Approximation of Finite Population Totals Using Lagrange Polynomial

Approximation of finite population totals in the presence of auxiliary information is considered. A polynomial based on Lagrange polynomial is proposed. Like the local polynomial regression, Horvitz Thompson and ratio estimators, this approximation technique is based on annual population total in order to fit in the best approximating polynomial within a given period of time (years) in this study. This proposed technique has shown to be unbiased under a linear polynomial. The use of real data indicated that the polynomial is efficient and can approximate properly even when the data is unevenly spaced.


Introduction
This study is using an approximation technique to approximate the finite population total called the Lagrange polynomial that doesn't require any selection of bandwidth as in the case of local polynomial regression estimator. The Lagrange polynomials are used for polynomial interpolation and extrapolation. For each given set of distinct points x j and y j , the Lagrange polynomial of the lowest degree takes on each point x j corresponding to y j (i.e. the functions coincide at each point). Although named after Joseph Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler as will be seen later on how it works.  [1] in the context of using auxiliary information from survey data to estimate the population total defined 1 2 , , , N U U U as the set of labels for the finite population. Letting ( ) ∈ be the first order inclusion probabilities. In 1940, Cochran made an important contribution to the modern sampling theory by suggesting methods of using the auxiliary information for the purpose of estimation in order to increase the precision of the estimates [2]. He developed the ratio estimator to estimate the population mean or the total of the study variable y. The ratio estimator of population Y is of the form The aim of this method is to use the ratio of sample means of two characters which would be almost stable under sampling fluctuations and, thus, would provide a better estimate of the true value. It has been well-known fact that r y is most efficient than the sample mean estimator y , where no auxiliary information is used, if ρ yx , the coefficient of correlation between y and x, is greater than half the ratio of coefficient of variation of x to that of y, that is, if Thus, if the information on an auxiliary variable is either already available or can be obtained at no extra cost and it has a high positive correlation with the main character, one would certainly prefer ratio estimator to develop more and more superior techniques to reduce bias and also to obtain unbiased estimators with greater precision by modifying either the sampling schemes or the estimation procedures or both. [3] further extended the work of [4] on systematic sampling.
[5] also dealt with the problem of estimation using the priori-information. Contrary to the situation of ratio estimator, if variables y and x are negatively correlated, then the product estimator of population mean Y is of the form that was proposed by [6]. It has been observed that the product estimator gives higher precision than the sample mean estimator y under the condition that is The expressions for bias and mean square errors of r y and q y have been derived by [7].  This estimator is biased, the bias being negligible for large samples.
The most common way of defining a more efficient class of estimators than usual ratio (product) and sample mean estimator is to include one or more unknown parameters in the estimators whose optimum choice is made by minimizing the corresponding mean square error or variance. Sometimes, such modifications or generalizations are made by mixing two or more estimators with unknown weights whose optimum values are then determined which generally depend upon population parameters. In order to propose efficient classes of estimators, [9] suggested a one-parameter family of factor-type (F-T) ratio estimators defined as where ( )( ) The literature on survey sampling describes a great variety of techniques of using auxiliary information to obtained more efficient estimators.
Keeping this fact in view, a large number of authors have paid their attention toward the formulation of modified ratio and product estimators using information on an auxiliary variate, for instance, see [10] and Singh et al. [11]. Suppose n is large and We assume that x and X are quite close such that so that the bias of R becomes quite small.
The concept of nonparametric models within a model assisted framework was first introduced by [ The first term in (1.8) is a design estimator which the second is model component. Therefore, when the sample comprises of the whole population, the model component reduces to zero since π i = 1 and s = N. We therefore have the actual population total. [13] proposed the super population model ξ, such that is a known function of x i . They proposed model calibration estimator for population total Y t to be In local polynomial regression, a lower-order weighted least squares (WLS) regression is fit at each point of interest, x using data from some neighborhood around x. Following the notation from [14], let the (X i , Y i ) be ordered pairs such is the variance of Y i at the point X i , and X i comes from some distribution, f. In some cases, homoscedastic variance is assumed, so we let It is typically of interest to estimate m(x). Using Taylor's expansion: We can estimate these terms using weighted least squares by solving the following for β: In (1.92), h controls the size of the neighborhood around x 0 , and K h (.) controls the weights, where ( ) , and K is a kernel function. Denote the solution to (1.92) as β . Then estimated . [15] proposed to use nonparametric method to obtain ( ) . µ . However, this estimator experiences a twin problem of how to determine the optimal degrees of the local polynomial. A higher degree polynomial yields a smoother ( ) . µ but worsens the boundary variance [16]. Such estimators are challenging to employ in cases of multiple covariates and when data is sparse. Another challenge is how to incorporate categorical covariates. It is therefore necessary to consider other methods to recover the fitted values such as splines. The term spline originally referred to a tool used by draftsmen to draw curves. According to [17], splines are piecewise regression functions we constrain to join at points called knots.
The Horvitz-Thompson (HT) estimator, which is originally discussed by [18] L. Kabareh N with units 1 2 3 , , , , N y y y y . Suppose we want to select sample s of size n s . Let π i be the probability of including i th unit of the population in sample s. This is called the inclusion probability or first order inclusion probability of i th unit in the sample.
Let π ij be the probability of including i th and j th units in the sample. This is called the joint inclusion probability or second order inclusion probability.
When the sample is obtained from a probability sampling design, an unbiased estimator for the Total is unbiased under design based approach [19] Variance ( ) The variance of this estimator can be minimized when π i ∝ y i . That is, if the first order inclusion probability is proportional to y i , the resulting HT estimator under this sampling design will have zero variance. However, in practice, we can't construct such design because we don't know the value of y i in the design stage. If there is a good auxiliary variable x i which is believed to be closely related with y i , then a sampling design with π i ∝ x i can lead to very efficient sampling design This method of estimating the finite population totals doesn't make use of the auxiliary information x i but instead uses only the study variable y i to obtain the population totals.
Research literature has revealed that the ratio estimator performs better than the local linear polynomial estimator when the population is linear no matter which variance is used. The local linear polynomial regression estimator becomes a better estimator when the population used is either quadratic or exponential especially with an increase in the sample size which increases the likelihood of outliers in the sample.
One of the most useful and well-known classes of functions mapping the set of real numbers into itself is algebraic polynomials, the set of functions of the form ( ) where n is a non-negative integer and 0 , , n a a are real constants. One reason for their importance is that they uniformly approximate continuous functions.
By this we mean that given any function, defined and continuous on a closed and bounded interval, there exists a polynomial that is as "close" to the given function as desired [20]. Open Journal of Statistics that, the best approximating polynomial for a quick convergence must be a linear one in order to give a better extrapolation.

