**1. Introduction**

In public health related studies, the virus of measles is considered as highly epidemic and responsible for serious diseases. According to the medical dictionary, measles virus infects the lungs in children, which can cause pneumonia in older children, can cause inflammation of the brain, called encephalitis, which can cause seizures and brain damage [1]. As a preventive measure, adequate vaccination is introduced in early childhood to acquire immunity against measles virus. According to the Integrated Child Development Services Program (ICDS) in India, a child should have received the basic vaccines (BCG, polio, DPT and measles) at 12–23 months of age.

In this regard, the World Health Organization (WHO) launched the Expanded Program on Immunization (EPI) in May 1974 to immunize children around the world. Since then, it has been widely used to access coverage. India launched the National Vaccination Program entitled Expanded Programme of Immunization in 1978 with the introduction of the BCG, OPV, DPT and anti-typhoid-paratyphoid vaccine. The EPI was renamed and launched, with a major change in list of vaccinations, as the Universal Vaccination Program on 19 November 1985. The measles vaccine [2] has been added to latest schedule.

To study the vaccination coverage a few among various sampling procedures and are the most popular *viz.,* lot quality assurance sampling (LQAS) [3, 4],

systematic sampling, cluster sampling [5] and stratified sampling [6], are based on simple random sampling (SRS) either at one stage or subsequent stage. In most of the practical situations to obtain actual measurement of an observation is not relatively easy and economical, but ranking a small subset of auxiliary information about a sample is relatively easy, economical and reliable. McIntyre in 1952 [7] have introduced the procedure of ranked set sampling (RSS), as an alternative, which is highly beneficial and much superior to the standard SRS, for estimating population parameters. In RSS procedure a set of units is randomly drawn from the population and the selected units are ranked by using judgment method or by other means which does not require actual measurement. Only the unit possessing lowest rank is measured for this set of units. Next, a second set of units of the same size is drawn as earlier and ranked; then the unit at the second lowest position is measured. This procedure of ranking and measuring is continued until we get as many observations as the size of the set. This entire procedure is regarded as a cycle. The process of cycle formation will be continued until the desired sample size is obtained for analysis.

Obviously, there is a high chance of having ties then in this situation, the observations are ordered systematically in sequence, as explained by Terpstra

*Estimation of Measles Immunization Coverage in Guwahati by Ranked Set Sampling*

sampled units are discarded from the data set.

0

*f x*ð Þ¼ <sup>j</sup>*<sup>p</sup> px*ð Þ <sup>1</sup> � *<sup>p</sup>*

*s*

1

0 @

� �*:*

entire target population. Here it can be easily shown that

*i*

child is administrated with measles vaccination or not is collected.

3.From the first set, the unit corresponds to the mother with the lowest age (in months) is selected. From the second set, the unit corresponds to the mother with the second lowest age is selected and so on. Finally, from *sth* set, the unit corresponding to the mother with the highest age is selected. The other *s*(*s* � 1)

4. Steps 1–3, called a cycle, are repeated *m* times to obtain a ranked set sample of

Corresponding to each selected mother, information regarding whether her

resent a ranked set sample of size *ms*, where *X*½ �*<sup>i</sup> <sup>j</sup>* takes the values "1" or "0" according as the *j*th child in the *i*th ranking class is vaccinated or not. By virtue of

Suppose *X* is the binary response that takes value "1" if the child is vaccinated with measles vaccine and "0" otherwise. Let *<sup>X</sup>*½ �*<sup>i</sup> <sup>j</sup>*, *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>s</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>m</sup>* � � rep-

*s*, *X*½ �*<sup>i</sup>* 1,*X*½ �*<sup>i</sup>* 2, ⋯, *X*½ �*<sup>i</sup> <sup>m</sup>* are independently and identically (i.i.d.) distributed. Here *X*½ �*<sup>i</sup>* <sup>1</sup> can be regarded as the *i*th order statistic corresponding to a simple random sample of *s* observations, say, ð Þ *X*1,*X*2, ⋯,*Xs* on *X*. Obviously, *Xi*s have the common prob-

1�*x*

where *p* is the probability that a child is vaccinated with measles vaccine in the

*P X*½ �*<sup>i</sup>* <sup>1</sup> <sup>¼</sup> <sup>1</sup> � � <sup>¼</sup> *<sup>P</sup>* among *Xi* ð Þ <sup>0</sup>*s*, at least *<sup>s</sup>* � *<sup>i</sup>* <sup>þ</sup> 1 X � values are 1

Obviously, *<sup>p</sup>*½ �*<sup>i</sup>* is the proportion, in *<sup>i</sup>*th class, of children who received the vaccination and *p* is the overall proportion of children receiving the vaccine in

