**5. Comparison between** *p*^*RSS* **and** *p*^*SRS*

It is easy to argue that the ML estimates *<sup>p</sup>*^½ � <sup>1</sup> , *<sup>p</sup>*^½ � <sup>2</sup> , <sup>⋯</sup>, *<sup>p</sup>*^½ �*<sup>s</sup>* are statistically independent as the variables *Zi* 0 *s* are independently distributed. Again, substituting the

*Estimation of Measles Immunization Coverage in Guwahati by Ranked Set Sampling DOI: http://dx.doi.org/10.5772/intechopen.84382*

value *Zi <sup>m</sup>* of *<sup>p</sup>*^½ �*<sup>i</sup>* in Eq. (5), the estimate *<sup>p</sup>*^*RSS* can be shown to be identical with the overall mean of the given ranked set sample. It is also readily verified that *p*^*SRS* is an unbiased estimator of *p*.

The comparison of effectiveness and efficiency of the estimators based on simple random samples and ranked set samples, is obtained on the basis of criteria *viz.,* relative precision (RP) and relative saving (RS). The expressions for RP, RS and MSE of the estimators are described below as

$$RP = \frac{E\left(\hat{p}\_{\text{SRS}} - p\right)^2}{E\left(\hat{p}\_{\text{RSS}} - p\right)^2} = \frac{V\left(\hat{p}\_{\text{SRS}}\right)}{V\left(\hat{p}\_{\text{RSS}}\right)}\tag{8}$$

$$\text{RS} = \frac{V(\hat{p}\_{\text{SRS}}) - V(\hat{p}\_{\text{RSS}})}{V(\hat{p}\_{\text{SRS}})}. \tag{9}$$

## **6. Result and discussion**

so that

*ai*s and *bi*s we have

follows from the fact that

in the above proof.

dent as the variables *Zi*

**132**

1 *s* X*s i*¼1

*Viruses and Viral Infections in Developing Countries*

**2.** For equal sample size, i.e., *n* ¼ *ms*

**Justification:** For all *i* ¼ 1, 2, ⋯, *s*, setting

Variance ð Þ¼ *<sup>p</sup>*^ <sup>1</sup>

*ai* <sup>¼</sup> <sup>X</sup>*<sup>s</sup> j*¼1

*Variance p*^*RSS*

*ms*<sup>2</sup>

<sup>¼</sup> <sup>1</sup> *ms*<sup>2</sup>

¼ 1 *ns* X*s i*¼1

Now, from the definition of *<sup>p</sup>*½ �*<sup>i</sup>* s, we argue that *<sup>p</sup>*½ �*<sup>i</sup>*

1 *s* X*s i*¼1

As we know that the variance of *p*^ in SRS is *<sup>p</sup>*ð Þ <sup>1</sup>�*<sup>p</sup>*

X*s i*¼1

**5. Comparison between** *p*^*RSS* **and** *p*^*SRS*

0

Also, for the above choice of **π**-matrix, it is verified that

X*s i*¼1 *p*∗

X*s i*¼1

*P X*½ �*<sup>i</sup>* <sup>¼</sup> *<sup>x</sup>* � � <sup>¼</sup> *P X*ð Þ <sup>¼</sup> *<sup>x</sup>* , *for x* <sup>¼</sup> 0, 1*:*

� � <sup>≤</sup>*Variance <sup>p</sup>*^*SRS*

*j*¼1

*<sup>π</sup>ijp*½ �*<sup>j</sup>* , *bi* <sup>¼</sup> <sup>X</sup>*<sup>s</sup>*

one can get under ranked set sampling with the presence of ranking error

X*s j*¼1

*aibi*, say,

*ai* <sup>¼</sup> *<sup>δ</sup>sp* <sup>þ</sup> ð Þ *<sup>ρ</sup>* � *<sup>δ</sup> <sup>p</sup>*½ �*<sup>i</sup>* and *bi* <sup>¼</sup> *<sup>δ</sup>s*ð Þþ <sup>1</sup> � *<sup>p</sup>* ð Þ *<sup>ρ</sup>* � *<sup>δ</sup>* <sup>1</sup> � *<sup>p</sup>*½ �*<sup>i</sup>*

non-decreasing and the other is non-increasing. So, from Chebyshev's inequality for

sequence and subsequently, among two sequences f g *ai <sup>i</sup>*¼1 1ð Þ*<sup>s</sup>* and f g *bi <sup>i</sup>*¼1 1ð Þ*<sup>s</sup>*

1 *s* X*s i*¼1 *ai* ( )

*ai* <sup>¼</sup> *sp*, <sup>X</sup>*<sup>s</sup>*

*i*¼1

The justification in case of perfect ranking follows automatically by taking *ρ* ¼ 1

It is easy to argue that the ML estimates *<sup>p</sup>*^½ � <sup>1</sup> , *<sup>p</sup>*^½ � <sup>2</sup> , <sup>⋯</sup>, *<sup>p</sup>*^½ �*<sup>s</sup>* are statistically indepen-

*aibi* ≤

½ �*<sup>i</sup>* <sup>1</sup> � *<sup>p</sup>*<sup>∗</sup> ½ �*i* � �

*<sup>π</sup>ijp*½ �*<sup>j</sup>* ! <sup>X</sup>*<sup>s</sup>*

� �

*<sup>π</sup>ij* <sup>1</sup> � *<sup>p</sup>*½ �*<sup>j</sup>* � �

*j*¼1

n o

1 *s* X*s i*¼1 *bi* ( )

*bi* ¼ *s*ð Þ 1 � *p :*

*s* are independently distributed. Again, substituting the

*i*¼1 1ð Þ*s*

*:*

*<sup>n</sup>* , the required justification

,

*<sup>π</sup>ij* <sup>1</sup> � *<sup>p</sup>*½ �*<sup>j</sup>* � � !

� �

*:* (7)

, one is

is a non-decreasing

**Table 1** shows the estimates of proportion of measles immunized children in Assam, under SRS and RSS are very different but RSS based estimates 0.80 and 0.92 are very close to Census report for Assam (2012) [20] true value, which is 0.84 (rural) and 0.90 (urban), and has less variability than the SRS estimator and are very distinct from the true values. The estimate based on RSS is found to be 58% and 142%, for slum and non-slum region, respectively, more precision than that of SRS. Here, smaller the value of *ρ* represents higher will be the ranking error in RSS. The performance of estimates even in imperfect situation as compare to SRS, for different choices of the ranking error probability *ρ* ¼ 0*:*2, 0*:*6, 0*:*9, shows the estimates based on RSS is 48%, 50% and 56% for slum region, and 139%, 140% and 141% for non-slum, respectively, more precision as compare to SRS. RSS also shows a saving of 37%(59%) under perfect and a minimum of 32%(58%) under imperfect in slum (non-slum) as compare to SRS.


#### **Table 1.**

*Estimate of proportion of immunized children (p), variance (in* 10�<sup>3</sup>*), relative precision and relative saving (in %), in different regions of Guwahati under SRS, RSS perfect and imperfect procedures.*

### **7. Conclusion**

The present study revealed that RSS based estimates in both of perfect and imperfect situations, performs better than SRS based estimates. It should also be emphasized in context of estimation of proportion of measles immunization coverage in slum and non-slum region of Assam, RSS based estimates for different choices of accuracy (ρ) are not only more accurate but are more precise and efficient than the SRS procedure, and also suggest that the procedure of RSS is better than the classical SRS. Therefore, based on the obtained results one can recommend to adopt RSS procedure in epidemiological application and in other health related studies so that it will help in planning to build a healthy and disease free environment.

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