Novel stability criterion for linear system with two additive time-varying delays using general integral inequalities

1.   Introduction

The basic objective of the survey practitioners in sample surveys is to obtain an efficient estimate of an unknown population parameter. Therefore, in sequence of improving the efficiency of estimators of parameters, the survey practitioners usually consider the additional information on an auxiliary variable $X$ that is correlated with the study variable $Y$. ^[1] suggested the traditional ratio estimator of population mean under simple random sampling $(SRS)$ provided the variable $Y$ is positively correlated with the variable $X$. ^[2] investigated the traditional product estimator of population mean provided the variable $Y$ is negatively correlated with the variable $X$. ^[3] mooted the exponential ratio and product estimators of population mean based on $SRS$. ^[4] introduced an improved mean estimation procedure under $SRS$. ^[5] proposed Kernel-based estimation of $P(X > Y)$ in ranked set sampling ($RSS$) whereas ^[6] developed an interval estimation of $P(X < Y)$ in $RSS$. ^[7] introduced entropy estimation from ranked set samples with application to test of fit. ^[8] suggested reliability estimation in multistage ranked set sampling ($MRSS$) whereas ^[9] investigated the estimation procedure of a symmetric distribution function in $MRSS$. Recently, ^[10,11,12] suggested various improved classes of estimators under $RSS$.

In real life scenarios, situations may also arise when the survey practitioners may be interested in evaluating the mean value of the variable being quantified for the non-sampled units with the help of available sample data. This approach is popularly established as predictive method of estimation which is based on superpopulation models and thus it is also established as model-based approach. This approach presumes that the parent population is a realization of random variables concerning to a superpopulation model. Under this superpopulation, the prior information about the population parameters namely variance, standard deviation, mean, coefficient of variation, etc is utilized to predict the non-sampled values of the study variable.

^[13] developed some predictive estimators of population mean based on conventional mean, ratio and regression estimators as predictors for the mean of unobserved units in the population. Later on, ^[14] constructed predictive estimator of population mean using classical product estimator as a predictor for the mean of an unobserved units in the population and compared it with the conventional product estimator. Further, ^[15] introduced predictive estimators based on ^[3] exponential ratio and product estimators as predictors for the mean of an unobserved units of the population. Readers may also refer to few recent related studies like, ^[16,17,18] for more detailed study of predictive estimation approach.

The objective of the present manuscript is to proffer few novel logarithmic type predictive estimators under $SRS$ for the mean of unobserved units of the population. The paper is organized in few sections. The Section 2 considers a thorough review of the existing predictive estimators and their properties. In Section 3, the proffered predictive estimators are given with their properties. The efficiency conditions are presented in Section 4 followed by a broad computational study given in Section 5. Lastly, the manuscript is ended in Section 6 with the conclusion.

2.   Conventional predictive estimators

Consider a finite population $\kappa = (\kappa_1, \kappa_2, ..., \kappa_N)$ consist of $N$ identifiable units labeled as $1$, $2$, ..., $N$. Let $(x_i, y_i)$ be the observations on $i^{th}$ population unit of the variables $(X, Y)$. Let $\bar{x}$, $\bar{y}$ and $\bar{X}$, $\bar{Y}$ respectively be the sample means and population means of variables $X$ and $Y$. It is presumed that the population mean $\bar{X}$ of variable $X$ is known and the population mean $\bar{Y}$ of variable $Y$ is computed by measuring a random sample of size $n$ from the population $\kappa$ utilizing simple random sampling with replacement $(SRSWR)$. Let $S$ be the aggregate of all possible samples from population $\kappa$ such that for any given $s\in S$, let $\vartheta(s)$ be the number of specified units in $s$ and $\bar{s}$ be the set of all those units of $\kappa$ that are not in $s$.

