1. 1 ON ESTIMATING THE URBAN POPULATIONS USING MINIMUM INFORMATION Arun Kumar Sinha Department of Statistics Central University of Bihar Patna 800 014, Bihar INDIA (arunkrsinha@yahoo.com )arunkrsinha@yahoo.com 08 October, 2014 IAOS 2014 Conference Da Nang, Vietnam

2. Abstract For implementing the programmes and policies of the urban planning department we need to know the current populations of the urban areas of interest. Many times the required information is not easily available. 2

3. In view of these scenarios the paper deals with some techniques of estimating the populations of urban areas when only the minimum information is available. The real data sets are used to illustrate the methods that provide some interesting and useful results. This paper could be of immense help to the planners and decision makers of the urban areas related issues. 3

4. 4 MAIN STEPS IN A SAMPLING Identification Acquisition Quantification

5. 5 FAMILIAR DILEMMA IN SAMPLING We need large sample sizes for adequately representing heterogeneous populations. [Desirability] But our budget permits only a limited number of measurements. [Affordability] We refer to this “best of both worlds” scenarios as observational economy.

6. 6 OBSERVATIONAL ECONOMY For observational economy to be feasible, identification and acquisition of sampling units should be inexpensive as compared with their quantification. Ranked set sampling (RSS) is one such method for achieving observational economy.

7. 7 RANKED SET SAMPLING McIntyre (1952) proposed a method of sampling for estimating pasture yields. He referred to it as “a method of unbiased selective sampling using ranked sets”. The present name, “ranked set sampling” (RSS), was coined by Halls and Dell (1966).

8. 8 Ranked set sampling is useful in situations in which measurements are difficult or costly to obtain but ranking of a subset of units is relatively easy. It aims to achieve what stratified sampling cannot in real-life situations, i.e. adequately representing a population.

9. 9 BASIC IDEAS A fairly large random collection of sampling units is portioned into small subsets, each subset being of size m, say. Each subset is ranked as accurately as possible with respect to the characteristic of interest without using its exact measurement, and exactly one member of each set with certain specification is quantified.

10. 10 Thus, only a fraction 1/m of the collection of sampling units is quantified for achieving observational economy.

11. 11 SOME USEFUL RANKING CRITERIA Visual perception / Inspection (Colour, Odour, Distressed Vegetation, Distressed Ground, Oily Sheens, Manmade Structures, Surface Water and Groundwater Flow Direction, Wind Direction, etc.) Prior Information Results of earlier sampling episodes Rank-correlated covariates Expert-opinion/expert-systems Some combination of these methods, etc.

12. 12 ILLUSTRATION Suppose we need to draw a sample of three trees from the nine randomly identified trees as shown on the next slide.

13. 13

14. 14 We need to rank the three trees of each group (set) visually with respect their heights and take the measurements of the smallest tree from the first group, of the second smallest from the second group and of the tallest tree from the last group. This yields three measurements representing three independent order statistics. The three trees whose heights are to be measured are shown on the next slide.

15. 15

16. 16 LARGER SAMPLE SIZE For a larger sample size (n) we repeat the process. With the set size as “m” and the number of cycles as “r”, we obtain the sample size, n = mr. For getting a sample of size 6 with set size 3, we need to have 2 cycles.

17. 17

18. 18

19. 19 FOR FIELD INVESTIGATIONS

20. 20

21. 21 STEPS FOR DRAWING A RANKED SET SAMPLE Randomly select m 2 units from the population. Allocate the m 2 selected units as randomly as possible into m sets each of size m. Rank the units within each set based on a perception of relative values of the variable of interest.

22. 22 Choose a sample by including the smallest ranked unit of the first set, then the second smallest ranked unit of the second set, continuing in this fashion until the largest ranked unit is selected from the last set. Repeat steps (1) through (4) for r cycles until the desired sample size, n = mr, is obtained for analysis.