For a dichotomous population, estimation of population proportion *p*, based on ranked set samples have already been investigated [8–14]. A method for estimation [15] of *p* using RSS for the situations where the binary variable is obtained from a

X*s r*¼1

<sup>A</sup>*p<sup>r</sup>* ð Þ <sup>1</sup> � *<sup>p</sup> <sup>s</sup>*�*<sup>r</sup>* <sup>¼</sup> *<sup>p</sup>*½ �*<sup>i</sup>* , say,

*s* are independently distributed. Here for any *i* from 1 to

, *x* ¼ 0, 1, 0 <*p*< 1, (1)

*<sup>p</sup>*½ �*<sup>r</sup>* <sup>¼</sup> *sp:* (2)

and Nelson (2005).

*DOI: http://dx.doi.org/10.5772/intechopen.84382*

size *ms*.

ranked set sampling all *X*½ �*<sup>i</sup> <sup>j</sup>*

ability mass function (p.m.f.) given by

population. Now we have, for any *i*, 1 ≤*i* ≤*s*

<sup>¼</sup> <sup>X</sup>*<sup>s</sup> r*¼*s*�*i*þ1

*P X*½ �*<sup>i</sup>* <sup>¼</sup> <sup>0</sup> � � <sup>¼</sup> <sup>1</sup> � *<sup>p</sup>*½ �*<sup>i</sup>*

**3. Estimation of parameter** *p*

**129**

Based on real-life primary data, in the present study, the procedure of RSS is investigated against SRS for estimating the proportion of children aged 12–23 months of slum and non-slum households of Guwahati, the capital city of state Assam, India, who are not immunized with the vaccine against measles. The information on a total of 500 (260 slum and 240 non-slum) households, were obtained after being ranked using mother's age 15–49 years (in months), as auxiliary variable is used as auxiliary variable. The data of the same size is obtained by following the SRS procedure, for the evaluation of the performance and effectiveness of the RSS estimator compared to SRS estimator.

## **2. Sampling design**

The study population is a representative cross-sectional sample of children aged 12–23 months to mothers aged 15–49 of Guwahati City, India. Five hundred households, following both SRS and RSS procedures, having the children of age from 6 months to 5 years were identified for the present study. Following the SRS technique, from both slum and non-slum regions, a sample 250 households were obtained. Among the selected 500 households under RSS, 260 were residents of slum region whereas remaining 240 are of non-slum parts of the Guwahati City. The variable considered in the study is the proportion of children in Guwahati City, India who are not immunized with the vaccine against measles. It is assumed that the case of receiving the vaccination of children usually depends on the awareness of the child's mother on immunization. Lower the age of a mother who has children aged 12–23 months, lower would be the consciousness as expected. Therefore, the age of the mother (in months) was used as a ranking variable in RSS for classification of the sample. Information on children was obtained by a face to face interview with the mothers. Mothers were selected through RSS conducted in Guwahati City.

The observation were divided into *m* sets of size four (*s* ¼ 4) each. The observations under RSS procedure are obtained through the following steps.


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

Obviously, there is a high chance of having ties then in this situation, the observations are ordered systematically in sequence, as explained by Terpstra and Nelson (2005).


Corresponding to each selected mother, information regarding whether her child is administrated with measles vaccination or not is collected.

Suppose *X* is the binary response that takes value "1" if the child is vaccinated with measles vaccine and "0" otherwise. Let *<sup>X</sup>*½ �*<sup>i</sup> <sup>j</sup>*, *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>s</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>m</sup>* � � represent a ranked set sample of size *ms*, where *X*½ �*<sup>i</sup> <sup>j</sup>* takes the values "1" or "0" according as the *j*th child in the *i*th ranking class is vaccinated or not. By virtue of ranked set sampling all *X*½ �*<sup>i</sup> <sup>j</sup>* 0 *s* are independently distributed. Here for any *i* from 1 to *s*, *X*½ �*<sup>i</sup>* 1,*X*½ �*<sup>i</sup>* 2, ⋯, *X*½ �*<sup>i</sup> <sup>m</sup>* are independently and identically (i.i.d.) distributed. Here *X*½ �*<sup>i</sup>* <sup>1</sup> can be regarded as the *i*th order statistic corresponding to a simple random sample of *s* observations, say, ð Þ *X*1,*X*2, ⋯,*Xs* on *X*. Obviously, *Xi*s have the common probability mass function (p.m.f.) given by

