Estimation of Some Proportion in a Clustered Population

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1 Nonlinear Analysis: Modelling and Control, 2009, Vol. 14, No. 4, Estimation of Some Proportion in a Clustered Population D. Krapavicaitė Institute of Mathematics and Informatics Aademijos str. 4, LT Vilnius, Lithuania rapav@tl.mii.lt Received: Revised: Published online: Abstract. The aim of this paper is to show that the traditional design-based estimator for the proportion of population units, associated with at least one subunit having an attribute of interest using the two-stage sampling design, is biased. We face such a situation in the Adult Education Survey of official statistics of the European countries when estimating the share of individuals in non-formal education, involved in job-related learning activities. The alternative design and model-based estimators are proposed. Keywords: finite population, proportion, two-stage sampling design, bias. 1 Introduction A new problem related to the estimation of a proportion has arisen in the Adult Education Survey of the European countries ( [1 3], hereinafter referred to as AES. The parameter of interest is the share of individuals in non-formal education involved in job-related learning activities. According to the sampling design, the individuals included into the first stage AES sample, present the second stage simple random sample of sie m 3 of the learning activities of non-formal education in which they have been involved during a year. Some of them are job-related, but some of them are not job-related. Even if there are no job-related learning activities in the sample, they can occur among non-sampled ones, and have to be taen into account. The problem has arisen in practical wor. The author has not met any similar problem solved, or at least touched, in the literature. In the paper, the problem is described in the general framewor, and it is shown by an example that the design-based estimator of this parameter is biased, and the sie of bias is demonstrated by an example. In order to tae into account possible non-sampled job-related learning activities for sampled individuals assumption on the distribution of the number of such learning activities for each individual is made. Alternative design and model-based estimators are proposed. Their application is shown by Examples 5,

2 D. Krapavicaitė 2 Population and parameters Let us denote by U 1 = {u 1, u 2,..., u N }, (or U 1 = {1, 2,..., N} without restriction of generality population of units, with each of which a cluster of subunits U 2i of sie M i, i = 1, 2,...,N, is associated. Thus, the population of all subunits U 2 consists of M = M M N elements: U 2 = N U 2i. Suppose that some of the subunits have an attribute of interest, some of them do not have it. Let us introduce an attribute indicator a study variable in population U 1 with the value i = 1, if there is at least one subunit with the attribute among M i subunits associated with the unit u i, and i = 0, otherwise, i = 1, 2,...,N. Then the number of units in the population associated with at least one subunit having an attribute is equal to the total of the variable : t = i. (1 The share (proportion of the units in U 1 having at least one subunit associated with the attribute is equal to the mean of the variable : µ = t /N. Let us consider estimation of parameters t and µ from the survey data. 3 Sample and the usual estimator The sample design of subunits that consistute population U 2 is described by a 2-stage sampling design with some probabilistic sample s I of n units from U 1, at the first stage, and a simple random sample s IIi of m i subunits in the cluster associated with the unit u i (or all of them if their number is smaller than m i, at the second stage: s = i s I s IIi U 2, s IIi U 2i. At the second stage, the sie m i of the sample s IIi can be any positive number, but, for simplicity, without loss of the generality, let us consider { M i, if M i = 0, 1, 2, m i = 3, if M i 3, for i s I. This is the case in the Lithuanian AES. Denote by d i = 1/π i the first stage sampling design weight with the first and second order inclusion probabilities π i = P(s I : i s I > 0, π ii = π i, π ij = P(s I : i s I & j s I > 0, i, j, U 1, i j. The Horvit-Thompson estimator of the population total t of the variable i π i s i I 474