23. 23 Let X 11, X 12, …, X 1m ; X 21, X 22, …, X 2m ; …, X m1, X m2, …, X mm be independent random variables having the same cumulative distribution function F(x). The X ij for the randomly drown units can be arranged as in the following diagram:

24. 24 Set 1 X 11 X 12 … X 1m 2 X 21 X 22 … X 2m. m X m1 X m2 … X mm After ranking the units appear as 1 X 1(1) X 1(2) … X 1(m) 2 X 2(1) X 2(2) … X 2(m). m X m(1) X m(2) … X m(m)

25. 25 The units to be quantified are presented below: 1X 1(1) * … * 2 * X 2(2) … *. m * * … X m(m)

26. 26 FIRST, SECOND AND THIRD ORDER OF NORMAL AND EXPONENTIAL DISTRIBUTIONS SOURCE:PATIL, SINHA AND TAILLIE (1994A)

27. 27 FIRST, SECOND AND THIRD ORDER OF LOGNORMAL AND UNIFORM DISTRIBUTIONS SOURCE:PATIL, SINHA AND TAILLIE (1994A)

28. 28 Let denote the ith order statistic in the jth cycle, where i = 1, 2,..., m and j = 1, 2,..., r. Here the sample size n = mr. Further, we observe that

29. 29 This reveals that all mr quantifications are independent but they are identically and independently distributed (iid) within each row only. The RSS estimator of the population mean is given by

30. 30 This expression is also expressed as

31. 31 RELATIVE PRECISION The relative precision (RP) of the ranked set sample (RSS) estimator of the population mean relative to the corresponding simple random sample (SRS) estimator with the same sample size (mr) is given by

32. 32

33. 33 where 1 ≤ RP ≤ (m+1)/2

34. 34 Relative Savings: As RS >= 0 => RSS is at least as cost efficient as SRS with the same number of quantifications.

35. 35 EFFECT OF SET SIZE

36. 36 IMPLEMENTATION OF RSS The implementation of RSS needs only the rankings of the randomly selected units, and in no way depends upon the method employed for determining the rankings. Thus, investigators can use any or all available information including subjective judgment for this purpose.

37. 37 Note that other sampling methods such as ratio and regression methods require very detailed model specifications for the outside information. These methods can be highly non-robust to violations of those specifications.

38. 38 One of the strengths of RSS is its flexibility and model robustness regarding the nature of the auxiliary information used to perform the ranking. For RSS to be cost effective, the quantification of sampling units rather than their identification, acquisition, and ranking should be the dominant factor of the total cost.

39. 39 RSS WITH CONCOMITANT RANKING RSS presumes that the sampling units are correctly ranked with respect to the variable of interest. But this may not be possible always while dealing with real life situations. In these cases one could take help of some other characteristic for ranking, which is supposedly inexpensive, easily available and highly correlated with the main characteristic of interest.

40. 40 RSS WITH CONCOMITANT RANKING The ranking so obtained may be referred to as concomitant ranking (CR) because of its dependence on a concomitant variable. In classical sampling this is called as an auxiliary variable, and we may denote it by Y while the main variable is represented by X. See Patil, Sinha and Taillie (1994a and b) and Sinha (2005) for a more detailed discussion.

41. 41 RSS WITH CONCOMITANT RANKING In order to obtain the expressions of the variances of the three RSS estimators under concomitant ranking we assume that there is a linear relationship between the main variable X and the concomitant variable Y. This yields that

42. 42 RSS METHODS WITH CONCOMITANT RANKING where X [i: m] denotes the ith order statistic of X based on a concomitant ranking whereas Y (i: m) shows the ith order statistic of Y based on perfect ranking. Thus, the expression

43. 43 RSS METHODS WITH CONCOMITANT RANKING This leads to the following expression for the var (X [i: m] ) for the standard bivariate normal distribution:

44. 44 The Takahasi’s method is slightly less efficient than the McIntyre’s method. But the former has a number of advantages that make it more suitable to the requirements of sampling and monitoring situations.