$$f(\mathbf{x}|p) = p^{\mathbf{x}}(\mathbf{1} - p)^{\mathbf{1} - \mathbf{x}}, \ \mathbf{x} = \mathbf{0}, \mathbf{1}, \ \mathbf{0} < p < \mathbf{1},\tag{1}$$

where *p* is the probability that a child is vaccinated with measles vaccine in the population. Now we have, for any *i*, 1 ≤*i* ≤*s*

$$P(\mathbf{X}\_{[i]1} = \mathbf{1}) = P(\text{among } X\_{i}s, \text{ at least } s - i + \mathbf{1} \text{ X} - \text{values are } \mathbf{1})$$

$$= \sum\_{r=s-i+1}^{s} \binom{s}{i} p^r \left(\mathbf{1} - p\right)^{s-r} = p\_{[i]}, \text{ say,}$$

$$P(\mathbf{X}\_{[i]} = \mathbf{0}) = \left(\mathbf{1} - p\_{[i]}\right).$$

Obviously, *<sup>p</sup>*½ �*<sup>i</sup>* is the proportion, in *<sup>i</sup>*th class, of children who received the vaccination and *p* is the overall proportion of children receiving the vaccine in entire target population. Here it can be easily shown that

$$\sum\_{r=1}^{s} p\_{[r]} = sp.\tag{2}$$

### **3. Estimation of parameter** *p*

For a dichotomous population, estimation of population proportion *p*, based on ranked set samples have already been investigated [8–14]. A method for estimation [15] of *p* using RSS for the situations where the binary variable is obtained from a

systematic sampling, cluster sampling [5] and stratified sampling [6], are based on simple random sampling (SRS) either at one stage or subsequent stage. In most of the practical situations to obtain actual measurement of an observation is not relatively easy and economical, but ranking a small subset of auxiliary information about a sample is relatively easy, economical and reliable. McIntyre in 1952 [7] have introduced the procedure of ranked set sampling (RSS), as an alternative, which is highly beneficial and much superior to the standard SRS, for estimating population parameters. In RSS procedure a set of units is randomly drawn from the population and the selected units are ranked by using judgment method or by other means which does not require actual measurement. Only the unit possessing lowest rank is measured for this set of units. Next, a second set of units of the same size is drawn as earlier and ranked; then the unit at the second lowest position is measured. This procedure of ranking and measuring is continued until we get as many observations as the size of the set. This entire procedure is regarded as a cycle. The process of cycle formation will be continued until the desired sample size is obtained for

Based on real-life primary data, in the present study, the procedure of RSS is investigated against SRS for estimating the proportion of children aged 12–23 months of slum and non-slum households of Guwahati, the capital city of state Assam, India, who are not immunized with the vaccine against measles. The information on a total of 500 (260 slum and 240 non-slum) households, were obtained after being ranked using mother's age 15–49 years (in months), as auxiliary variable is used as auxiliary variable. The data of the same size is obtained by following the SRS procedure, for the evaluation of the performance and effectiveness of the RSS

The study population is a representative cross-sectional sample of children aged 12–23 months to mothers aged 15–49 of Guwahati City, India. Five hundred households, following both SRS and RSS procedures, having the children of age from 6 months to 5 years were identified for the present study. Following the SRS technique, from both slum and non-slum regions, a sample 250 households were obtained. Among the selected 500 households under RSS, 260 were residents of slum region whereas remaining 240 are of non-slum parts of the Guwahati City. The variable considered in the study is the proportion of children in Guwahati City, India who are not immunized with the vaccine against measles. It is assumed that the case of receiving the vaccination of children usually depends on the awareness of the child's mother on immunization. Lower the age of a mother who has children aged 12–23 months, lower would be the consciousness as expected. Therefore, the age of the mother (in months) was used as a ranking variable in RSS for classification of the sample. Information on children was obtained by a face to face interview with the mothers. Mothers were selected through RSS conducted in Guwahati City. The observation were divided into *m* sets of size four (*s* ¼ 4) each. The obser-

vations under RSS procedure are obtained through the following steps.

2. In each set *s*, the units are ranked according to the age of the mother. Since the samples in different sets are ranked according to the age of the mother (1, 2, 3, 4).

is distributed randomly in sets *s*, each with *s* units.

<sup>2</sup> is selected from the target population and

1.A simple random sample of units *s*

**128**

analysis.

estimator compared to SRS estimator.

