }}{{N^{2} }}\left\{ {\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{t} } + \sum\limits_{{\overline{s}}} {x_{t} } } \right\}.$$
(22)
$$M_{D} \left[ {T_{1}^{*} \left( \alpha \right)} \right]_{VI} = \left[ {\beta_{0} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - 1} \right\} + \beta_{1} \left\{ {\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)\left( {\frac{{n_{1} \overline{x}_{{s_{1} }} + n_{2} \overline{x}_{{s_{{h_{2} }} }} }}{n}} \right) - \overline{X}} \right\}} \right]^{2}$$
$$+ \left[ {\frac{{\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{2} \sigma^{2} \sum\limits_{{s_{1} }} {x_{t}^{2} } + \left[ {\frac{{f_{2} }}{n}\psi^{*} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{2} \sigma^{2} \sum\limits_{{s_{{h_{2} }} }} {x_{t}^{2} }$$
$$+ \frac{{\sigma^{2} }}{{N^{2} }}\left[ {\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{t}^{2} } + \sum\limits_{{\overline{s}}} {x_{t}^{2} } } \right].$$
(23)

7 Some existing strategies with their MSEs

Two families of estimators in the presence of non-response was discussed Singh et al. (Singh et al. 2017) and compared them under different PRMs. The estimators and their MSEs under \(D\left[ {\delta_{0} ,\delta_{1} ,...,\delta_{J} :v\left( x \right)} \right]\) are as follows

$$\left( {\text{i}} \right)\,T_{s}^{*} \left( \alpha \right) = \overline{y}_{w}^{*} \psi^{{}} \left( {\alpha ,\overline{X},\overline{x}} \right)$$
(24)

where \(\psi^{{}} \left( {\alpha ,\overline{X},\overline{x}} \right) = \exp \left[ {\alpha \left( {\frac{{\overline{X} - \overline{x}_{{}} }}{{\overline{X} + \overline{x}_{{}} }}} \right)} \right]\), and

$$M_{D} \left( {T_{s}^{*} \left( \alpha \right)} \right) = \left[ {B_{D} \left( {T_{s}^{*} \left( \alpha \right)} \right)} \right]^{2} + \left[ {\frac{{\psi^{{}} \left( {\alpha ,\overline{X},\overline{x}} \right)}}{n} - \frac{1}{N}} \right]^{\,\,2} \sigma^{2} \sum\limits_{{s_{1} }} {v\left( {x_{t} } \right)}$$
$$+ \left[ {\frac{{f_{2} }}{n}\psi^{{}} \left( {\alpha ,\overline{X},\overline{x}} \right) - \frac{1}{N}} \right]^{\,\,2} \sigma^{2} \sum\limits_{{s_{{h_{2} }} }} {v\left( {x_{t} } \right)} + \frac{{\sigma^{2} }}{{N^{2} }}\left[ {\sum\limits_{{\overline{s}_{{h_{2} }} }} {v\left( {x_{t} } \right)} + \sum\limits_{{\overline{s}}} {v\left( {x_{t} } \right)} } \right]$$
(25)
$$\left( {{\text{ii}}} \right)t_{NR}^{*} (\alpha ) = \overline{y}_{w}^{**} \frac{{\overline{X}}}{{\overline{x}_{{}} }},$$
(26)

