Improved estimation of mean in randomized response models

The present investigation considers the problem of estimating the mean of a sensitive quantitative variable μA in a human population survey, using the scrambled response technique suggested by Ryu, Kim, Heo and Park (On stratified randomized response sampling, Model Assisted Statistics and Application 1(1), 31–36, 2005–2006). Specifically, using the prior estimate (or guessed mean) of the mean of a population, a family of estimators μAk is presented to estimate the population mean μA, and its properties are examined. The optimum value of the de- gree k(0 $geq k geq$ 1) of the belief in the prior estimate depends, besides others, on the unknown population parameters, e.g. mean and vari- ance, so the proposed family of estimators may have limited practical applications. In an attempt to overcome this problem, another esti- mator based on the estimated optimum value of k has been proposed. The proposed estimator has been compared with the Ryu et al. and Hussain and Shabbir (Improved estimation procedure for the mean of a sensitive variable using randomized response model, Pakistan Journal of Statistics 25(2), 205–220, 2009) estimators assuming simple random sampling with replacement.

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[1] Ahmed, S. E. and Rohatgi, V.K. Shrinkage estimation of proportion in randomized response, Metrika 43, 17–30, 1996.
[2] Arnab, R. and Dorffner, G. Randomized response technique for complex survey designs, Statistical Papers 48, 131–141, 2006.
[3] Bar-Lev, S.K., Bobovitch. E. and Boukai, B. A note on randomized response models, Metrika 60, 255-260, 2004.
[4] Eichhorn, B.H. and Hayre, L. S. Scrambled randomized response methods for obtaining sensitive quantitative data, Journal of Statistical Planning and inference 7, 307–316, 1983.
[5] Espejo R,M. and Singh, H.P. Protection of privacy with objective prior distribution in randomized response, Statistica Anno LXIII (4), 697–701, 2003.
[6] Greenberg, B.G., Kuebler, R.R. Jr., Abernathy, J.R. and Hovertiz, D.G. Application of the randomized response techniques in obtaining quantitative data, Journal of the American Statistical Associations 66, 243–250, 1971.
[7] Gupta S., Gupta, B. and Singh, S. Estimation of sensitivity level of personal interview survey questions, Journal of Statistical Planning and Inference 100, 239–247, 2002.
[8] Gupta S. and Shabbir J. Sensitivity estimation for personal interview survey question, Sta- tistica Anno LXIV (4), 643–653, 2004.
[9] Hussain, Z. and Shabbir, J. On estimation of mean of a sensitive quantitative variable, InterStat. July (6), 1–14, 2007.
[10] Hussain, Z. and Shabbir, J. Improved estimation procedure for the mean of a sensitive variable using randomized response model, Pakistan Journal of Statistics 25 (2), 205–220, 2009.
[11] Hussain, Z., Shabbir, J. and Gupta, S. An alternative to Ryu randomized response model, Journal of Statistics & Management Systems 10 (4), 511–517, 2007.
[12] Mangat, N. S. and Singh, R. An alternative randomized response procedure, Biometrika 77, 439–442, 1990.
[13] Mathur, N. and Singh, H.P. Estimation of population mean with prior information using scrambled response technique, Brazilian Journal of Probability and Statistics 22 (2), 165– 181, 2008.
[14] Mehta, J. S. and Srinivasan, R. Estimation of mean by shrinkage to a point, Journal of the American Statistical Association 66, 86–90, 1971.
[15] Ryu, J. -B., Kim, J. -M., Heo, T. -Y. and Park, C.G. On stratified randomized response sampling, Model Assisted Statistics and Application 1 (1), 31–36, 2005–2006.
[16] Saxana, S. Estimation of exponential mean life in complete and failure censored samples with prior information, The Philippine Statistician 53 (1-4), 35–45, 2004.
[17] Searls, D.T. The utilization of a known coefficient of variation in the estimation procedure, Journal of the American Statistical Association 59, 1225–1226, 1964.
[18] Shirke, D.T. and Nalawade, K.T. Estimation of the parameter of the binomial distribution in the presence of prior point information, Journal of the Indian Statistical Association 41 (1), 117–128, 2003.
[19] Singh, H.P. and Mathur, N. A revisit to alternative estimator for randomized response technique, Journal of Indian Society of Agricultural Statistics 55 (1), 79–87, 2002.
[20] Singh, H.P. and Mathur, N. An alternative to an improved randomized response strategy, Statistics in Transition 5 (5), 873–886, 2002.
[21] Singh, H.P. and Mathur, N. An optimally randomized response technique, Aligarh Journal of Statistics 23, 1–5, 2003.
[22] Singh, H.P. and Mathur, N. Modified optimal randomized response sampling technique, Journal of Indian Society of Agricultural Statistics 56 (2), 199–206, 2003.
[23] Singh, H.P. and Mathur, N. Unknown repeated trials in the unrelated question randomized response model, Biometrical Journal 46 (3), 375–378, 2004.
[24] Singh, H.P. and Mathur, N. Estimation of population mean with known coefficient of vari- ation under optimal randomized response model using scrambled response technique. Sta- tistics in Transition 6 (7), 1079–1093, 2004.
[25] Singh, H. P. and Mathur, N. Improved estimation of population proportion possessing sensi- tive attribute with unknown repeated trial in randomized response sampling, Statistica Anno LXIV (3), 537–544, 2004.
[26] Singh, H.P. and Mathur, N. Estimation of population mean when coefficient of variation is known using scrambled randomized response technique, Journal of Statistical Planning & Inference 131 (1), 135–144, 2005.
[27] Singh, H. P. and Mathur, N. An improved estimation procedure for estimating the proportion of a population possessing sensitive attribute in unrelated question randomized response technique, Brazilian journal of Probability and Statistics 20, 93–110, 2006.
[28] Singh, H.P. and Mathur, N. An improved estimator for proportion of sensitive group of population using optional randomized response technique, Assam Statistical Review 21 (1), 64–72, 2007.
[29] Singh, H.P. and Shukla, S.K. A class of shrinkage estimators for the variance of exponential distribution with type-I censoring, IAPQR Transitions 27, 119–141, 2002.
[30] Thompson, J.R. Some shrinkage techniques for estimating the mean, Journal of the Amer- ican Statistical Association 63, 113–123, 1968.
[31] Tracy, D. S., Singh, H. P. and Raghuvansh, H. S. Some shrinkage estimators for the variance of exponential density, Microelectronics Reliability 36 (5), 651–655, 1996.
[32] Tse, S. and Tse, G. Shrinkage estimation of reliability of reliability for exponentially dis- tributed lifetimes, Communication in Statistics-Simulation and Computation 25 (2), 415– 430, 1996.
[33] Warner, S. L. Randomized response: a survey technique for eliminating evasive answer bias, Journal of the American Statistical Association 60, 63–69, 1965.