*Viruses and Viral Infections in Developing Countries*

**2. Sampling design**

continuous variable. Let *XRSS* ¼ **X**½ � <sup>1</sup> , ⋯, **X**½ �*<sup>i</sup>* , ⋯, **X**½ �*<sup>s</sup>* � �, **<sup>X</sup>**½ �*<sup>i</sup>* <sup>¼</sup> *<sup>X</sup>*½ �*<sup>i</sup>* 1, <sup>⋯</sup>, *<sup>X</sup>*½ �*<sup>i</sup> <sup>m</sup>* � �*<sup>T</sup>* and **<sup>p</sup>** <sup>¼</sup> *<sup>p</sup>*½ � <sup>1</sup> <sup>⋯</sup>, *<sup>p</sup>*½ �*<sup>i</sup>* , <sup>⋯</sup>, *<sup>p</sup>*½ �*<sup>s</sup>* � �. Here *<sup>X</sup>*½ �*<sup>i</sup>* 1, <sup>⋯</sup>,*X*½ �*<sup>i</sup> <sup>m</sup>* are i.i.d. Binomial 1, *<sup>p</sup>*½ �*<sup>i</sup>* � �, *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>s</sup>*. So the joint p.m.f. of *X*½ �*<sup>i</sup> <sup>j</sup>*, *j* ¼ 1, 2, ⋯, *m* is given by

$$f\left(\mathbf{x}\_{[i]}|p\_{[i]}\right) = \prod\_{j=1}^{m} p\_{[i]}^{\mathbf{x}\_{[i]}} \left(\mathbf{1} - p\_{[i]}\right)^{1 - \mathbf{x}\_{[i]}} = p\_{[i]}^{\mathbf{x}\_{i}} \left(\mathbf{1} - p\_{[i]}\right)^{m - \mathbf{z}\_{i}} \tag{3}$$

mechanism is very rare in practice and hence some error in judgment ranking is inevitable. So it is necessary to study how much robust the proposed procedure is against when the rankings are not perfect. Estimation of *p* under perfect and

*Estimation of Measles Immunization Coverage in Guwahati by Ranked Set Sampling*

statistic while a set of *s* units are ordered. In the presence of ranking error, *X*½ �*<sup>i</sup>* is not

statistic actually have the true rank *j*, for *i* ¼ 1 1ð Þ*s*, *j* ¼ 1 1ð Þ*s*. Assume that *πij*s satisfy

*j*¼1

That is, **<sup>π</sup>** <sup>¼</sup> *<sup>π</sup>ij* � � is a doubly stochastic symmetric matrix of order *<sup>s</sup>* � *<sup>s</sup>*. Under

*<sup>p</sup>***<sup>∗</sup>** <sup>¼</sup> **<sup>π</sup>***<sup>p</sup>*

1

CCCCCCA

Here "*ρ* ¼ 1" corresponds to the case of perfect ranking. Under the above probability model for imperfect ranking some consequential facts are justified below.

½ �*<sup>i</sup>* <sup>¼</sup> *sp*, that is, the present imperfect ranking mechanism is consistent

X*s i*¼1

*<sup>π</sup>ij* !*p*½ �*<sup>j</sup>* <sup>¼</sup> <sup>X</sup>*<sup>s</sup>*

*j*¼1

*<sup>p</sup>*½ �*<sup>j</sup>* <sup>¼</sup> *sp*

and *<sup>p</sup>***<sup>∗</sup>** <sup>¼</sup> *<sup>p</sup>*<sup>∗</sup>

*<sup>π</sup>ijp*½ �*<sup>j</sup> :*

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

½ � <sup>2</sup> , <sup>⋯</sup>, *<sup>p</sup>*<sup>∗</sup> ½ �*s*

, 0<*<sup>ρ</sup>* <sup>&</sup>lt;1, *<sup>δ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ρ</sup>*

*<sup>s</sup>* � <sup>1</sup> *:*

� �<sup>0</sup>

this assumption the distribution of *<sup>X</sup>*½ �*<sup>i</sup>* would be changed to Binomial (1, *<sup>p</sup>*<sup>∗</sup>

*p*∗ ½ �*<sup>i</sup>* <sup>¼</sup> <sup>X</sup>*<sup>s</sup> j*¼1

th judgment order and *i*

*<sup>π</sup>ij* <sup>¼</sup> 1, <sup>∀</sup>*i*, <sup>X</sup>*<sup>s</sup>*

*i*¼1

*πij* ¼ 1, ∀*j:*

th true order

½ �*<sup>i</sup>* ), for

th judgment order

. For the present pur-

imperfect unbalanced RSS [13, 16, 17] are discussed. Let *X*½ �*<sup>i</sup>* and *X*ð Þ*<sup>i</sup>* denote, respectively, the *i*