The usual mean estimator of population mean $\bar{Y}$ consist of sampled units is given by

The usual mean estimator of population mean $\bar{Y}$ consist of non-sampled units is given by

^[13] mooted a model based predictive approach in which a model is defined to predict the non-sampled values. Thus, under $SRS$ for any given $s\in S$, we have the following model:

Under $SRS$ with size $\vartheta(s) = n$, the predictor for overall population mean is stated as

Thus, the estimator for estimating the population mean $\bar{Y}$ is stated as

where $T$ is the predictor of the mean $\bar{Y}_{\bar{s}}$ of unobserved values which is given as

where $\bar{x}_s = n^{-1}\sum_{i \in s}x_i$ and $\bar{X}_{\bar{s}} = (N-n)^{-1}\sum_{i \in \bar{s}}x_i = (N\bar{X}-n\bar{x}_s)/(N-n)$. Also, $b$ is the regression coefficient of $Y$ on $X$, $\beta_1$ and $\beta_2$ are duly opted scalars.

Now, corresponding to every predictors $T_i, \; i = 1, 2, ..., 8$, we obtain the predictive estimators $t_i, \; i = 1, 2, ..., 8$ using (2.5) as

where $f = n/N$.

^[13] demonstrated that while using the usual mean estimator, ratio estimator and regression estimator as predictor $T_i, \; i = 1, 2, 3$ respectively, the predictive estimator $t_i, \; i = 1, 2, 3$ becomes the corresponding usual mean estimator $T_1$, ratio estimator $T_2$ and regression estimator $T_3$ respectively. Further, ^[14] demonstrated that when product estimator $T_4$ is used as predictor, the predictive estimator $t_4$ is rather different from the usual product estimator $T_4$. Later on, ^[15] demonstrated that when ^[3] exponential ratio and product estimators are used as predictor, the corresponding predictive estimators are rather different from the natural estimators $T_i, \; i = 5, 6$ respectively. It is also observed that when the log type estimators envisaged by ^[15] are used as predictor, the corresponding predictive estimators are found to be rather different from the customary estimators $T_i, \; i = 7, 8$.

To enhance the efficiency of the conventional estimators, ^[20] investigated a technique by multiplying a regulating constant $\phi\; (say)$ whose optimum value depend on the coefficient of variation which is a fairly stable quantity. Using ^[20] procedure, ^[16] defined the following improved estimators corresponding to the predictive estimators $t_{i}, \; i = 1, 2, 4$ as

where $\phi_i, \; i = 1, 2, 3$ are duly opted scalars to be determined.

Further, ^[16] developed the ^[20] based predictive estimators corresponding to the predictive estimators $t_{i}, \; i = 5, 6$ as

where $\phi_4$ and $\phi_5$ are duly opted scalars to be determined.

^[17] suggested regression type predictive estimator corresponding to the predictive estimator $t_3$ as

where $\phi_6$ is a duly opted scalar to be determined.

The readers may refer to appendix $A$ for the properties like, bias and mean square error $(MSE)$ of the above predictive estimators.

3.   Proposed predictive estimators

The motivation of this study is to examine an efficient alternative to survey practitioners under $SRS$. These predictive estimators provide a better alternative to the existing predictive estimators discussed in the previous section. In our proposal, motivated by ^[21], we suggest few novel logarithmic predictive estimators corresponding to the predictive estimators $t_i, \; i = 1, 2$ for the population mean $\bar{Y}$ as

where $\phi_7$, $\phi_8$ and $\beta_i, \; i = 1, 2$ are duly opted scalars.

Theorem 3.1. The bias and minimum $MSE$ of the proffered predictive estimators $t_{sb_i}, \; i = 1, 2$ are given by

where $\phi_{j(opt)} = \frac{Q_i}{P_i}$, $P_1 = 1+f_1C_y^2+\left\{\beta_1(\beta_1-1)+\beta_1f+\frac{\beta_1f^2}{(1-f)}+\frac{\beta_1(\beta_1-1)}{(1-f)}\right\}$$f_1C_x^2+4\beta_1f_1\rho_{xy}C_xC_y$, $Q_1 = 1+\beta_1f\rho_{xy}C_xC_y$$-\frac{\beta_1}{2}\left\{\frac{(1-2f)}{(1-f)}-\frac{(\beta_1-1)}{(1-f)}\right\}f_1C_x^2$, $P_2 = 1+f_1C_y^2+\beta_2\left\{\beta_2-\frac{(1-2f)}{(1-f)}\right\}f_1C_x^2+4\beta_2f_1\rho_{xy}C_xC_y$ and $Q_2 = 1+\beta_2f_1\rho_{xy}C_xC_y-\frac{\beta_2(1-2f)}{2(1-f)}f_1C_x^2$.