3 Estimation of Some Proportion in a Clustered Population cannot be used to estimate the number of units, associated at least one subunit with the attribute in the population U 1 because for m i < M i the values i may be not observable. The often used design-based estimator for the number of units associated with at least one subunit having an attribute is ˆt = i: i s I d i ẑ i, (2 where ẑ i is the design-based estimator of i : { 1, if at least one subunit with the attribute belongs to s IIi, ẑ i = 0, otherwise. (3 For the share of units associated with at least one subunit having an attribute, the suggested design-based estimator is µ = ˆt /N. (4 This estimator is usually used in the pilot AES of statistical offices in European community. Hypothesis: estimators (2 and (4 are biased, e. g., Eˆt t, E µ µ, the expectation is taen here with respect to the two-stage sampling design. The following examples confirm the hypothesis. 4 Bias We show by example 1 that using design-based approach to the problem we unavoidably obtain the biased estimator of a parameter. Example 1 (Existence of a bias of estimator ˆt. Let us study a Small Population U 1 = {u 1, u 2, u 3 } consisting of N = 3 units. The unit u 1 is associated with one subunit without an attribute, denoted as nonattr; the unit u 2 is associated with one subunit with an attribute, denoted as attrib; the unit u 3 is associated with two subunits: one with an attribute (attrib and one without an attribute (nonattr. For this population, the number of units with an attribute and their share is equal to t = = = 2, µ = 2/3. Let us draw the first-stage simple random sample s I of n = 2 elements from population U 1. The possible realiations of the sample according to this sampling design and their sampling probabilities are: s I1 = (u 1, u 2, s I2 = (u 1, u 3, s I3 = (u 2, u 3, P(s I1 = P(s I2 = P(s I3 =

4 D. Krapavicaitė Let us simplify the sample design, taing for the sample of subunits { M i, for M i = 0, m i = 1, for M i 1. The second stage sampling design probabilities are as follows: P(nonattr u 1 = 1, P(attrib u 1 = 0, P(nonattr u 2 = 0, P(attrib u 2 = 1, P(nonattr u 3 = P(attrib u 3 = 1 2. Let us estimate t using estimator (2 and the data of these samples: s I1 = (u 1, u 2 : ˆt (1 = N n ( = 3 2 (0 + 1 = 3 2, µ(1 = 1 2. For the element u 3, we estimate ẑ 3 = 1, if a unit with an attribute is selected for the second-stage sample, and ẑ 3 = 0, otherwise. s I2 = (u 1, u 3 : if s II3 = {nonattr}, then ˆt (2 if s II3 = {attrib}, then ˆt (3 s I3 = (u 2, u 3 : if s II3 = {nonattr}, then ˆt (4 if s II3 = {attrib}, then ˆt (5 = N n ( 1 + ẑ 3 = 3 (0 + 0 = 0, µ(2 = 0, 2 = N n ( 1 + ẑ 3 = 3 2 (0 + 1 = 3 2, µ(3 = 1 2, = N n ( 2 + ẑ 3 = 3 2 (1 + 0 = 3 2, µ(4 = 1 2, = N n ( 2 + ẑ 3 = 3 (1 + 1 = 3, µ(5 = 1. 2 Let us calculate the expectation of ˆt with respect to the sampling design: Eˆt = ˆt (1 P(s (2 I1 + (ˆt P(nonattr u 3 + ˆt (3 P(attrib u 3 P(s I2 (ˆt (4 + P(nonattr u 3 + ˆt (5 P(attrib u 3 P(s I3 = 3 ( ( = = 3 2 t = 2. It means that the estimator ˆt is biased. Consequently, E µ = Eˆt N = 1 2 µ = 2 3, 476

5 Estimation of Some Proportion in a Clustered Population and the estimator µ of the proportion of the units with an attribute is also biased. It is clear by intuition that estimator (2 underestimates the true number of the units with an attribute, because the cases are possible, where a sampled unit based on the sampled subunits is classified as without an attribute (cases s I2, s I3 with s II3 = {nonattr}, while in reality there exists a non-sampled subunit with an attribute associated with it. On the other hand, there are no possible cases where a sampled unit is classified as being associated with the subunit with an attribute, as in reality it is not so. The situation is visualied in Fig. 1. Big circles mean units, all the small circles mean subunits; the small blac circles mean subunits with an attribute. A subunit joined with the unit means a sampled subunit. + means ẑ i = 1, means ẑ i = 0,? means ẑ i = 0 and a source of bias. Fig. 1 show that estimator (2 underestimates the number of units with an attribute. Fig. 1. Sample of subunits. Example 2 (Sie of a bias. Let us write an expression for the bias in some special case. Denote by X i the number of subunits with the attribute associated with the ith unit, and by Y i the number of sampled subunits with the attribute. Consider M i = M, m i = m, X i = > 0 are fixed numbers for all i = 1,...,N. It means that each population unit is associated with exactly subunits having attributes. The numbers of sampled subunits with the attribute Y i are independent identically distributed random variables. Suppose they have the same distribution as the random variable Y. For self-weighting 1st stage sampling design, the population sie can be expressed in such a way: N = t = NP(Y = 0 + NP(Y > 0. For the estimator ˆt in our example we have: Eˆt = E i s I d i ẑ i = NP(Y > 0. From the last two expressions we obtain Bias(ˆt = Eˆt t = NP(Y =

