Introduction

Today's society is filled with many delicate topics, like cases of rape, sexual practices, earned income that is not legal, etc. It can be challenging to gather information on these topics when using the direct questioning technique, which will ultimately lead to unreliable information gathered from respondents and unreliable estimates from measuring the study variable's parameter that is sensitive in nature. We employ a covert technique to gather information from interview subjects while maintaining their privacy in order to produce more accurate data and a trustworthy estimate. When data on delicate subjects are tainted by response bias and non-response, estimation becomes problematic. The tacit means of acquiring private data is employed to counter this. ³⁴ was the one who first presented the technique. Subsequently, ^{24, 34, 35} introduced the quantitative version of their questionnaire to gather quantitative data on delicate topics for mean estimation. The use of additive models to get respondents to give randomized or scrambled responses was first introduced by ⁹. ²⁴ presented and applied the multiplicative scrambling model. Additional references are ^{2,6,8,10,13,14,15,16,17,20,22,26,28,31.}

Numerous authors, including ^{1,3,4,5,7,12,18,21,27,29,33,36}. ^{11,19,23,25,30} are among the researchers who have worked on sampling surveys and have calculated the population mean of the study variable that is sensitive when the auxiliary variable that is not sensitive is present.. In order to achieve precision, we plan to propose a few log-type estimators for the population mean estimation of sensitive study variables in the current study.

Literature Review

Let Z₁, Z₂,.....,Z_n be randomized replies from n respondents who were sampled from population of size N with units Z₁, Z₂,.....,Z_N using the ²⁴ additive model Z = Y + S , where S is the jumbled variable, the distribution of which the researcher knows, with mean and variance as

, the study variable’s mean and variance

is unknown to the investigator but the mean and variance of the auxiliary variable

is provided by the respondents truthfully. The coefficient of variation for the scrambled response is given by

and the correlation coefficient between Z and X is given by .

We established the following relationships for the relative error in order to derive the estimators' properties:

Various Current Randomized Response Estimators in the Absence of Auxiliary Information

The first quantitative randomized response technique (RRT) model, known as the additive model, was described by ³⁵ for the purpose of estimating the mean of the quantitatively sensitive study variable. ²⁴ further examined estimation along this line.

When asking respondents for sensitive information, ²⁴ thought that an additive RRT model would be the best option.

An estimator of u_y and its variance are given by

In this case, the distribution of scramble variable S is known beforehand.

An RRT model with multiplicative properties was presented by ⁹ to extract sensitive data from respondents as

An estimator of  and it’s variance are given by

An optional RRT model with one stage multiplicative was proposed by ¹³. Respondents will give a jumbled response (YS) in this model if they believe the question is sensitive; otherwise, they will directly respond to the sensitive survey question (Y) with their genuine response. In this model, they assume that both Y and S are positive valued random variables and that the mean of the scrambling variable u_s = 1 and variance o_s² . Under this model, the reported response Z is given by

The mean of Z is given by

The variance of this unbiased estimator of the population mean is given by:

It should be noted that W increases as Var(u_y) increases and hence there is gain in efficiency compared to non-optional model when W=1. They also gave an estimator for the sensitivity level W as:

⁹Multiplicative model was modified by ⁶ to propose the forced quantitative RRT (FQRRT) model (1983). According to this model, each respondent is asked to use a randomization tool like a spinner to complete a Bernoulli trial. Let P represent the percentage of the delicate question that is answered. In all honesty, the researcher fixes this proportion if the spinner stops in the shaded area; if it stops in the unshaded area, (1-P) represents the proportion of reporting a scrambled response YS. The responses are distributed as follows:

The unbiased estimator for u_y and its variance are given by

An information-gathering model for Y, the sensitive variable, was presented by ²⁸ using the randomized response technique (RRT). This model is similar to ⁹ model, but he considered incorporating two scrambled variables S₁ and S₂ whose distributions are assumed to be known and independent of Y in the model. According to this model, each respondent is requested to provide the randomized answer as:

The unbiased estimator of u_y is given by

The variance of the estimator of u_y is given by

are the population means and population variances of the variable of interest Y and scrambled variables and population second moment.

The optional scrambling model was presented by ¹⁰; they identify Y as the quantitative sensitive variable, S as an additive random variable, and the observed responses from respondents are expressed as

Where, a and B denote the constants are determined by the interviewer.

An estimator for u_y using the ¹⁰ scrambling procedure may be expressed as

The variance of u_GS can be derived as

are the population means and population variances of the variable of interest Y and random variable S.                                                       

¹⁵Proposed a two stage additive optional RRT model with two scrambling variables S_i (i = 1,2) and are uncorrelated with mean u_si = 0 and variance o_s² . Two independent subsample approach of size n_i (i = 1,2) , are drawn from the population, such that n = n₁ + n₂ . In the i^th sample, a fixed predetermined proportion T₀ of respondent is instructed to tell the truth and the remaining proportion (1 - T₀ ) of respondents have an option to scramble their response additively as Y + S_i if they consider the sensitive question as sensitive or report truthfully if they consider the sensitive question non-sensitive. The distribution of their responses is given by

The mean and variance of Z are

The unbiased estimator for u_y and w are given as well as their variances as

¹⁵Proposed a two stage multiplicative optional RRT model with two scrambling variables S_i (i = 1,2) and are uncorrelated with mean u_si = 1 and variance o_si² . Two independent subsample approach of size n_i (i = 1,2) , are drawn from the population, such that n = n₁ + n₂ . In the i^th sample, a fixed predetermined proportion (T₀) of respondent is instructed to tell the truth and the remaining proportion (1 - T₀ ) of respondents have an option to scramble their response additively as YS_i if they consider the sensitive question as sensitive or report truthfully if they consider the sensitive question non-sensitive. The distribution of their responses is given by