Proof. To derive the expressions of bias and $MSE$ of various predictive estimators, let us assume that $\bar{y} = \bar{Y}(1+\epsilon_0)$, $\bar{x} = \bar{X}(1+\epsilon_1)$, such that $E(\epsilon_0) = E(\epsilon_1) = 0$, $E({\epsilon_0}^2) = f_1{C_y^2}$, $E({\epsilon_1}^2) = f_1{C_x^2}$ and $E({\epsilon_0, \epsilon_1}) = f_1\rho_{xy}{C_xC_y}$.

where $f_1 = (n^{-1}-N^{-1})\cong1/n$. Also, $C_x$ and $C_y$ are respectively the population coefficient of variations of variables $X$ and $Y$ and $\rho_{xy}$ is the population coefficient of correlation between variables $X$ and $Y$.

Using the above notations, we convert $t_{sb_1}$ in $\epsilon's$ as

Taking expectation both the sides of (3.5), we get

Similarly, we can obtain bias of predictive estimator $t_{sb_2}$.

Now, squaring and applying expectation both the sides of (3.5), we get

which can be written as

On differentiating the above $MSE$ expression regarding $\phi_7$ and equating to zero, we get

Putting the value of $\phi_{7(opt)}$ in the $MSE(t_{sb_1})$, we get

Similarly, the derivations of $MSE$ of the estimator $t_{sb_2}$ can be obtained. In general, we can write

We note that the simultaneous optimization of $\phi_j$ and $\beta_i$ of the $MSE$ equation is not possible. So, we get the optimum values of $\beta_i$ = $\beta_{i(opt)}$ given $\phi_j$ = 1 and put it inside $\phi_j$ = $\phi_{j(opt)}$ to get (3.4). The optimum values of scalars $\phi_j$ are given by

where

The optimum values of $\beta_i, \; i = 1, 2$ are given by

We would like to note that the $MSE$ expression stated in (3.4) is important in order to determine the efficiency conditions of next sections.

Corollary 3.1. The proposed predictive estimator $t_{sb_1}$ dominate the proposed predictive estimator $t_{sb_2}$, iff

and contrariwise. Otherwise, both are equally efficient when equality holds in (3.15).

Proof. On comparing the minimum $MSE$ of both the proffered estimators, we get (3.15). We can merely obtain (3.15) whether it retains in practice is through a computational study carried out in Section 5.

4.   Efficiency conditions

In the present section, the efficiency conditions are derived by comparing the minimum $MSE$ of the proffered predictive estimators $t_{sb_i}, \; i = 1, 2$ from (3.4):

$(1)$ with the $MSE$ of the predictive estimator $t_1$ from (A.1) and get,

$(2)$ with the $MSE$ of the predictive estimator $t_2$ from (A.3) and get,

$(3)$ with the minimum $MSE$ of the predictive estimator $t_3$ from (A.4) and get

$(4)$ with the $MSE$ of the predictive estimator $t_4$ from (A.8) and get

$(5)$ with the minimum $MSE$ of the predictive estimator $t_5$ from (A.10) and get

$(6)$ with the minimum $MSE$ of the predictive estimator $t_6$ from (A.12) and get

$(7)$ with the minimum $MSE$ of the predictive estimator $t_7$ from (A.15) and get

$(8)$ with the minimum $MSE$ of the predictive estimator $t_8$ from (A.18) and get

$(9)$ with the minimum $MSE$ of the predictive estimator $t_{9}$ from (A.19) and get