6 D. Krapavicaitė For our sampling design we calculate: Then P(Y = 0 = P(Y = 0 X = = C0 Cm M = Bias(ˆt = N ( 1 ( 1 M M 1 C m M... ( 1. M m + 1 ( 1 ( 1 ( M M 1 M m + 1 Some numerical values of the bias in the case the parameters close to the Lithuanian AES ones are given in Table 1. Table 1. Values of the Bias(ˆt for the case N = , M = 10, m = 3, = 1, 2,..., M Bias(ˆt `Bias(ˆt /N 100 (% , 9, We see that the higher the number of subunits with an attribute in the population, the lower the bias of estimator (2 is for the number of units associated with the subunits with an attribute. Bias is unavoidable when M 3. In order to adjust estimator to the bias, we introduce a superpopulation model for the distribution of the number of subunits with the attribute. 5 Alternative estimators We propose some design and model-based estimator for the proportion of the first-stage sampling elements associated with the subunits with an attribute under the two-stage sampling design. Some auxiliary assumptions on the superpopulation of subunits have to be stated. 1. Suppose that the number M i of subunits associated with the unit u i is fixed and nown, but the number of subunits X i with an attribute is random, 0 X i M i, i = 1,...,N. Let us define the probabilities p Mi ( = P(X i = M i, M i p Mi ( = 1, i = 1, 2,..., N. =0 478

7 Estimation of Some Proportion in a Clustered Population We consider these probabilities (distribution of the variable X i to be nown. 2. The values of the study variable become random because they depend on the values of the random variables X i, i = 1,...,N. The population total t is also random. 3. The number of sampled subunits with an attribute, Y i, is random, 0 Y i min(3, X i. The values Y i, Y j are independent for i j. For estimator (3 of the value i of the attribute indicator (study variable, the following relationship is valid: 1, if Y i > 0 { X i > 0, i = 1, ẑ i = X i > 0, i = 1, (5 0, if Y i = 0 X i = 0, i = 0. Conditional distribution of Y i under the condition that the value of X i is nown, is also nown due to the nown simple random sampling design of subunits. Taing into account (5, we obtain an expression for the probability, denoted by p i, that the variable i obtains value 1: p i = P( i = 1 = P(X i > 0, or, equivalently, p i = P(Y i > 0 + P(X i > 0 Y i = 0P(Y i = 0. (6 In order to obtain the new estimator of the total t, the value of ẑ i in (2 is changed by p i from (6. The expectation of t with respect to the distribution of X i, i = 1,...,N, is E X t = E X i = P(X i > 0 = p i. We are going to estimate this expectation. Estimator A. Let us estimate t by ˆt (A = i s I d i p i. (7 This is a Horvit-Thompson type estimator of the total of the study variable with the values p i, i = 1, 2,...,N, and we use further the well nown result [4, p. 43], for this estimator. Proposition A. Suppose the probabilities p Mi (, = 1, 2,...,M i, i = 1, 2,...,N, M i > 0, are fixed and nown. Then (i the estimator ˆt (A given in (7 is unbiased for E X t under the sampling design described in Section 3: Eˆt (A = p i, 479

8 D. Krapavicaitė (ii its variance V ar (A (ˆt = N 1 π i π i p 2 i + i,j=1 i j (iii the estimator of variance (A 1 π i V ar (ˆt = π 2 p 2 i + i s I i i,j s I i j (A is unbiased for V ar (ˆt. (π ij π i π j p ip j π i π j, π ij π i π j π ij p i π i p j π j Estimator B. Let us introduce a design and model-based estimator of a value i { 1, if Y i > 0, ẑ i = P(X i > 0 Y i = 0, if Y i = 0. and define a new estimator of the total t : ˆt (B = i s I d i ẑ i. (8 The probability P(X i > 0 Y i = 0 in the case Y i = 0 is included here in the definition of ẑ i in comparison to ẑ i in (3. Proposition B. Assume the distribution of random variables X i, i = 1, 2,..., N, to be nown, and M i, M i > 0 to be fixed and nown. Then (i the estimator ˆt (B, given in (8, is unbiased for E X t under the sampling design described in Section 3 and model: Eˆt (B = (ii its variance V ar (B (ˆt p i, (9 = V ar (ˆt (A N ( + d i pi p 2 i P(X i > 0 Y i = 0p Mi (0, (iii the suggested estimator of variance is (B V ar (ˆt = V ar (ˆt (A + i s I d 2 i ( pi p 2 i P(X i > 0 Y i = 0p Mi (0. The first term in the expression of V ar(ˆt (B is due to the sampling design, and second term is due to the distribution of X i, i = 1, 2,...,N. 480