45. 45 RSS WITH SKEWED POPULATIONS For skewed populations, RSS with the unequal allocation following Neyman’s criterion gives better performance than that with equal allocation. (See Takahasi and Wakimoto (1968). Accordingly, if r i denotes the number of quantifications of units having rank i then

46. 46 Then, where r 1 + r 2 + … + r m = n.

47. 47 Setm Units 1● ○ … r1r1 2...r1...r1 ● ○ …

48. 48 Setm Units 1○ ● … r2r2 2...r2...r2

49. 49...... Setm Units 1... ● rmrm 2...rm...rm... ●

50. 50 RELATIVE PRECISION

51. 51 SYMMETRIC POPULATIONS Under one family of symmetric distributions the variances of the order statistics increase with the rank order until the middle position, and then they decrease to the end (for example, Uniform (0,1), Unfolded Weibull (2,0,1), Symmetric Beta (2) etc.) while for the other family the variances decrease with the rank order until the middle and then they increase to have a symmetric pattern (for example Normal (0,1), Logistic (0,1), Laplace (0,1), etc.). Kurtosis seems to discriminate between the two families of symmetric distributions.

52. 52 For the first family of distributions we need to quantify the extreme order statistics to obtain the minimum variance of the estimator of the population mean. For getting the minimum variance in the second family of distributions we quantify either the middle order statistic or the two closest middle order statistics in 1:1 proportion depending on whether the set size is odd or even.

53. 53 VARIANCE ORDER STATISTICS OF UNIFORM DISTRIBUTION SOURCE:PATIL, SINHA AND TAILLIE (1994A)

54. 54 VARIANCE ORDER STATISTICS OF NORMAL DISTRIBUTION SOURCE:PATIL, SINHA AND TAILLIE (1994A)

55. 55 VARIANCE ORDER STATISTICS OF LOGISTIC DISTRIBUTION SOURCE:PATIL, SINHA AND TAILLIE (1994A)

56. 56 VARIANCE ORDER STATISTICS OF DOUBLE EXPONENTIAL DISTRIBUTION SOURCE:PATIL, SINHA AND TAILLIE (1994A)

57. 57 For the first family of the distributions the RP ext denotes the RP of the RSS estimator of the population mean as compared with the corresponding SRS estimator. It is defined below:

58. 58 For the second family of symmetric distributions RP mid gives the RP of the MRSS estimator of the population mean as compared with the SRS estimator and it is defined below: denotes the variance of the middle order statistic when the set size is odd and that of the any of the two closest middle order statistics when the set size is even.

59. ESTIMATING THE URBAN POPULATIONS 59

60. 60

61. Illustration 1 provides the male, female and the total populations of 145 urban places of Bihar (India) in 1991, which are 5372380, 4533326 and 9905706 respectively according the Census of India 1991 Reports. For estimating the total population of these places one could either estimate the total population directly or estimate the male and female population separately and then by adding the two one could find of the total population. 61 Illustration 1

62. As ranked set sampling uses ranking with respect to the variable of interest, the ranking of the urban places based on the male population may not be the same as that based on the female population if the sex ratio fluctuates from place to place. Considering this fact, we have shown both the estimation methods of the total population. Obviously, if the sex ratio does not differ much from place to place the two methods would provide almost the same value. 62

63. We estimate their total population with the help of a sample of 27 towns / cities. In sampling terminology, here N = 145 and n = 27 (18.62%). Estimation of theTotal Population

64. For obtaining an RSS we need to specify the set size and the number of replications in the beginning. As the ranking of three places with respect to their populations is supposed to be convenient for the field personnel we take the set size as three in this illustration. 64

65. One could consider any number for the set size. In this illustration for getting a sample of size 27 the number of replications is taken as nine because the set size is three. For this sample 81 out of 145 places are located first following SRSWR. Thus, we have m = 3 and r = 9. These places along with their populations under parentheses are presented in 27 sets (rows) each having three places in Table 1.

