Ratio Estimator

We also generalize the ratio estimator to two extensions of the quantification problem. In the first scenario, some labels are available in the target population. The combined estimator extends the ratio estimator in order to incorporate these labels and obtain a larger effective sample size. The second scenario considers that the prevalence of each label varies according to additional covariates. This generalization allows one to use unlabeled data to identify e.g. how the approval of a product varies with age. In this scenario, we introduce the regression ratio estimator, which offers improvements over the standard methods that are used in sentiment analyses (Wang et al., 2012).

The rest of the article is arranged as follows. Section 2 suggests a new model- based variance estimator of the ratio estimator. A Monte Carlo comparison of the suggested estimator with the available estimators is made in Section 3. The conclusions are given in Section 4.

ratio estimator of population mean using information on auxiliary attribute and the estimation of population mean in double sampling in ratio form. The proposed modified ratio estimator is a family of estimator which results to different estimators at different value of alpha. For the proposed estimator, when 𝛼 = 0, 0.5 and 1, the estimators of Naik and Gupta (1996), Nirmala Sawan (2010) and Sample mean were recovered respectively. When the auxiliary attribute is a variable, the estimator result to that of Subhash et al. (2016). When α = 0 and the auxiliary attribute is a variable, it results to conventional double sampling. The expression for the Bias, and Mean Square Error of the proposed modified estimator were obtained up to the first order of approximation. An efficiency comparison of both the theoretical and empirical was carried out with some related existing estimators in double sampling. It has been established that the proposed modified ratio estimator is more efficient when compared with the existing ones at optimum value of alpha.

has in general led a relatively smaller error compared to the usual separate ratio estimator. We can therefore conclude that nonparametric regression approach in stratified sampling using the modified kernel smoothing yields very good results.

The biases, the mean squared errors and the constants of the first 5 modified ratio estimators ̅̂ ̅̂ listed in the Table 1.1 are classified into a single class (say, Class 1), which will be very much useful for comparing with that of proposed modified ratio estimator and are given below:

Jhajj, Sharma and Grover [9] proposed a family of estimators using information on auxiliary attribute. They used known information of population proportion possessing an attribute that is highly correlated with study va- riable Y. The attribute is normally used when the auxiliary variable is not available e.g. an amount of milk pro- duced and a particular breed of cow or an amount of yield of wheat and a particular variety of wheat. Rajesh, Pankaj, Nirmala and Florentins [10] used the information on auxiliary attribute in ratio estimator in estimating population mean of the variable of interest using known attributes such as coefficient of variation, coefficient kurtosis and point bi-serial correlation coefficient. The estimator performed better than the usual sample mean and Naik and Gupta [11] estimator. Rajesh, Pankaj, Nirmala and Florentins [10] also used the auxiliary attribute in regression, product and ratio type exponential estimator following the work of Bahl and Tuteja [12].

The bootstrap approach to statistical inference is described in Efron (1982). The method has wide applicability and has seen considerable development in recent years. However, use of the bootstrap in sample survey inference has been somewhat limited. Rao and Wu (1988), describe an application of the bootstrap under the design-based approach to sample survey inference. Sitter (1992a, 1992b), has extended their results to more complex survey designs. More recently, Booth, Butler and Hall (1991) and Booth and Murison (1992) describe a rather different approach to constructing a design-based bootstrap. In this paper we describe how this approach to the bootstrap can be applied under model-based sample survey inference, focussing on an application where the popular ratio estimator is the estimator of choice.

Olkin [1] proposed a ratio estimator considering p auxiliary variables under simple random sampling. As is expected, Simple Random Sampling comes with relatively low levels of precision especially with regard to the fact that its vari- ance is greatest amongst all the sampling schemes. We extend this to stratified random sampling and we consider a case where the strata have varying weights. We have proposed a Multivariate Ratio Estimator for the population mean in the presence of two auxiliary variables under Stratified Random Sampling with L strata. Based on an empirical study with simulations in R statistical software, the proposed estimator was found to have a smaller bias as compared to Olkin’s estimator.

In this study, we proposed a ratio type population mean estimator. The performance of the proposed estimator over the usual ratio estimator and some selected existing estimators using three natural populations in which their properties (Bias and Mean Square Errors (MSEs)) were established and comparing their PREs. Tables 4 and 5 show the results of Mean Square Error (MSE) and Percentage Relative Efficiency (PRE) of the proposed and some related estimators considered in the study for all the data sets I, II, and III. The results revealed that the proposed estimator has minimum MSE and highest PRE than other estimators. This results implies that the average of dispersion of the proposed estimators from the population mean is smaller and this indicate that the proposed estimator give better estimates on the average than other estimators in the study.

However, in many practical situations it is more favourable to use the ratio estimator not only on the grounds of efficiency but also Another advantage of using ratio estimator over regression estimator is that the optimum matched portion, the portion which minimizes the variance of the pooled estimator, is larger in case of ratio estimator In this present work we have proposed a pooled estimator for population mean at the current occasions by using ratio type estimator involving a suitably chosen scalar . We have calculated its variances upto the first order of approximations and obtained the optimum replacement policy. We also have investigated the efficiency of the proposed estimator compared with other conventional estimators with and without cost considerations.

The proposed estimator has been given in (2.1) with the mean square error in (2.4). Now we are comparing proposed estimator with the estimators given by Singh and Vishwakarma (2007), Singh et al. (2008), Noor-ul-Amin and Hanif (2012), Sanaullah et al. (2012) and Yadav et al. (2013). The comparison is done on the basis of MSE and checked the efficiency of proposed estimator with others.