9 Estimation of Some Proportion in a Clustered Population Remar 1. If X i is non-random, then { 1, if X i > 0, p i = 0, if X i = 0, and Eˆt (A = Eˆt (B = t is the number of population units associated with at least one subunit with an attribute. Remar 2. Calculation of the probability P(X i > 0 Y i = 0 used for V ar(ˆt (B for m = 3: P(X i > 0 Y i = 0P(Y i = 0 Hence, = = = M i 3 =1 M i 3 =1 M i 3 =1 P(X i > 0 Y i = 0 = P(X i = Y i = 0P(Y i = 0 P(Y i = 0 X i = P(X i = (1 Mi ( 1 1 P(Y i = 0 M i 3 =1 ( 1 M i 1 (1 Mi ( 1 p Mi (. (10 M i 2 ( 1 M i 1 p Mi (. M i 2 Estimator C. In practice, distribution of X i is not nown and it is estimated. Suppose that estimators p Mi ( are used for the probabilities p Mi (. Then we define P(X i > 0 Y i = 0 = 1 P(Y i = 0 M i 3 =1 (1 Mi ( 1 p i = P(Y i > 0 + P(X i > 0 Y i = 0. ( 1 M i 1 p Mi (, M i 2 Denote E bp (, V ar bp ( expectation and variance with respect to the distribution of the estimators p Mi (, = 1,..., M i, i = 1,..., N, { 1, if Y i > 0, ẑ i = P(X i > 0 Y i = 0, if Y i = 0, as well as the estimator of the total t ˆt (C = d i ẑ i. (11 i s I 481

10 D. Krapavicaitė Proposition C. Assume that the estimators p Mi ( are defined for the probabilities p Mi (, i = 1, 2,..., N, = 1,...,M i. Then (i the expectation ( under model, design and distribution of p Mi ( of estimator ˆt (C, given in (11, is Eˆt (C = E bp p i, (ii its variance is expressed by ( (C V ar (ˆt =V ar d i E bp p i + d i V ar bp ( p i i si + d i E bp ( pi p 2 i + P(Y i = 0 P(X i > 0 Y i = 0 ( P(Xi > 0 Y i = 0 1. The first term in the expression of variance is due to the sampling design, the third term is is due to the distribution of X i, i = 1, 2,..., N, and the second term is due to estimation of the superpopulation distribution probabilities. Remar 3. The situation can occur that the clusters of subunits are associated only with some, but not all the elements of the population. Then the number of units n in the sample associated with some subunits may be random, and n n. This invoes one more source of randomness in the estimators of the number of population units, associated with the subunits having attributes, which is not considered here. Estimation of the proportion. From the equalities µ = ˆt /N, V ar( µ = V ar(ˆt /N 2, V ar( µ = V ar(ˆt /N 2 we can obtain the estimator needed for a proportion, using any estimator of the total presented above. 6 Possible distributions of X i Example 3. Let any subunit attached to some unit have an equal probability p (0, 1 of bearing the attribute, and subunits have attributes independently of one another. Then the number of subunits X i is distributed according to the binomial distribution with the parameter p, and p Mi ( = P(X i = M i = C M i p (1 p Mi, = 0, 1,...,M i. 482