66. 66 Set 1 2 3 Maner (24343) Warisaliganj (22773) Patratu (33131) Saunda (76691) Ara (157082) Madhupur (39257) Barauli (28311) Bhagalpur (253225) Chapra (136877) 4 5 6 Hasanpur (23941) Chatra (31147) Darbhanga (218391) Hilsa (29923) Buxar (55753) Madhepura (32838) Loyabad (31297) Bairgania (28516) Nawada (53174) 7 8 9 Jagdishpur (21384) Hilsa (29923) Bakhtiarpur (26867) Jugsalai (38623) Jamalpur (86112) Jhajha (31013) Jharia (69641) Loyabad (31297) Barauli (28311) 10 11 12 Chaibasa (56729) Purnia (114912) Dumraon (35068) Samastipur (58952) Simdega (23750) Mango (108100) Kishanganj (64568) Lohardaga (31761) Aurangabad (47565) 13 14 15 Ramgarh Cant. (51264) Patratu (33131) Khagaria (34190) Nawada (53174) Madhubani (53747) Buxar (55753) Chaibasa (56729) RamgarhCant.(51264) Biharsharif (201323) 16 17 18 Barh (45285) Loyabad (31297) Biharsharif (201323) Saunda (76691) Lakhisarai (53360) Khagaria (34190) Mokameh (59528) Purnia (114912) Banka (27369) 19 20 21 Gaya (291675) Darbhanga (218391) Bhabua (27041) Bodh Gaya (21692) Katihar (154367) Marhaura (20630) Bokaro S.City (333683) Araria (45257) Chapra (136877) 22 23 24 Musabani (36909) Darbhanga (218391) Sheikhpura (34429) Mango (108100) Jamalpur (86112) Dinapur Niz. (84616) Bhagatdih (30174) Dumka (38096) 25 26 27 Bakhtiarpur (26867) Barbigha (30148) Adityapur (77803) Jhanjharpur (20019) Loyabad (31297) Sasaram (98122) Supaul (40588) Maner (24343) Table1. Randomly selected 81 towns / cities with their total populations in parentheses

67. Now, we rank the three places of each row separately on the basis of their populations. The rank one is given to the place having the lowest population, the rank two to the place possessing the next higher population and the rank three is accorded to the place with the highest population. 67

68. After ranking, we select the place with the minimum population from the first set, the place with the second rank from the second set, and finally the place with the highest population from the third set. 68

69. Thus, Maner, Ara and Chapra get selected from the set one, two and three respectively. This process is continued for each group of three sets. The selected places with their populations and their ranks are mentioned in Table 2. 69

70. 70 Ranks 123 Maner (24343) Hasanpur (23941) Jagdishpur (21384) Chaibasa (56729) Ramgarh Cant. (51264) Barh (45285) Bodh Gaya (21692) Musabani (36909) Bakhtiarpur (26867) Ara (157082) Chatra (31147) Loyabad (31297) Lohardaga (31761) Ramgarh Cant. (51264) Lakhisarai (53360) Katihar (154367) Jamalpur (86112) Barbigha (30148) Chapra (136877) Darbhanga (218391) Jhajha (31013) Mango (108100) Biharsharif (201323) Chapra (136877) Jamalpur (86112) Loyabad (31297) Table 2: Ranked set sample of 27 towns / cities with their names and total populations

71. The arithmetic means of the towns / cities having ranks one, two and three are obtained as 34268.22222, 69615.33333 and 127923.66670 respectively while their standard deviations are computed as 13719.46743, 52015.78893 and 70701.10526 respectively. Thus, the mean population of the towns / cities of Bihar in 1991 is estimated as = 77269.07405 and the estimate of the total population is obtained as = 11204015.74. 71

72. As the actual population of 145 urban places is 9905706, the difference between the actual and the RSS based estimated population is 1298299, which is 13% as compared with the actual population.

