Estimation of Survey Accuracy for Percentile Estimators in Exponential Populations

Determining accuracy of survey results is important due to their application in planning. In the previous work, estimation of survey accuracy for median estimator in populations with normal distribution is assessed. In this paper, a right skewed population that is exponential distribution is considered. Besides, instead of median, percentile estimators is studied. Furthermore, under some assumptions, accuracy of percentiles as well as median estimator is assessed under some assumptions. Mathematics Subject Classification: 62D05


Introduction
Estimation of survey precision and survey accuracy of median and the other percentiles is different from many of the other nonlinear estimators.Alimohammadi [1] present some equations to compute survey accuracy of median estimator in surveys for normal populations.Alimohammadi [2] assess survey precision of nonlinear estimators and present some relations to calculate survey precision based on taylor's approximation and the proposed response error model.
In this paper, survey accuracy of percentile estimators as well as median estimator in exponential distribution is studied.Therefore, total variance, including sampling as well as nonsampling variance should be estimated.Response error is an unavoidable part of nonsampling error.In the next sections, some response error model is considered.One of the applications of the model is to quantify response error in surveys.

Response Error Models and Survey Accuracy
To study of total survey error, Mean Square Error (MSE) criterion may be applied.Then bias and variance of the estimators should be determined.Total survey variance has an inverse relation with survey precision.Therefore, survey precision is an essential component of survey accuracy.In this study, precision and accuracy of percentile estimators in exponential populations is computed under some assumptions.These assumptions included the cases that the other nonsampling errors are ignorable except that response error.In the other words, processing error may be ignorable now, because of technology's development.If a proper frame is applied in the survey, then coverage error does not occurred.On the other hand, there are several methods to reduce nonresponse rate.Usually response error is an unavoidable part of nonsampling error.
Biemer and Trewin [1] have considered a general models for response error.Alimohammadi and Navvabpour [2] proposed a model of response error in face to face surveys.In this paper, the following response error model is considered: Where µ ls is the true value and y ls is the observed value of lsth unit, D l is the effect of lth interviewer on any interview and R ls is respondent error.Usually, interviewer effect D l and respondent effect R ls are random effects in surveys.In model ( 1), effect of respondents nested in interviewers.Assume that these random variables distributed as N (0, σ 2 D ) and N (0, σ 2 R ) respectively and µ ls ∼ N (µ, σ 2 M ).In this paper, variance of ȳ under model ( 1) is called response variance of ȳ.It is denoted by V ar z (ȳ).Two-stage sampling design is a commonly used method in surveys.It can be shown that, for two-stage cluster sampling design, V ar z (ȳ) equals to: Where n is the number of sampled clusters, L is the number of interviewers, h is the number of clusters assigned to each interviewer such that n=hL, m is the number of sampled units from each of the sampled clusters and t = σ 2 R + σ 2 M .σ 2 D and σ 2 R are variance of effects D and R in model (1), respectively.
3 Survey Accuracy for Percentiles as well as Median Estimator Percentiles of some survey variables may have many applications.Median as a percentile, is used quite often in social science and social policies.For example, median of population income is a more proper central tendency criterion than the mean.Because of asymmetric distribution of income and it is well known that income has a right skewed distribution, in this paper exponential distribution is considered.
The method of estimating survey accuracy is based on the straight definition of percentiles in continues populations.
In exponential distribution function with parameter λ, as follows: kth percentile equals to: For k = 50 in equation ( 3), median may be obtained as: On the other hand, E(X) = µ = 1 λ .Easily Maximum Likelihood estimator of kth percentile may be obtained as αML = (ln(100) − ln(100 − k)) Ȳ .Where Ȳ is the sample mean estimator of observations.Maximum Likelihood estimator of median estimator is med M L = ln(2) Ȳ .
Total variance of percentile estimator is For median, k = 50 and V ar T ( med M L ) = (ln(2)) 2 V ar T ( Ȳ ).
Alimohammadi [4] estimates Maximum Likelihood (ML) estimator for effects of a proposed response error model.
In this paper, ML estimator of model ( 1) effects are obtained as follows: Where L is the number of interviewers, N is the sample size, SSD and SSR are the sum of squares of effects D and R respectively.By replacing the above relations in equation ( 3), ML estimator of response variance of kth percentile estimator in exponential distribution may be obtained as follow: For (1 Where MSD and MSR are mean squares of effects D and R respectively.Under the mentioned assumptions about nonsampling errors, and approximating it by response error as the main part of nonsampling error, sampling variance may be added to response variance for computing total variance.
On the other hand, ML estimators of kth percentile in exponential distribution is an unbiased estimator.Since, Besides, computing expectation under model (1) indicates that, ML estimator of percentile is unbiased.Then, total error with MSE criterion equals to total variance.Finally, the obtained relations may be applied to estimate survey accuracy of percentile estimators under the mentioned assumptions.

Conclusion
Estimation of survey precision and accuracy of survey results is important due to their application in planning and decision making.Median and some percentiles have considered in many area.In this paper, accuracy of percentiles estimators is assessed under some assumptions.Besides, maximum likelihood estimators of the considered response error model is obtained.Furthermore, some relations is presented to estimate precision and accuracy in surveys based on the considered response error model and two stage cluster sampling design.
The results of this paper may be applied to estimate precision and accuracy of median estimator and the other percentiles in surveys.