11 Estimation of Some Proportion in a Clustered Population Then p i = P(X i > 0 = 1 P(X i = 0 = 1 (1 p Mi. For some case of the binomial distribution of X i, the probabilities p Mi ( are given in Table 2. They have a pea for = 0 (0, M i with fixed M i. Table 2. Probabilities p Mi ( for the binomial distribution of X i and p = 0.6 M i Example 4. Each subunit associated with the ith population unit has an attribute with its own probability p (i,j, j = 1, 2,..., M i. Attributes are obtained by the subunits independently of one another. Then the probabilities needed are as follows: P(X i = 0 = ( 1 p (1,i( 1 p (2,i... ( 1 p (Mi,i, p i = 1 P(X i = 0 = 1 ( 1 p (1,i( 1 p (2,i... ( 1 p (Mi,i, p Mi (u = p (,i (1 p (,i. ω U 2i ω =u ω U 2i\ω Example 5. Let us suppose the superpopulation distribution of the number of subunits having attribute (variables X i, i = 1, 2,..., N to be nown, e. g. probabilities p i ( = p Mi ( = P(X i = M i, = 1, 2,...,M i, i = 1, 2,...,N, to be nown. We apply estimator B for the Small Population described in Example 1. Probabilities defining distribution of X 1, X 2, X 3 Model 1 are p 1 (0 = 1, p 1 (1 = 0, p 2 (0 = 0, p 2 (1 = 1, p 3 (0 = 0, p 3 (1 = 1, p 3 (2 = 0. Then we calculate according to (10 P(X 3 > 0 Y 3 = 0P(Y 3 = 0 = ( 1 1 p 3 (1 = (12 We can find easily P(Y 3 > 0 = 1/2. Hence, P(Y 3 = 0 = 1 P(Y 3 > 0 = 1/2. From (12 we obtain P(X 3 > 0 Y 3 = 0 = 1. Then we can calculate all the estimates: ˆt (B1 = 3 2, ˆt (B2 = 1, ˆt (B3 = 3 2, ˆt (B4 = 3, ˆt (B5 =

12 D. Krapavicaitė The average of the estimator (8 with respect to the design and model is Eˆt B = 1 3 (ˆt (B1 and we obtain Eˆt (B + ˆt (B2 P(Y 3 = 0 + ˆt (B3 P(Y 3 > 0 + ˆt (B4 P(Y 3 = 0 + ˆt (B5 P(Y 3 > 0 (13 On the other hand, we have that = 2. It means Eˆt (B = t, and unbiasedness of the estimator B. p 1 + p 2 + p 3 = P(X 1 > 0+P(X 2 > 0+ ( P(Y 3 > 0+P(X 3 > 0 Y 3 = 0P(Y 3 = 0 = /2 + 1/2 = 2 coincides with Eˆt (B, as it is said in Proposition B. Example 6. Let us suppose other models for superpopulation distribution of X i in Small Population of Example 1 for 0 < ε < 1. The results of estimation are presented in Table 3. Table 3. Results of the total estimation in the case of Small Population and various superpopulation models Model 2 Model 3 p 1(0 = 1, p 1(0 = 0, p 1(0 = 1, p 1(0 = 0, p 2(0 = 0, p 2(1 = 1, p 3(0 = ε, p 2(0 = 0, p 2(1 = 1, p 3(0 = ε 2, p 3(1 = 1 2ε, p 3(2 = ε p 3(1 = 2ε, p 3(2 = 1 2ε ε 2 P(Y 3 > 0 1/2 1 ε ε 2 P(Y 3 = 0 1/2 ε + ε 2 P(X 3 > 0 Y 3 = 0 1 2ε 1/(1 + ε ˆt (B1 3/2 3/2 ˆt (B2 3(1 2ε/2 3/`2(1 + ε ˆt (B3 ˆt (B4 ˆt (B5 3/2 3/2 3(1 ε 3`1 + 1/(1 + ε /2 3 3 Eˆt (B 2 ε 2 ε 2 Average of the estimator ˆt (B with respect to the design and model is calculated according to the formula (13. We see how average of the estimator ˆt (B depends on the probability of the unit to have at least one subunit with the attribute. We see also the expectation of the estimator ˆt (B for changed model assumptions (distribution of X 3. Model 3 shows distribution of X 3 for small ε of the type similar to the distribution in Lithuanian AES (compare Table 4, and this is, of course, non-linear function. 484