73. 73

74. As in a real life situation, we would not draw a ranked set sample and a simple random sample separately from the given population; we estimate  2 on the basis of the RSS. This estimator is denoted by and its value is obtained as 3927160189. In this case, RP = 1.49273 and RS = 33 %. 74

75. 75

76. This is carried out following the Neyman’s criterion, which is also used in the case of stratified sampling with unequal allocation. 76

77. Here, we obtain r 1 = 3, r 2 = 10 and r 3 = 14 on the basis of the Neyman’s criterion. We, then, note down the populations of three towns/cities with rank one, ten towns/cities with rank two and fourteen three towns/cities with the rank three. The selected places with their populations are given in Table 3.

78. 78 Table 3: Ranked set sample (using unequal allocation) of 27 towns / cities with their total populations Ranks 123 Maner (24343) Warisaliganj (22773) Patratu (33131) Hilsa (29923) Chatra (31147) Nawada (53174) Jugsalai (38623) Loyabad (31297) Barauli (28311) Samastipur (58952) Lohardaga (31761) Aurangabad (47565) Nawada (53147) Madhubani (53747) Biharsharif (201323) Saunda (76691) Purnia (114912) Biharsharif (201323) Bokaro S.City (333683) Darbhanga (218391) Chapra (136877) Mango (108100) Darbhanga (218391) Jamalpur (86112) Sasaram (98122) Supaul (40588) Loyabad (31297)

80. Finally, we compute an estimate of the population mean of town/cities classified according to three ranks as and an estimate of the population total as N = 9872224.19. 80

81. The difference between the actual and the estimated population is 33623, which comes to be 0.339% with respected to the actual population.

82. 82

83. We, thus, get relative savings of 65 %. Next, we estimate the male and the female populations separately to obtain the estimate of the total population. This investigation takes into account the varying sex ratios in different urban places, which could influence the ranking of a place in a set. Also, this, in turn, could help obtain a better estimate of the total population. 83

84. 84 Estimation of Male Population

85. 85 Set 1 2 3 Maner (12919) Warisaliganj (12059) Patratu (18692) Saunda (45011) Ara (84740) Madhupur (21121) Barauli (14320) Bhagalpur (136547) Chapra (73934) 4 5 6 Hasanpur (12533) Chatra (16322) Darbhanga (116915) Hilsa (16178) Buxar (30168) Madhepura (19365) Loyabad (18428) Bairgania (15112) Nawada (28294) 7 8 9 Jagdishpur (11150) Hilsa (16178) Bakhtiarpur (14339) Jugsalai (20699) Jamalpur (47057) Jhajha (16495) Jharia (38500) Loyabad (18428) Barauli (14320) 10 11 12 Chaibasa (30788) Purnia (62265) Dumraon (18729) Samastipur (32428) Simdega (12333) Mango (57763) Kishanganj (34678) Lohardaga (16195) Aurangabad (25827) 13 14 15 Ramgarh Cant. (29201) Patratu (18692) Khagaria (18582) Nawada (28294) Madhubani (28774) Buxar (30168) Chaibasa (30788) RamgarhCant.(29201) Biharsharif (106905) 16 17 18 Barh (24148) Loyabad (18428) Biharsharif (106905) Saunda (45011) Lakhisarai (28513) Khagaria (18582) Mokameh (31274) Purnia (62265) Banka (14721) 19 20 21 Gaya (156909) Darbhanga (116915) Bhabua (14629) Bodh Gaya (11423) Katihar (83977) Marhaura (10995) Bokaro S. City (186338) Araria (24626) Chapra (73934) 22 23 24 Musabani (19650) Darbhanga (116915) Sheikhpura (18782) Mango (57763) Jamalpur (47057) Dinapur Niz. (45474) Bhagatdih (17516) Dumka (20671) 25 26 27 Bakhtiarpur (14339) Barbigha (16099) Adityapur (42413) Jhanjharpur (10344) Loyabad (18428) Sasaram (51928) Supaul (21852) Maner (12919) Table 1.1. Randomly selected 81 towns / cities with their male populations in parentheses