Neurons have very strong biophysical properties such as the absolute and relative refractory periods and bursting propensity. These properties affect how the neuron represents and transmits information about a stimulus (signal). In most current analyses of neural responses to a signal these properties are not considered. Nevertheless, these biophysical properties contribute in a structured, non-random way to the fluctuations in the neural response. Hence, they should not be considered as noise but must be taken into account (corrected for) to assess properly the effect of the signal on the neural response. By incorporating the bias correction our SNR estimator (17) does not increase with the addition of unimportant covariates. Our bias-corrected SNR estimator can give negative values when the true SNR is at or close to zero, suggesting a very low SNR system. The SNR for the simulated neuron of 0.0289 (-15.4 dB) is typical of preliminary results we have obtained from the analysis of actual spiking neurons [25].

In order to improve the efficiency of the estimators, auxiliary information is used at both selection as well as estimation stage. While Cochran, (1940) used auxiliary information at estimation stage and proposed ratio estimator, Murthy(1964) envisaged product estimator and Searl(1964), Sisodia and Dwibedi(1981) utilised co- efficient of variation of auxiliary variable in their respective ratio and product method of estimation. Srivenkataraman(1980) first proposed dual to ratio estimator, Singh and Tailor(2005) and Tailor and Sharma(2009) worked on ratio cum product estimator. Deriving inspiration from the above works completed with the estimator due to Mallick and Tailor(2013), we have proposed a new product -cum-dual to product estimator of finite population mean.

Several authors have used prior value of certain population parameter(s) to find more precise estimates. Sisodiya and Dwivedi (1981), Sen (1978) and Upadhyaya and Singh (1984) used the known coefficient of variation (CV) of the auxiliary character for estimating population mean of a study character in ratio method of estimation. The use of prior value of coefficient of kurtosis in estimating the population variance of study character y was first made by Singh et. al. (1973). Later used by Singh and Kakaran (1993) in the estimation of population mean of study character. Singh and Tailor (2003) proposed a modified ratio estimator by using the known value of correlation coefficient. Kadilar and Cingi (2006), Khosnevisan et. al. (2007), Singh et. al. (2007) Singh and Kumar (2009) and Singh et. al. (2009) have suggested modified ratio estimators by using different pairs of known value of population parameter(s).

to the sample mean y , the asymptotic minimum mean square error of the resultant estimator cannot be reduced further than that given in (1.3). Thus the usual ratio estimator, product estimator and power transformation estimator are the special cases of the class of estimators defined in (1.1). While the regression estimator and difference estimator are not special cases of the general class of estimators. Then Srivastava (1980) t g  y H   u (1.1)

( ( ) [ Under case I and II] are more efficient than the usual unbiased estimator y , ratio estimator in two- phase sampling y R (d ) , two-phase sampling product estimator y P (d ) , two-phase sampling versions of Singh et al. (2004) ratio and product type estimators Y ˆ SR ( d ) and

Simple Random Sampling with sample size (462 to 561) gave moderate variances both by Jacknife and Bootstrap. By applying Systematic Sampling, we received moderate variance with sample size (467). In Jacknife with Systematic Sampling, we obtained variance of regression estimator greater than that of ratio estimator for a sample size (467 to 631). At a sample size (952) variance of ratio estimator gets greater than that of regression estimator. The most efficient design comes out to be Ranked set sampling compared with other designs. The Ranked set sampling with jackknife and bootstrap, gives minimum variance even with the smallest sample size (467). Two Phase sampling gave poor performance.

that it has less variance as compared to usual ratio estimator in SRS. Khan and Shabbir (2015) suggested a class of Hartley-Ross type unbiased estimator in RSS. Khan and Shabbir (2016) have also suggested Hartley-Ross type unbiased estimators in RSS and stratified ranked set sampling (SRSS). Khan et al. (2016) proposed unbiased ratio estimator of finite population mean in SRSS.

sitively correlated with the variable under study, using a linear combination of ratio estimator based on each auxiliary variable. Raj [4] suggested a method of using multi-auxiliary information in sample survey. Using this idea, Singh [5] proposed a multivariate expression of product estimator where the study variable was negatively correlated with the multi-auxiliary variable. In the same year, Singh [6] proposed a ratio-cum-product estimator and its multi-variable expression. Singh and Tailor [7] proposed a ratio-cum-product estimator for finite popula- tion mean in simple random sampling using coefficient of variation and coefficient of kurtosis which was more efficient than the previous ratio-cum product estimator.

Jhajj, Sharma and Grover [11] proposed a family of estimators using information on auxiliary attribute. They used known information of population proportion possessing an attribute (highly correlated with study variable Y). The attribute are normally used when the auxiliary variables are not available e.g. amount of milk produced and a particular breed of cow or amount of yield of wheat and a particular variety of wheat. Jhajj, Sharma and Grover [11] used the information on auxiliary attributes in ratio estimator in estimating population mean of the variable of interest using known attributes such as coefficient of variation, coefficient kurtosis and point bi-serial correlation coefficient. The estimator performed better than the usual sample mean and Naik and Gupta [12] es- timator. Jhajj, Sharma and Grover [11] also used the auxiliary attribute in regression, product and ratio type ex- ponential estimator following the work of Bahl and Tuteja [13].