13 Estimation of Some Proportion in a Clustered Population Example 7. Let us try to find an approximation of the distribution of the number of the subunits with an attribute attached with some unit, which is met in the Lithuanian AES. n = individuals participated in non-formal education in the sample of the year 2007 of Lithuanian AES. The estimated probabilities p Mi ( = P(X i = M i are given in Table 4. They are increasing with an increase of for fixed M i and are far from those given in Table 2. Table 4. Relative frequencies bp Mi ( of the number of job-related learning activities in non-formal education in the Lithuanian AES M i The analytical expression of the function can be used for approximating the probabilities p Mi ( for real data: f(x = (1 + cx α, α > 0, c > 0, x 0. Choosing the proper parameters α, c we derive p Mi ( = P(X i = M i = (1 + c α Mi j=0 (1 + cjα, = 0, 1,...,M i. (14 The probabilities p Mi ( = P(X i = M i estimated, using this function with α = 6 and c = 2, are given in Table 5. They seem to be quite close to the values of the real survey in Table 4. Table 5. Estimated probabilities bp Mi ( = b P(X i = M i using the function proposed in (14 M i The empirical results show that probability approximations of the type (14 can be used in estimator C (11 for AES. We were successful in the Lithuanian AES of the year 2007: there are no cases in the sample with M i > 3 and Y i = 0. Anyway, there can be the case in subsequent survey and the estimators proposed may be needed. 485

14 D. Krapavicaitė 7 Conclusion The alternative estimators have been proposed for the proportion of the population units associated with at least one subunit with the attribute of interest, using the two-stage sampling design and assumptions on the superpopulation distribution of the number of subunits having the attribute. The Examples 1, 5 and 6 show that estimator B allows to obtain unbiased estimates to the problem. The success of usage of the estimators proposed depends on nowledge of the distribution of the number of subunits with the attribute associated with the population units. Acnowledgements The author is thanful to Statistics Lithuania for a possibility to analye the survey data and to anonymous referees for useful comments. Appendix Proof of the Proposition B. The expectation of the estimator under the design and model is as follows: Eˆt (B = E i s I d i E(ẑ i s I = E i s I d i ( P(Yi > 0 + P(X i > 0 Y i = 0P(Y i = 0 = E i s I d i p i = p i. The variance is calculated taing into account that sampling designs at both stages are independent, and Y i, i s I, are independent random variables: ( (B V ar (ˆt = V ar E(ˆt (B s I + E ( V ar(ˆt (B s I ( ( = V ar d i E(ẑ i s I + E d 2 i V ar(ẑ i s I. i s I i s I Hence, V ar (B (ˆt = V ar (ˆt (A N + d i V ar(ẑ i. (15 For V ar(ẑ i = Eẑ 2 i (Eẑ i 2, we find: Eẑ i = P(Y i > 0 + P(X i > 0 Y i = 0P(Y i = 0 = p i, 486

15 Estimation of Some Proportion in a Clustered Population Eẑ 2 i = P(Y i > 0 + P(X i > 0 Y i = 0 2 P(Y i = 0 = p i + P(X i > 0 Y i = 0P(Y i = 0 ( P(X i > 0 Y i = 0 1 = p i P(X i > 0 Y i = 0P(Y i = 0P(X i = 0 Y i = 0 = p i P(X i > 0 Y i = 0P(X i = 0 = p i P(X i > 0 Y i = 0p Mi (0. Hence it follows that V ar(ẑ i = p i p 2 i P(X i > 0 Y i = 0p Mi (0. By substituting V ar(ẑ i in (15, we obtain the expression of variance. The estimator of variance is obtained using the expression of (iii Proposition A for the first term and the unbiased Horvit-Thompson estimator of the total for the second term of the variance. References 1. Eurostat, Adult Education survey AES, cache/ity SDDS/EN/ trng aes base.htm access/. 2. Eurostat, CIRCA a collaborative worspace with partners of the European Institutions, 3. Eurostat, Database, pa geid=1996, & dad=portal& schema=portal &screen=welcomeref &open=/edtr/trng/trng aes&language=en&product=eu MASTER education training&root=eu MASTER education training&scrollto=0. 4. C.-E. Särndal, B. Swensson, J. Wretman, Model Assisted Survey Sampling, New Yor, Springer,

NONLINEAR CALIBRATION. 1 Introduction. 2 Calibrated estimator of total. Abstract

NONLINEAR CALIBRATION. 1 Introduction. 2 Calibrated estimator of total. Abstract NONLINEAR CALIBRATION 1 Alesandras Pliusas 1 Statistics Lithuania, Institute of Mathematics and Informatics, Lithuania e-mail: Pliusas@tl.mii.lt Abstract The definition of a calibrated estimator of the