86. 86 Ranks 123 Maner (12919) Hasanpur (12533) Jagdishpur (11150) Chaibasa (30788) Nawada (28294) Barh (24148) Bodh Gaya (11423) Musabani (19650) Bakhtiarpur (14339) Ara (84740) Chatra (16322) Loyabad (18428) Lohardaga (16195) Madhubani (28774) Lakhisarai (28513) Katihar (83977) Jamalpur (47057) Barbigha (16099) Chapra (73934) Darbhanga (116915) Jhajha (16495) Mango (57763) Biharsharif (106905) Chapra (73934) Jamalpur (47057) Loyabad (18428) Table 2.1: Ranked set sample (using equal allocation) of 27 towns / cities with their male populations

87. 87 Ranks 123 Maner (12919) Warisaliganj (12059) Patratu (18692) Hilsa (16178) Chatra (16322) Nawada (28294) Jugsalai (20699) Loyabad (18428) Bakhtiarpur (14339) Samastipur (32428) Lohardaga (16195) Aurangabad (25827) Ramgarh Cant. (29201) Biharsharif (106905) Saunda (45011) Purnia (62265) Biharsharif (106905) Bokaro S.City (186338) Darbhanga (116915) Chapra (73934) Mango (57763) Darbhanga (116915) Jamalpur (47057) Sasaram(51928) Supaul (21852) Loyabad (18428) Table 3.1: Ranked set sample (using unequal allocation) of 27 towns / cities with their male populations

88. 88

89. The difference between the actual and the estimated population is 3.578% with respect to the actual population of 145 places. Further, and the RP ua =2.143, which yields the relative savings as 53%.

90. 90 Estimation of Female Population

91. 91 Set 1 2 3 Maner (11424) Warisaliganj (10714) Patratu (14439) Saunda (31680) Ara (72342) Madhupur (18136) Barauli (13991) Bhagalpur (116678) Chapra (62943) 4 5 6 Hasanpur (11408) Chatra (14825) Darbhanga (101476) Hilsa (13745) Buxar (25585) Madhepura (13473) Loyabad (12869) Bairgania (13404) Nawada (24880) 7 8 9 Jagdishpur (10234) Hilsa (13745) Bakhtiarpur (12528) Jugsalai (17924) Jamalpur (39055) Jhajha (14518) Jharia (31141) Loyabad (12869) Barauli (13991) 10 11 12 Chaibasa (25941) Purnia (52647) Dumraon (16339) Samastipur (26524) Simdega (11417) Mango (50337) Kishanganj (29890) Lohardaga (15566) Aurangabad (21738) 13 14 15 Ramgarh Cant. (22063) Patratu (14439) Khagaria (15608) Nawada (24880) Madhubani (24973) Buxar (25585) Chaibasa (25941) Ramgarh Cant(22063) Bihar Sharif (94418) 16 17 18 Barh (21137) Loyabad (12869) Bihar Sharif (94418) Saunda (31680) Lakhisarai (24847) Khagaria (15608) Mokameh (28254) Purnia (52647) Banka (12648) 19 20 21 Gaya (134766) Darbhanga (101476) Bhabua (12412) Bodh Gaya (10269) Katihar(70390) Marhaura (9635) Bokaro S. City (147345) Araria (20631) Chapra (62943) 22 23 24 Musabani (17259) Darbhanga (101476) Sheikhpura (15647) Mango (50337) Jamalpur (39055) Dinapur Niz. (39142) Bhagatdih (12658) Dumka (17425) 25 26 27 Bakhtiarpur (12528) Barbigha (14049) Adityapur(35390) Jhanjharpur (9675) Loyabad (12869) Sasaram (46194) Supaul (18736) Maner (11424) Table 1.2. Randomly selected 81 towns / cities with their female populations in parentheses