*DOI: http://dx.doi.org/10.5772/intechopen.84382*

the conditions

each *i* ¼ 1 1ð Þ*s*, where

Equivalently,

**1.** P*<sup>s</sup> <sup>i</sup>*¼<sup>1</sup>*p*<sup>∗</sup>

Justification:

and hence

**131**

X*s i*¼1 *p*∗ ½ �*<sup>i</sup>* <sup>¼</sup> <sup>X</sup>*<sup>s</sup> i*¼1

where *<sup>p</sup>* <sup>¼</sup> *<sup>p</sup>*½ � <sup>1</sup> , *<sup>p</sup>*½ � <sup>2</sup> , <sup>⋯</sup>, *<sup>p</sup>*½ �*<sup>s</sup>*

**π** ¼

� �<sup>0</sup>

0

BBBBBB@

pose we take the particular form of **π** as in [18, 19],

*ρδδ* … *δ ρ δ* … *δ ρ* … *δ* ⋱ ⋮ *ρ*

> X*s j*¼1

> > X*s i*¼1

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

<sup>1</sup> � *<sup>p</sup>*<sup>∗</sup> ½ �*i* � � <sup>¼</sup> *<sup>s</sup>*ð Þ <sup>1</sup> � *<sup>p</sup>* ,

*j*¼1

necessarily equal to *X*ð Þ*<sup>i</sup>* . Let *πij* denote the probability that the *i*

<sup>0</sup> <sup>≤</sup>*πij* <sup>≤</sup>1, *<sup>π</sup>ij* <sup>¼</sup> *<sup>π</sup>ji*, <sup>∀</sup>ð Þ *<sup>i</sup>*, *<sup>j</sup>* , <sup>X</sup>*<sup>s</sup>*

where *zi* <sup>¼</sup> <sup>P</sup>*<sup>m</sup> <sup>i</sup>*¼1*x*½ �*<sup>i</sup> <sup>j</sup>*, the number of children getting vaccinated observed in *<sup>i</sup>*th ranking class of the given ranked set sample. Obviously, *Zi* <sup>¼</sup> <sup>P</sup>*<sup>m</sup> <sup>i</sup>*¼1*X*½ �*<sup>i</sup> <sup>j</sup>* � Binomial *<sup>m</sup>*, *<sup>p</sup>*½ �*<sup>i</sup>* � �, independently for all *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>s</sup>*. Then the joint p.m.f. of the whole sample **XRSS** is of the form

$$f(\mathbf{x}\_{\text{RSS}}|\mathbf{p}) = \prod\_{i=1}^{s} f\left(\mathbf{x}\_{[i]}|p\_{[i]}\right) = \prod\_{i=1}^{s} p\_{[i]}^{x\_i} \left(1 - p\_{[i]}\right)^{m - x\_i}.\tag{4}$$

Applying standard maximum likelihood (ML) principle the ML estimate of *p* under ranked set sampling, is given by

$$
\hat{p}\_{\rm RSS} = a \text{rg } \max \,\, f(\mathbf{x}\_{\rm RSS}|p) \,\,.
$$

Given the RSS data, the form of likelihood function of *p* is complicated and hence the MLE of *p* is difficult to obtain directly. Alternatively, for *i* ¼ 1, 2, ⋯, *s*, one can separately derive ML estimate, say, *<sup>p</sup>*^½ �*<sup>i</sup>* of based on the likelihood function (3) and then MLE of *p* can be formulated by using the relation (2) as

$$
\hat{p}\_{\rm RSS} = \frac{1}{s} \sum\_{i=1}^{s} \hat{p}\_{[i]}.
$$

Here, it can be shown that, for each *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>s</sup>*, *Zi <sup>m</sup>* would be the maximum likelihood estimator of *<sup>p</sup>*½ �*<sup>i</sup>* and hence we get

$$
\hat{p}\_{\rm RSS} = \frac{1}{s} \sum\_{i=1}^{s} \frac{Z\_i}{m} = \overline{X}\_{\rm RSS}, \quad \text{say,} \tag{5}
$$

where *XRSS* is the overall mean of the ranked set sample.

Let *Y*1, *Y*2, ⋯, *Yn* are the observations drawn according to SRS design and *p*^*SRS* be an unbiased estimator of *p*, then the corresponding unbiased estimator of *p* can be obtained, as

$$
\hat{p}\_{\text{SRS}} = \sum\_{i=1}^{n} \frac{\mathcal{Y}\_i}{n} \tag{6}
$$

#### **4. Effect of imperfect ranking**

The discussion done in the previous sections are based on the assumption that the ranking procedure produces the correct order statistics. But, a perfect ranking *Estimation of Measles Immunization Coverage in Guwahati by Ranked Set Sampling DOI: http://dx.doi.org/10.5772/intechopen.84382*

mechanism is very rare in practice and hence some error in judgment ranking is inevitable. So it is necessary to study how much robust the proposed procedure is against when the rankings are not perfect. Estimation of *p* under perfect and imperfect unbalanced RSS [13, 16, 17] are discussed.