and

$$\overline{y}_{w}^{**} = \frac{{n_{1} \overline{y}_{{s_{1} }} + n_{2} \,t_{{s_{2} }}^{*} (\alpha )}}{n}$$
(27)
$$t_{{s_{2} }} (\alpha ) = \overline{y}_{{s_{{h_{2} }} }} \exp \left[ {\alpha \left( {\frac{{\overline{x}_{{s_{2} }} - \overline{x}_{{s_{{h_{2} }} }} }}{{\overline{x}_{{s_{2} }} + \overline{x}_{{s_{{h_{2} }} }} }}} \right)} \right] \,$$
(28)
$${M_D}\left[ {t_{NR}^*\left( \alpha \right)} \right] = {\left[ {\sum\limits_{j = 0}^J {{\delta _j}} {\beta _j}\left( {\frac{{\bar X}}{{n\bar x}}\left( {{n_1}\bar x_{{s_1}}^{(j)} + {n_2}\psi \left( {\alpha {{, }}{{{{\bar {\text{x}}}}}_{{{\text{s}}_{{2}}}}},{{\bar x}_{{s_{h2}}}}} \right)\bar x_{{s_{{h_2}}}}^{(j)}} \right) - {{\bar X}^{(j)}}} \right)} \right]^2} + {\left( {\frac{{\bar X}}{{n\bar x}} - \frac{1}{N}} \right)^2}{\sigma ^2}{\sum\limits_{{s_1}} {v({x_t}) + \left[ {\frac{{{n_2}}}{{n{h_2}}}\psi \left( {\alpha {{, }}{{{{\bar {\text{x}}}}}_{{{\text{s}}_{{2}}}}},{{\bar x}_{{s_{h2}}}}} \right)\frac{{\bar X}}{{{{\bar x}_{}}}} - \frac{1}{N}} \right]} ^2} {\sigma ^2}\sum\limits_{{s_{{h_2}}}} {v({x_t}) + \frac{1}{{{N^2}}}} {\sigma ^2}\sum\limits_{{{\bar s}_{{h_2}}}} {v({x_t}) + \frac{1}{{{N^2}}}} {\sigma ^2}\sum\limits_{\bar s} {v({x_t})} .$$
(29)

8 Robustness of the estimator \(T_{1}^{*} \left( \alpha \right)\) and its comparison with \(T_{s}^{*} \left( \alpha \right)\) and \(t_{NR}^{*} (\alpha )\)

8.1 In remark 2, we have seen, the D-MSE of the strategies is affected by the deviations in \(h(x_{t} )\) and function \(v(x_{t} )\), if it a D-biased, while the bias is affected only by the function \(h(x_{t} )\) and is completely do not dependent in \(v(x_{t} )\). Therefore, it would be desirable to consider change in the amount of MSE with the deviation of the model, either due to misspecification in \(h(x_{t} )\) or in \(v(x_{t} )\) or both.

Royall and Herson [17&18], therefore, considered an estimator ‘robust’ if there is a nominal change in the amount of D—MSE due to the deviation of the model, that is, if the optimality of an estimator vitiates slightly under the deviation of model, it could be termed as robust one. Thus, it can be stated that the general aim of the model approach is to find out a strategy which performs well in some broad sense allowing for our uncertainty about the assumed model, that is, a strategy which is almost insensitive to errors in the model. Since a large number of theoretical models may be thought of, it is quite impossible to examine the robustness of an estimator theoretically under the deviations of the models. It is, therefore, advisable to examine the robustness of the estimator under some working models with known parameters.

8.2 It is also appropriate to have a study of comparison of MSEs of estimators \(T_{1}^{*} \left( \alpha \right)\),\(T_{s}^{*} \left( \alpha \right)\) and \(t_{NR}^{*} (\alpha )\), all of which are developed under same set-up. Based upon an empirical data, such a comparison has also been presented in the next section.

9 Empirical data and results

9.1 We have taken a real data Singh et al. (Singh et al. 2017). We have considered two different (15% and 30%) non—response rate. The number of dwelling is \(x_{t}\) and dwelling occupied with \(y_{t}\). The following values are as:

\(N\) = 90, \(\beta_{0}\) = 0.8787, \(\beta_{1}\) =—4.9157, \(\overline{X}\) = 41.4556, \(\sigma^{2}\) = 0.7998.

(i) The values of non-response rate at 15% are:

\(n\) = 20, \(n_{1}\) = 17, \(n_{2}\) = 3, \(h_{2}\) = 2, \(f_{2}\) = 1.5, \(\overline{x}\) = 39.55, \(\overline{x}_{{s_{1} }}\) = 39.824, \(\overline{x}_{{s_{2} }}\) = 38.0,\(\overline{x}_{{s_{{h_{2} }} }}\) = 30.5, \(\sum\limits_{{s_{1} }} {x_{t} }\) = 677, \(\sum\limits_{{s_{h2} }} {x_{t} }\) = 61, \(\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{t} }\) = 53, \(\sum\limits_{{\overline{s}}} {x_{t} }\) = 2940 \(\sum\limits_{{s_{1} }} {x_{t}^{2} } =\) 36,729, \(\sum\limits_{{s_{{h_{2} }} }} {x_{t}^{2} } =\) 2081,

\(\sum\limits_{{\overline{s}}} {x_{t}^{2} } =\) 179,614,\(\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{t}^{2} } =\) 2809.