92. 92 Ranks 123 Maner (11424) Hasanpur(11408) Jagdishpur (10234) Chaibasa (25941) Ramgarh Cant. (22063) Barh(21137) Bodh Gaya (10269) Musabani (17259) Bakhtiarpur (12528) Ara (72342) Chatra (14825) Hilsa (13745) Lohardaga (15566) Ramgarh Cant. (22063) Lakhisarai (24847) Katihar (70390) Jamalpur (39055) Barbigha (14049) Chapra (62943) Darbhanga (101476) Jhajha (14518) Mango (50337) Biharsharif (94418) Chapra (62943) Jamalpur (39055) Barbigha (14049) Table 2.2: Ranked set sample (using equal allocation) of 27 towns / cities with their female populations

93. 93 Ranks 123 Maner (11424) Warisaliganj (10714) Patratu (14439) Loyabad (12869) Chatra (14825) Nawada (24880) Jugsalai (17924) Hilsa (13745) Barauli (13991) Samastipur (26524) Lohardaga (15566) Aurangabad (21738) Nawada (24880) Madhubani (24973) Bihar Sharif (94418) Saunda (31680) Purnia (52647) Bihar Sharif (94418) Bokaro S. City (147345) Darbhanga (101476) Chapra (62943) Mango (50337) Darbhanga (101476) Jamalpur (39055) Sasaram (46194) Supaul (18736) Barbigha (14049) Table 3.2: Ranked set sample (using unequal allocation) of 27 towns / cities with their female populations

94. 94

95. Thus, we get the RP under equal allocation as 1.496 with RS as 33% while under unequal allocation these values are obtained as 2.940 and 66% respectively. 95

96. Illustration 2 96 Estimating the total population for Class I and II cities/towns of Bihar for the Census 2001

97. 97 Figure 1. Boxplot of Class I and II cities/towns population

98. 98 SetClass I and II populations, Census 2001 Set 1 9698323207191467 (Lowest) 305525109919119412 (Middle) 61998190873 (Highest)81503 Set 2 12516785590 (Lowest)171687 188050 (Middle)34076781891 131172 (Highest)11905756615 Procedure of drawing a ranked set sample of size 6, (n = mr = 3x2 = 6)

99. 99 Set Lowest, Middle and Highest Class I and II Population Lowest (1)Middle (2)Highest (3) Set 191467119412190873 Set 285590188050131172 Mean88529153731161023 Variance1726956523555875221782104701 Ranked set samples of the populations of class I and II cities/towns

100. 100

101. 101

102. 102

103. 103 Estimating the total population for Class III cities/towns of Bihar for the Census 2001

104. 104 SetClass III population using Census 2001 Set 1 32526 (Lowest)3855454449 36447 (Middle)3143238408 45806 (Highest)4149941958 Set 2 382474844229873 (Lowest) 416102518732293 (Middle) 301093404238014 (Highest) Set 3 2749222936 (Lowest)38672 7939348306 (Middle)30082 66797 (Highest)2986843113 Set 4 373703483634653 (Lowest) 2552428085 (Middle)29991 3110633738 (Highest)33490 Procedure of selection of ranked set samples of size, n = mr = 3x4 = 12

105. 105 Set Lowest, Middle and Highest class III Population Lowest (1)Middle (2)Highest (3) Set 1325263644745806 Set 2298733229338014 Set 3229364830666797 Set 4346532808533738 Mean299973628346089 Variance2598242575902243215551393 Ranked set sample of size n = mr = 3x4 = 12

106. 106

107. 107

108. 108 Estimating the total population for Class IV and V cities/towns of Bihar for the Census 2001