Approximation of Finite Population Totals
In this section, we are basically introducing an approximator that is the Lagrange polynomial approximate of the finite population totals.

Proposed Lagrange Polynomial
From the population U, a simple random sample of size n is drawn without replacement. Then, the Lagrange interpolating polynomial is the polynomial p(x) of degree ≤ (n − 1) that passes through the n points ( ) ( ) and is given by:

Asymptotic Properties of Polynomial Approximations
In order to obtain a best approximating polynomial that has less error, one needs to choose a linear interpolating points that is closest to the target point

Data Exploration
The plot showed an upward growth in the population of Kenya. This could be attributed to good health services causing a reduction in the maternal death, deaths as a result of disease outbreak, a boost in socio-economic growth and political stability (Figure 1). iformly approximate the function f(x) representing the total population trend per year. The chart has clearly shown that, the black dotted line depicted the best Figure 2. This chart was obtained from a set of data ranging from [1969,1979] in yellow to [1969,1989] in blue and the green function (f(x)). Figure 3. This chart was obtained from a set of data ranging from [1969,1999] in red to [1969,2009] in blue and the green function (f(x)).  . This chart was obtained from a set of data ranging from [1979,1989] in green dotted line to [1979,1999] in red and the green function(f(x)). Figure 5. This chart was obtained from a set of data ranging from [1979,2009] in black to [1989,1999] in blue and the green function(f(x)). Figure 6. This chart was obtained from a set of data ranging from [1989,2009] in red dotted line to [1999,2009] in black dotted line and the green function (f(x)).
approximate on its entire interval which is [1999,2009]

Conclusion
In this work, the Lagrange polynomial has proven to be a good technique in approximating the population total from data obtained from the Kenya National Bureau of Statistics (KNBS). The research revealed that, subsequent population totals can better be approximated using a sample closest to the target population being approximated. Therefore, the best approximating polynomial must be a linear form in order to obtain convergence with a diminishing variation in a given interval. The precision of this technique can better be measured with the outcome obtained in the interpolation of missing values shown in the results above to extrapolate the population total in 2009 which was equal to the exact population obtained in that census. We therefore conclude that, the population of Kenya for the 2019 census will be forty-eight million five hundred and thirty-three thousand five hundred and eighty-seven.