(ii) The values of non-response rate at 30% are:

\(n\) = 20, \(n_{1}\) = 14, \(n_{2}\) = 6, \(h_{2}\) = 4, \(f_{2}\) = 1.5, \(\overline{x}\) = 39.55, \(\overline{x}_{{s_{1} }}\) = 36.5, \(\overline{x}_{{s_{2} }}\) = 46.6667, \(\overline{x}_{{s_{{h_{2} }} }}\) = 50, \(\sum\limits_{{s_{1} }} {x_{t} }\) = 511, \(\sum\limits_{{s_{h2} }} {x_{t} }\) = 200, \(\sum\limits_{{\overline{s}_{{h_{2} }} }} {x_{t} }\) = 80, \(\sum\limits_{{\overline{s}}} {x_{t} }\) = 2940, \(\sum\limits_{s1} {x_{t}^{2} } =\) 25,337, \(\sum\limits_{{\overline{s}}} {x_{t}^{2} } =\) 179,614,\(\sum\limits_{{\overline{s}h2}} {x_{t}^{2} } = 3328\).

9.2 Tables 1, 2, 3, 4, 5, 6 details the MSEs of stretegies \(T_{1}^{*} \left( \alpha \right)\), \(T_{s}^{*} \left( \alpha \right)\) & \(t_{NR}^{*} (\alpha )\) for \(\alpha = 0,1\) and -1 with 15% and 30% non response rates over Models I-VI.

Table 1 The D-MSE of the estimators (\(\alpha\) = 0) for the Models I to VI in presence of 15% of non-response rate
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Table 2 The D- MSE of the estimators (\(\alpha\)= 0) for the Models I to VI in presence of 30% of non-response rate
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Table 3 The D-MSE of the estimators (\(\alpha\)= 1) for the Models I to VI in presence of 15% of non-response rate
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Table 4 The D-MSE of the estimators (\(\alpha\)= 1) for the Models I to VI in presence of 30% of non-response rate
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Table 5 The D-MSE of the estimators (\(\alpha\)=—1) for the Models I to VI in presence of 15% of non-response rate
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Table 6 The D-MSE of the estimators (\(\alpha\)=—1) for the Models I to VI in presence of 30% of non-response rate
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10 Conclusions

We obtained the following point from the tables:

(i) The proposed estimator \(T_{1}^{*} \left( \alpha \right)\) is nearly robust for models I, II, IV and V regardless of the choice of \(\alpha\) of a given nonresponse rate. There is a significant change in the MSE value of the estimator for models III and VI compared to models I, II, IV and V, but for a fixed non-response rate. The estimator can again be considered robust under the variance function \(x^{2}\), regardless of the choice of \(\alpha\). Thus, the estimator is not affected under misspecification of \(h\left( {x_{t} } \right)\) when the variance function is \(x_{t}^{g}\) for \(g = 0\) and 1. Similarly, the conclusion for the variance function \(x^{g}\) for \(g = 2\).

(ii) \(T_{1}^{*} \left( \alpha \right)\) is an efficient or sometimes better than the estimators \(T_{s}^{*} \left( \alpha \right)\) and \(t_{NR}^{*} (\alpha )\) in terms of estimators accuracy.

(iii) It is interesting to note that for \(\alpha\) =0, \(T_{s}^{*} \left( \alpha \right)\) coincides with the estimator \(T_{1}^{*} \left( \alpha \right)\) under all the models and non-response rates. This is because of the reason that for \(\alpha\) =0, both estimators reduces to \(T_{1}^{*} (0) = \overline{y}_{w}^{*} = T_{s}^{*} (0)\).

However, the conclusions drawn above are based on solely on empirical data and a particular configuration of the sample for different non-response rates. Therefore, a comparison of the estimators at different non-response rates would not be possible due to changing in the sample configuration. Furthermore, the results presented here are limited to the data at hand so no consistent conclusions can be drawn. Clearly, results may change with other data. The presentation made here is only an attempt to get an idea of the nature of the proposed family on the misspecifications of PRMs.