109. 109 SetClass IV and V population, Census 2001 Set 1 258112126218710 (Lowest) 14469 (Middle)1376917982 1905020177 (Highest)9366 Set 2 2195715300 (Lowest)20196 17912 (Middle)2357617621 22354 (Highest)1388219567 Set 3 7745 (Lowest)860820871 178402086020741 (Middle) 1452624992 (Highest)19928 Procedure of selecting ranked set sample of size, n = mr = 3x3 = 9

110. 110 Ranked set sample of size, n = mr = 3x3 = 9 Set Lowest, Middle and Highest class IV and V Population SetLowest (1)Middle (2)Highest (3) Set 1187101446920177 Set 2153001791222354 Set 377452074124992 Mean139181770722508 Variance3148955898659125813766

111. 111

112. Therefore, the estimated population of 38 Cities = 703728 and the actual population = 695552. Error = |Actual population – Estimated population | = 8176 112

113. 113

114. 114 The results obtained for RP, RC and RS for the population of Bihar according to the Census 2001 for the class I and II, class III and class IV and V are mentioned below. ClassRPRCRS Class I and II population (2001) 1.430.700.30 Class III population (2001) 1.250.800.20 Class IV and V population (2001) 1.560.640.36

115. Concluding Remarks Though all the estimates are based on specific samples, the performance of RSS with unequal allocation is better for estimating the total population. This is true when the total population is estimated directly on the basis of the total populations of the sampled places as well as when the estimated total population is obtained as the sum of the separately estimated male and female populations. 115

116. The directly obtained estimate of the total population is 9872228 while the estimated population based on the estimated male and female population is 9882234. The latter estimate is closer to the actual total population. 116

117. Further, illustration 2 shows that the RSS method based on only 20 % of the population gives much better estimate than the SRS method. We could save almost 80 % of the expenses by the new method. 117

118. 118 REFERENCES Census of India (1991). Series -5, Part II -A, Bihar Published by the Registrar General of Census, Gov. of India, New Delhi“ General Population Tables”, pp.306-348. Kumar, Vijay (2013). Some Contributions to Theory and Designs of Survey Sampling for Estimating Multiple Characteristics. Ph.D. thesis in Statistics, Patna University, Patna, India.

119. 119 Halls, L. K. and Dell, T. R. (1966). Trial of ranked set sampling for forage yields. Forest science, 12, 22-26. McIntyre, G. A. (1952). A method of unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 3, 385-390.

120. Norris, R. C., Patil, G. P. and Sinha, A. K. (1995). Estimation of multiple characteristics by ranked set sampling methods. Coenoses, 10, 95-111. Patil, G. P., Sinha, A. K. and Taillie, C. (1994a). Ranked set sampling. In: Handbook of Statistics, vol. 12 (Environmental Statistics), G. P. Patil and C. R. Rao eds, North Holland/Elsevier Science, B. V., New York, 167-200. 120

121. 121 Patil, G. P., Sinha, A. K. and Taillie, C. (1994b). Ranked set sampling for multiple characteristics. International Journal of Ecology and Environmental Sciences 20, 357-373. Sinha, A. K. (2005). On some recent developments in ranked set sampling. Bulletin of Informatics and Cybernetics, 37, 137-160. Sinha, A. K., Perez-Abreu, V., Patil, G. P. and Taillie, C. (2001). On the effectiveness of the Takahasi’s ranked set sample estimator as compared with the regression estimator. American Journal of Mathematical and Management Sciences, 20, Nos. 1 & 2, 145-163.

122. 122 Stokes, S. L. (1980). Estimation of variance using judgment ordered ranked set samples. Biometrics 36, 35-42. Takahasi, K. and Wakimoto, K. (1968). On unbiased estimates of the population mean based on the sample stratified means of ordering. Annals of the Institute of Statistical Mathematics, 20, 1-31.

123. 123 THANK YOU VERY MUCH!