$(10)$ with the minimum $MSE$ of the predictive estimator $t_{10}$ from (A.20) and get

$(11)$ with the minimum $MSE$ of the predictive estimator $t_{14}$ from (A.28) and get

$(12)$ with the minimum $MSE$ of the predictive estimator $t_{11}$ from (A.21) and get

$(13)$ with the minimum $MSE$ of the predictive estimator $t_{12}$ from (A.24) and get

$(14)$ with the minimum $MSE$ of the predictive estimator $t_{13}$ from (A.27) and get

Under the above conditions, the proffered predictive estimators dominate the reviewed predictive estimators in $SRS$. Further, these conditions hold in practice is verified through a broad computational study using various real and artificially generated symmetric and asymmetric populations. Also, it is worth mentioning that the population coefficient of variations and coefficient of correlation are stable quantities and therefore, the optimum values of both proposed and existing estimators can be estimated using sample data.

5.   Computational study

In tandem of the theoretical results, a broad computational study is carried out under the four heads namely, numerical study using real populations, simulation study using real populations, simulation study using artificially generated symmetric and asymmetric populations and discussion of computational results.

5.1.   Numerical study using real populations

We consider six natural populations to perform the numerical study. The source of the populations, the nature of the variables $Y$ and $X$ and the values of different parameters are described below.

Population 1: Source: (^[22], pp. 1115), $Y$ = season average price per pound during 1996, $X$ = season average price per pound during 1995, $N$ = 36, $n$ = 12, $\bar{Y}$ = 0.2033, $\bar{X}$ = 0.1856, $S_y^2$ = 0.006458, $S_x^2$ = 0.005654 and $\rho_{xy}$ = 0.8775.

Population 2: Source: (^[22], pp. 1113), $Y$ = duration of sleep (in minutes), $X$ = age of old persons ($\ge$ 50 years), $N$ = 30, $n$ = 8, $\bar{Y}$ = 384.2, $\bar{X}$ = 67.267, $S_y^2$ = 3582.58, $S_x^2$ = 85.237 and $\rho_{xy}$ = -0.8552.

Population 3: Source: (^[23], pp. 228), $Y$ = output for 80 factories in a region, $X$ = number of workers for 80 factories in a region, $N$ = 80, $n$ = 35, $\bar{Y}$ = 5182.637, $\bar{X}$ = 285, $S_y^2$ = 3369642, $S_x^2$ = 73188.3 and $\rho_{xy}$ = 0.9150.

Population 4: Source: (^[24], pp. 653-659), $Y$ = real estate values according to 1984 assessment (in millions of kroner), $X$ = number of municipal employees in 1984, $N$ = 284, $n$ = 75 $\bar{Y}$ = 3077.525, $\bar{X}$ = 1779.063, $S_y^2$ = 22520027, $S_x^2$ = 18089178 and $\rho_{xy}$ = 0.94.

Population 5: Source: (^[22], pp. 1116), $Y$ = number of fish caught by marine recreational fisherman in 1995, $X$ = number of fish caught by marine recreational fisherman in 1993, $N$ = 69, $n$ = 28 $\bar{Y}$ = 4514.89, $\bar{X}$ = 4591.07, $S_y^2$ = 37199578, $S_x^2$ = 39881874 and $\rho_{xy}$ = 0.9564.

Population 6: The data is chosen from ^[25] based on apple production and number of apple trees in 7 regions of Turkey during 1999. However, we take only the data of South Anatolia region consist of 69 villages. (Origin: Institute of Statistics, Republic of Turkey). The essential statistics are presented as, $Y$ = amount of apple yield in South Anatolia region, $X$ = quantity of apple trees in South Anatolia region, $N$ = 69, $n$ = 22 $\bar{Y}$ = 71.347, $\bar{X}$ = 3165.029, $S_y^2$ = 12289.72, $S_x^2$ = 15723128 and $\rho_{xy}$ = 0.9177.