Let *X*½ �*<sup>i</sup>* and *X*ð Þ*<sup>i</sup>* denote, respectively, the *i* th judgment order and *i* th true order statistic while a set of *s* units are ordered. In the presence of ranking error, *X*½ �*<sup>i</sup>* is not necessarily equal to *X*ð Þ*<sup>i</sup>* . Let *πij* denote the probability that the *i* th judgment order statistic actually have the true rank *j*, for *i* ¼ 1 1ð Þ*s*, *j* ¼ 1 1ð Þ*s*. Assume that *πij*s satisfy the conditions

$$\mathbf{0} \le \pi\_{\vec{\imath}\vec{\jmath}} \le \mathbf{1}, \ \pi\_{\vec{\imath}\vec{\jmath}} = \pi\_{\vec{\jmath}\vec{\imath}}, \forall (i, j), \ \sum\_{j=1}^{s} \pi\_{\vec{\imath}\vec{\jmath}} = \mathbf{1}, \forall i, \ \sum\_{i=1}^{s} \pi\_{\vec{\imath}\vec{\jmath}} = \mathbf{1}, \forall j.$$

That is, **<sup>π</sup>** <sup>¼</sup> *<sup>π</sup>ij* � � is a doubly stochastic symmetric matrix of order *<sup>s</sup>* � *<sup>s</sup>*. Under this assumption the distribution of *<sup>X</sup>*½ �*<sup>i</sup>* would be changed to Binomial (1, *<sup>p</sup>*<sup>∗</sup> ½ �*<sup>i</sup>* ), for each *i* ¼ 1 1ð Þ*s*, where

$$p\_{[i]}^{\*} = \sum\_{j=1}^{s} \pi\_{ij} p\_{[j]}.$$

Equivalently,

continuous variable. Let *XRSS* ¼ **X**½ � <sup>1</sup> , ⋯, **X**½ �*<sup>i</sup>* , ⋯, **X**½ �*<sup>s</sup>*

*Viruses and Viral Infections in Developing Countries*

So the joint p.m.f. of *X*½ �*<sup>i</sup> <sup>j</sup>*, *j* ¼ 1, 2, ⋯, *m* is given by

*<sup>f</sup>*ð Þ¼ **xRSS**j**<sup>p</sup>** <sup>Y</sup>*<sup>s</sup>*

<sup>¼</sup> <sup>Y</sup>*<sup>m</sup> j*¼1 *p x*½ �*<sup>i</sup> <sup>j</sup>* ½ �*<sup>i</sup>* <sup>1</sup> � *<sup>p</sup>*½ �*<sup>i</sup>* � �1�*x*½ �*<sup>i</sup> <sup>j</sup>*

ranking class of the given ranked set sample. Obviously, *Zi* <sup>¼</sup> <sup>P</sup>*<sup>m</sup>*

*i*¼1

and then MLE of *p* can be formulated by using the relation (2) as

Here, it can be shown that, for each *<sup>i</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup>, *<sup>s</sup>*, *Zi*

*<sup>p</sup>*^*RSS* <sup>¼</sup> <sup>1</sup> *s* X*s i*¼1

where *XRSS* is the overall mean of the ranked set sample.

likelihood estimator of *<sup>p</sup>*½ �*<sup>i</sup>* and hence we get

**4. Effect of imperfect ranking**

*<sup>f</sup>* **<sup>x</sup>**½ �*<sup>i</sup>* <sup>j</sup>*p*½ �*<sup>i</sup>* � �

*<sup>f</sup>* **<sup>x</sup>**½ �*<sup>i</sup>* <sup>j</sup>*p*½ �*<sup>i</sup>* � �

under ranked set sampling, is given by

**<sup>p</sup>** <sup>¼</sup> *<sup>p</sup>*½ � <sup>1</sup> <sup>⋯</sup>, *<sup>p</sup>*½ �*<sup>i</sup>* , <sup>⋯</sup>, *<sup>p</sup>*½ �*<sup>s</sup>* � �

where *zi* <sup>¼</sup> <sup>P</sup>*<sup>m</sup>*

sample **XRSS** is of the form

*<sup>m</sup>*, *<sup>p</sup>*½ �*<sup>i</sup>* � �

obtained, as

**130**

� �, **<sup>X</sup>**½ �*<sup>i</sup>* <sup>¼</sup> *<sup>X</sup>*½ �*<sup>i</sup>* 1, <sup>⋯</sup>, *<sup>X</sup>*½ �*<sup>i</sup> <sup>m</sup>*

