Ratio Estimation

INTRODUCTION:

In Survey Sampling, we often use information on some auxiliary variable, to improve our estimator of the finite population parameter, by giving our estimator protection against selection of bad sample. One such estimator is the ratio estimator introduced as follows:

In practice, we are often interested to estimate the ratio of the type:

R= \frac{\bar{Y}}{\bar{X}} = \frac{\sum Y_i}{\sum X_i} 

For example, in different socio-economic survey, we may be interested in per-capita expenditure on food-items, infant mortality rate, literacy rate, etc. So, the estimation of R itself is of interest to us and beside that, we can get an improved estimate of Y ̄, as follows:

NOTATIONS:

Population Size N
Population U=(U_1,U_2,\dots,U_N)
Study variable Y=(Y_1,Y_2,\dots,Y_N)
Auxilliary Variable X=(X_1,X_2,\dots,X_N)
Mean of Y(study variable) \bar{Y} = \frac{1}{N} \sum Y_\alpha
Mean of X( auxilliary variable) \bar{X} = \frac{1}{N} \sum X_\alpha
Sample size n
Sample
(drawn by SRSWOR from U)
s=(i_1,i_2,\dots,i_n)
Sample Mean of Y(study variable) \bar{y} = \frac{1}{n} \sum y_i
Sample Mean of X( auxilliary variable) \bar{x} = \frac{1}{n} \sum x_i

Since,  \bar{Y} = R \bar{X}  , where R is unknown but  \bar{X}  is known, we can take

\hat{\bar{Y}}_R = \hat{R} \bar{X} = \frac{\bar{y}}{\bar{x}} \bar{X}

as an estimate of  \bar{Y}  and this estimator is called the ratio estimator of  \bar{Y}  .

RESULT 1:

\hat{\bar{Y}_R} is not an unbiased estimator of \hat{\bar{Y}} and its approximate bias is given by:

B(\hat{\bar{Y_R}}) = \frac{1}{\bar{X}} \frac{1-f}{n} \left( RS_X^2 - S_{XY} \right) 

;where \quad S_X^2= \frac{1}{N-1} \sum_{i=1}^{N} \left( X_i - \bar{X} \right)^2 , \\
\quad \quad  S_{XY}= \frac{1}{N-1} \sum_{i=1}^{N} \left( X_i - \bar{X} \right) \left(Y_i - \bar{Y} \right)

RESULT 2:

\frac{\left| B( \hat{\bar{Y}}_R) \right|}{ \sigma_{\hat{\bar{Y}}_R}} \leq \left| C.V.(\bar{x}) \right|

RESULT 3:

Mean square error of \hat{\bar{Y}_R} is given by:

MSE(\hat{\bar{Y}}_R) = \frac{1-f}{n} \left[ S_Y^2 + R^2 S_X^2 - 2 R S_{XY}   \right]

RESULT 4:

In SRSWOR, for large n, an approximation of the variance of \hat{\bar{Y}_R} is given by:

V(\hat{\bar{Y}}_R) = \frac{1-f}{n}  \frac{1}{N-1} \sum_{i=1}^{N} U_i^2 \\
\textit{; where} \quad U_i= Y_i-RX_i, \forall i=1(1)N 

Linear Algebra

Definition 1 (Vector Space). A vector space over a field F is a quadruple (V, +, ., F )) satisfying the following axioms for all α, β ∈ F and x, y, z ∈ V :
1. (V, +) is a commutative group, that is,
(a) ’+’ is map from V x V to V. [Closure]
(b) (x + y) + z = x + (y + z). [Associative]
(c) there exists an element 0 of V such that x + 0 = 0 + x = x. [Existence of 0]
(d) for each x in V there exists an element −x in V such that x + (−x) = (−x) + x = 0.
[Existence of Negative]
(e) x + y = y + x. [Commutative]
2. ’.’ is a map from F xV to V. [Closure wrt . ]
3. α.(β.x) = (α.β).x.
4. 1.x=x
5. (α + β).x = (α.x) + (β.x). [Distributivity]
6. α.(x + y) = (α.x) + (α.y). [Distributivity]
Remark 1. The elements of V are called vectors and the elements of F are called scalars. F is called the base field or ground field of the vector space. ’0’ in axiom 1(c) is called the null vector or the zero vector. It is to noted that, the bold faced Roman letters are treated as vectors whereas the lower case Greek letters are treated as scalars throughout the course.