For the above populations, we have calculated the percent relative efficiency $(PRE)$ of different predictive estimators $T$ with respect to $(w.r.t.)$ the usual mean estimator $t_1$ as follows.

The results of the numerical study calculated for the above discussed populations are displayed in Table 1 by $MSE$ and $PRE$.

5.2.   Simulation study using real populations

In order to generalize the findings of numerical study, a simulation study is carried out using some real populations. The steps involved in the simulation study are as follows:

$\textbf{Step 1}.$ Consider the real populations discussed in subsection 5.1.

$\textbf{Step 2}.$ Draw a simple random sample of size given in the respective populations using $SRSWR$ scheme.

$\textbf{Step 3}.$ Compute the necessary statistics.

$\textbf{Step 4}.$ Iterate the above steps 10,000 times and compute the $MSE$ and $PRE$ of various estimators.

The simulated $PRE$ is computed as

The outcomes of the simulation study consist of the real populations are reported in Table 2 by $MSE$ and $PRE$.

5.3.   Simulation study using artificially generated populations

Following ^[26], we accomplish a simulation study using some artificially rendered populations. The simulation steps are are given as follows:

$\textbf{Step 1.}$ Generate two families of symmetric populations such as Normal and Logistic and two families of asymmetric populations such as Gamma and Weibull each of size $N$ = 500. The data on variables $X$ and $Y$ are generated through the models $Y = 8.4+\sqrt{(1-\rho_{xy}^2)}\; Y^*+\rho_{xy}\left(S_y/S_x\right)X^*$ and $X = 4.4+X^*$ with particular values of parameters given in Tables 3 and 4.

$\textbf{Step 2.}$ Draw a bivariate simple random sample of size $n$ = 50 using $SRSWR$ scheme from each population.

$\textbf{Step 3.}$ Compute the required statistics.

$\textbf{Step 4.}$ Iterate the above steps 10,000 times.

We have taken different values of correlation coefficient $\rho_{xy} = 0.3, 0.5, 0.7, 0.9$ to observe the deportment of the proffered predictive estimators. The $MSE$ and simulated $PRE$ of different predictive estimators $T$ regarding the usual mean estimator $t_1$ are computed using the expression given in (5.2).

The simulation results for both the populations are displayed in Tables 3 and 4 by $MSE$ and $PRE$ for various values of correlation coefficient $\rho_{xy}$.

5.4.   Discussion of computational results

The following discussion is drawn from the computational results displayed from Tables 1 to 4.

(i) From Table 1 consists of the results of numerical study of six real populations, the proposed predictive estimators $t_{sb_i}, \; i = 1, 2$ show their ascendancy over the existing predictive estimators $t_i, \; i = 1, 2, ..., 14$ by minimum $MSE$ and maximum $PRE$. The dominance of the proposed predictive estimators can also be observed from the histogram drawn from Figures 1 to 6 for $MSE$ and $PRE$.

(ii) The similar inclination can be observed from the findings of simulation study of Table 2 consist of the six real populations.

(iii) From Table 3 based on the simulation results for symmetric populations such as Normal and Logistic with different values of $\rho_{xy}$ also exhibit the ascendancy of the proposed predictive estimators $t_{sb_i}, \; i = 1, 2$ over the existing predictive estimators $t_i, \; i = 1, 2, ..., 14$ by minimum $MSE$ and maximum $PRE$.

(iv) The similar conclusion can be drawn from Table 4 based on the asymmetric populations such as Gamma and Weibull.

(iv) From Tables 3 and 4 consist of the simulation results using artificially generated populations, it can be observed that the $MSE$ of the proffered predictive estimators gradually declines as the value of correlation coefficient $\rho_{xy}$ increases and contrariwise in sense of $PRE$ in each population.

(v) Furthermore, from Tables 1 to 4 the proffered predictive estimator $t_{sb_1}$ is found to be superior than the proposed predictive estimator $t_{sb_2}$.