<sup>¼</sup> *pzi*

*<sup>i</sup>*¼1*x*½ �*<sup>i</sup> <sup>j</sup>*, the number of children getting vaccinated observed in *<sup>i</sup>*th

½ �*<sup>i</sup>* <sup>1</sup> � *<sup>p</sup>*½ �*<sup>i</sup>* � �*m*�*zi*

½ �*<sup>i</sup>* <sup>1</sup> � *<sup>p</sup>*½ �*<sup>i</sup>* � �*<sup>m</sup>*�*zi*

. Here *<sup>X</sup>*½ �*<sup>i</sup>* 1, <sup>⋯</sup>,*X*½ �*<sup>i</sup> <sup>m</sup>* are i.i.d. Binomial 1, *<sup>p</sup>*½ �*<sup>i</sup>*

, independently for all *i* ¼ 1, 2, ⋯, *s*. Then the joint p.m.f. of the whole

Applying standard maximum likelihood (ML) principle the ML estimate of *p*

*p*^*RSS* ¼ *arg max f*ð Þ **xRSS**j*p*

Given the RSS data, the form of likelihood function of *p* is complicated and hence the MLE of *p* is difficult to obtain directly. Alternatively, for *i* ¼ 1, 2, ⋯, *s*, one can separately derive ML estimate, say, *<sup>p</sup>*^½ �*<sup>i</sup>* of based on the likelihood function (3)

> *<sup>p</sup>*^*RSS* <sup>¼</sup> <sup>1</sup> *s* X*s i*¼1 *p*^½ �*i :*

> > *Zi*

*<sup>p</sup>*^*SRS* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

Let *Y*1, *Y*2, ⋯, *Yn* are the observations drawn according to SRS design and *p*^*SRS* be an unbiased estimator of *p*, then the corresponding unbiased estimator of *p* can be

*i*¼1

The discussion done in the previous sections are based on the assumption that the ranking procedure produces the correct order statistics. But, a perfect ranking

*yi n*

<sup>¼</sup> <sup>Y</sup>*<sup>s</sup> i*¼1 *pzi*

*<sup>p</sup> :*

� �*<sup>T</sup>* and

, *i* ¼ 1, 2, ⋯, *s*.

*<sup>i</sup>*¼1*X*½ �*<sup>i</sup> <sup>j</sup>* � Binomial

*:* (4)

*<sup>m</sup>* would be the maximum

(6)

*<sup>m</sup>* <sup>¼</sup> *XRSS*, say, (5)

(3)

� �

$$p^\* = \pi p$$

where *<sup>p</sup>* <sup>¼</sup> *<sup>p</sup>*½ � <sup>1</sup> , *<sup>p</sup>*½ � <sup>2</sup> , <sup>⋯</sup>, *<sup>p</sup>*½ �*<sup>s</sup>* � �<sup>0</sup> and *<sup>p</sup>***<sup>∗</sup>** <sup>¼</sup> *<sup>p</sup>*<sup>∗</sup> ½ � <sup>1</sup> , *<sup>p</sup>*<sup>∗</sup> ½ � <sup>2</sup> , <sup>⋯</sup>, *<sup>p</sup>*<sup>∗</sup> ½ �*s* � �<sup>0</sup> . For the present purpose we take the particular form of **π** as in [18, 19],

$$
\mathfrak{m} = \begin{pmatrix}
\rho & \delta & \delta & \dots & \delta \\
& \rho & \delta & \dots & \delta \\
& & \rho & \dots & \delta \\
& & & \ddots & \vdots \\
& & & & \rho
\end{pmatrix}, \quad 0 < \rho < 1, \ \delta = \frac{1-\rho}{\mathfrak{s}-1}.
$$

Here "*ρ* ¼ 1" corresponds to the case of perfect ranking. Under the above probability model for imperfect ranking some consequential facts are justified below.

**1.** P*<sup>s</sup> <sup>i</sup>*¼<sup>1</sup>*p*<sup>∗</sup> ½ �*<sup>i</sup>* <sup>¼</sup> *sp*, that is, the present imperfect ranking mechanism is consistent

Justification:

$$\sum\_{i=1}^{s} p\_{[i]}^{\*} = \sum\_{i=1}^{s} \sum\_{j=1}^{s} \pi\_{\vec{\eta}} p\_{[j]} = \sum\_{j=1}^{s} \left( \sum\_{i=1}^{s} \pi\_{\vec{\eta}} \right) p\_{[j]} = \sum\_{j=1}^{s} p\_{[j]} = sp$$

and hence

$$\sum\_{i=1}^{s} \left(\mathbf{1} - p\_{[i]}^{\*}\right) = s(\mathbf{1} - p),$$

so that

$$\frac{1}{s} \sum\_{i=1}^{s} P(X\_{[i]} = x) = P(X = x), \text{ for } x = 0, 1.$$

value *Zi*

unbiased estimator of *p*.

**6. Result and discussion**

in slum (non-slum) as compare to SRS.

**7. Conclusion**

**Table 1.**

**133**

MSE of the estimators are described below as

*DOI: http://dx.doi.org/10.5772/intechopen.84382*

*<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

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

*<sup>E</sup> <sup>p</sup>*^*RSS* � *<sup>p</sup>* <sup>2</sup> <sup>¼</sup> *<sup>V</sup> <sup>p</sup>*^*SRS*

*RS* <sup>¼</sup> *<sup>V</sup> <sup>p</sup>*^*SRS* � *<sup>V</sup> <sup>p</sup>*^*RSS*

**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

**Region Criteria** *n* **SRS** *s m* **RSS perfect RSS imperfect**

Slum Estimate 250 0.64 4 65 0.80 0.80 0.80 0.80

Non-slum Estimate 250 0.74 4 60 0.92 0.92 0.92 0.92

*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.*

Variance 0.92 0.58 0.62 0.62 0.59 RP 1.58 1.48 1.50 1.56 RS 36.83 32.29 33.27 35.70

Variance 0.76 0.32 0.32 0.32 0.32 RP 2.42 2.39 2.40 2.41 RS 58.60 58.22 58.30 58.51

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

*<sup>V</sup> <sup>p</sup>*^*RSS* (8)

*ρ* ¼**0.2** *ρ* ¼**0.6** *ρ* ¼**0.9**

*<sup>V</sup> <sup>p</sup>*^*SRS :* (9)

*RP* <sup>¼</sup> *<sup>E</sup> <sup>p</sup>*^*SRS* � *<sup>p</sup>* <sup>2</sup>

*Estimation of Measles Immunization Coverage in Guwahati by Ranked Set Sampling*

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

$$\text{Variance}(\hat{p}\_{\text{RSS}}) \le \text{Variance}(\hat{p}\_{\text{SRS}})$$

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

$$a\_i = \sum\_{j=1}^s \pi\_{ij} p\_{[j]}, \quad b\_i = \sum\_{j=1}^s \pi\_{ij} \left(1 - p\_{[j]}\right),$$

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

$$\begin{aligned} \text{Variance } (\dot{p}) &= \frac{1}{ms^2} \sum\_{i=1}^s p\_{[i]}^\* \left( \mathbf{1} - p\_{[i]}^\* \right) \\ &= \frac{1}{ms^2} \sum\_{i=1}^s \left( \sum\_{j=1}^s \pi\_{ij} p\_{[j]} \right) \left( \sum\_{j=1}^s \pi\_{ij} \left( \mathbf{1} - p\_{[j]} \right) \right) \\ &= \frac{1}{ms} \sum\_{i=1}^s a\_i b\_i, \quad \text{say}, \end{aligned}$$

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

$$a\_i = \delta p + (\rho - \delta) p\_{[i]} \quad \text{and} \quad b\_i = \delta \mathbf{s} (\mathbf{1} - p) + (\rho - \delta) \left(\mathbf{1} - p\_{[i]}\right). \tag{7}$$

Now, from the definition of *<sup>p</sup>*½ �*<sup>i</sup>* s, we argue that *<sup>p</sup>*½ �*<sup>i</sup>* n o *i*¼1 1ð Þ*s* is a non-decreasing 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>* , one is non-decreasing and the other is non-increasing. So, from Chebyshev's inequality for *ai*s and *bi*s we have

$$\frac{1}{s}\sum\_{i=1}^{s}a\_{i}b\_{i} \le \left\{\frac{1}{s}\sum\_{i=1}^{s}a\_{i}\right\}\left\{\frac{1}{s}\sum\_{i=1}^{s}b\_{i}\right\}.$$

As we know that the variance of *p*^ in SRS is *<sup>p</sup>*ð Þ <sup>1</sup>�*<sup>p</sup> <sup>n</sup>* , the required justification follows from the fact that

$$\sum\_{i=1}^{s} a\_i = sp, \quad \sum\_{i=1}^{s} b\_i = s(1-p).$$

The justification in case of perfect ranking follows automatically by taking *ρ* ¼ 1 in the above proof.