6.   Conclusions

In this manuscript, we have developed few novel logarithmic predictive estimators of population mean in $SRS$. The properties like bias and $MSE$ of the proffered logarithmic predictive estimators are determined to the first order of approximation. The efficiency conditions have been obtained which are successively enhanced by a broad computational study using various real and artificially generated symmetric and asymmetric populations. From the computational results listed from Tables 1 to 4, we observe that:

(i) The proffered predictive estimators $t_{sb_i}, \; i = 1, 2$ are found to be most efficient than the usual unbiased, ratio and regression predictive estimators due to Basu (1971), product predictive estimator due to Srivastava (1983), Bahl and Tuteja (1991) exponential ratio and product type predictive estimators, logarithmic type predictive estimators, Searls (1964) based predictive estimators defined and proposed by Singh et al. (2019) and Bhushan et al. (2020) predictive estimator.

(ii) The correlation coefficient $\rho_{xy}$ demonstrate adverse effect over the $MSE$ and favorable effect over the $PRE$ of the proffered predictive estimators $t_{sb_i}, \; i = 1, 2$ which can be seen from the simulation results of Tables 3 and 4.

(iii) The proffered predictive estimator $t_{sb_1}$ performs better than the proposed predictive estimator $t_{sb_2}$ in each real and simulated populations.

Thus, we enthusiastically recommend the utilization of the proffered predictive estimators to the survey professionals in real life. Moreover, in forthcoming studies, we are intended to develop the proposed predictive estimators using ranked set sampling.

Conflict of interest

The authors have no conflict of interest.

Appendix A

The variance of predictive estimator $t_1$ is given by

The bias and $MSE$ of predictive estimator $t_2$ are given by

The $MSE$ of predictive estimator $t_3$ is given by

The optimum value of $b$ is obtained by minimizing $(A.4)$ w.r.t. $b$ as

The minimum $MSE$ at optimum value of $b$ is given by

The bias and $MSE$ of predictive estimator $t_4$ are given by

where $f = n/N$.

The bias and $MSE$ of predictive estimator $t_5$ are given by

The bias and $MSE$ of predictive estimator $t_{6}$ are given by

The $MSE$ of predictive estimator $t_7$ is given by

The optimum value of $\beta_1$ is obtained by minimizing $(A.13)$ w.r.t. $\beta_1$ as

The minimum $MSE$ at optimum value of $\beta_1$ is

The $MSE$ of predictive estimator $t_8$ is given by

The optimum value of $\beta_2$ is obtained by minimizing $(A.16)$ w.r.t. $\beta_2$ as

The minimum $MSE$ at optimum value of $\beta_2$ is

The minimum $MSE$ of predictive estimator $t_9$ under $SRS$ is given by

The minimum $MSE$ of predictive estimator $t_{10}$ under $SRS$ is given by

where $\phi_{2(opt)} = {(\bar{Y}^2+\bar{Y}Bias(t_2))}/{(\bar{Y}^2+MSE(t_2)+2\bar{Y}Bias(t_2)})$.

The minimum $MSE$ of predictive estimator $t_{11}$ is given by

where $\phi_{3(opt)} = {(\bar{Y}^2+\bar{Y}Bias(t_4))}/{(\bar{Y}^2+MSE(t_4)+2\bar{Y}Bias(t_4))}$.

The $MSE$ of predictive estimator $t_{12}$ is given by

The optimum value of $\phi_4$ is obtained by minimizing $(A.22)$ w.r.t. $\phi_4$ as

The minimum $MSE$ at the optimum value of $\phi_4$ is given by

The $MSE$ of predictive estimator $t_{13}$ is given by

The optimum value of $\phi_5$ is obtained by minimizing $(A.25)$ w.r.t. $\phi_5$ as

The minimum $MSE$ at the optimum value of $\phi_5$ is given by

The minimum $MSE$ of predictive estimator $t_{14}$ under $SRS$ is given by

where $\phi_{6(opt)} = {(\bar{Y}^2+\bar{Y}Bias(t_3))}/{(\bar{Y}^2+MSE(t_3)+2\bar{Y}Bias(t_3))}$.