The mean and variance of Z are

The unbiased estimator foru_y and w are given as well as their variances as

⁸Presented a randomization strategy that combines additive and multiplicative approaches. They believe that this combination can increase respondents' confidence regarding privacy protection because their model takes into account two scrambling variables, S and T. The observed responses based on this model are provided by:

They also assumed that E(T) = u_T =1 and E(S) = u_S = 0 then the mean and variance of Z are given by

The unbiased estimator u_y and its variance are given by

²⁰Proposed a three stage additive optional RRT model with two scrambling variables S_i (i = 1,2) and are uncorrelated. Two independent subsample approach of size n_i (i = 1,2) , are drawn from the population, such that n = n₁ + n₂ . In the i^th sample, a fixed predetermined proportion (T₀) of respondent is instructed to tell the truth and a fixed predetermined proportion (F₀) of respondents is instructed to scrambled additively Y + S_i and the remaining proportion (1 - T₀ - F₀ ) of respondents have an option to scramble their response additively if they consider the sensitive question as sensitive or report truthfully if they consider the sensitive question non-sensitive. The distribution of their responses is given by

The mean and variance of Z are

The unbiased estimator for u_y and W are given as well as their variances as

When gathering data on a quantitatively sensitive variable for the purpose of estimating the population mean, ¹⁷ took into consideration the subtractive randomized response technique (RRT) model. The model asks respondents to deduct the value of their random or scrambled variable (S) from a known distribution from the true value of the sensitive response or true response. As indicated by Z, Y is the observed/scrambled response and is provided by:

The scrambled variable S is distributed independently of the sensitive variable Y, whose mean u_S = 0 and variance o_s² are known, then the mean and variance of Z are given by

If Z₁, Z₂,....., Z_n are the observed responses of sample size n then an unbiased estimator for u_y is given by:

The variance of u_y is given by

Following their analysis of the quantitative model put forth by ^{10, 22} proposed an optional scrambling model. The observed responses can be expressed as:

Where a, B and Y denote the constants and are determined by the interviewer before the survey is conducted.

An estimator for u_y using the ²²scrambling procedure may be expressed as

The variance of u_NS can be derived as:

are the population means and population variances of the variable of interest Y and random variable S.

Four randomized response models were compared by ² in order to extract sensitive information. The models are provided by

An expression for the unbiased estimator is

The variance expressions at E(S) = 0 = 0 for all the models are reformulated as:

Comparing the likelihoods found in ²² with those found in ¹³. For the study variable that is sensitive, the mean estimator is provided by

The Variance of Z is given by

Some Existing Randomized Response Estimators with Auxiliary Information

In order to obtain information about Y indirectly,²⁵ employed the additive model and presented the ratio method of estimation as:

The mean square error of Z_s is given by

In order to obtain information about Y indirectly, ³⁰ employed the additive model and presented the exponential ratio estimation method as:

The mean square error of Z_sh is given by

Materials and Methods

Proposed Estimator for Sensitive Study Variable

In this segment, we modified and suggested the estimator created by ¹ when sensitive study variables were present, along with alternative estimator classes as

The general form of the estimators (3.2) through (3.8) is as follows:

Where, i = 1, 2, 3, 4, 5, 6, 7 and a, b, c, d are real numbers.

Applying the relative definitions from section one to (3.2) and (3.9), we get

By streamlining 3.10 and 3.11, we arrive at (3.12) and (3.13).

Subtracting Z from both sides of (3.12) and (3.13), we obtained,

Taking the square root of (3.14) and (3.15) and using expectation on both sides to get the estimators' mean square as:

Differentiating (3.8) with respect to u_i and v_i, equate to zero and solve, we have,

Efficiency Comparisons

Provided certain conditions are met, the suggested estimators outperform estimators found in the literature.

Results and Discussion

This section presents empirical investigations that were carried out using three data sets to show how well the suggested estimators performed in comparison to some existing ones.

Population 1: [Source: ³²]

Population 2: [Source: ³²]

Population 3: [Source: ³²]

Table 1: MSE of proposed estimator and the existing ones using the populations

Table 2: PRE of proposed estimator and the existing ones using the populations

Using the three data sets, Tables 1 and 2 present the empirical results of the MSE and PRE of the suggested and current estimators. The suggested estimators exhibit lower PRE and minimum MSE across all populations. This illustrates how the highly efficient estimators that have been suggested can yield more accurate population mean estimates when sensitive surveys are present than the estimators that have been taken into consideration in this study.

Conclusion

In this work, we proposed several log-type mean estimators for research variables that are sensitive, and we computed the mean square errors of these estimators. Based on the empirical results, the proposed estimators were seen to be more effective than the alternatives considered for this investigation. Accordingly, in situations where a sensitive issue is present, the suggested estimators ought to be applied when estimating the population mean.

Acknowledgment

The authors would like to thank Statistics Department of Usman Danfodiyo University for the successful completion of this research.

Funding Sources

The author(s) received no financial support for the research, authorship and publication of this article.

Conflict of Interest

The authors declare no conflict of interest.

Data Availability Statement

This statement does not apply to this article

Ethics Statement

This research did not involve human participants, animal subjects, or any material that requires ethical approval.

Authors' Contribution

Each author mentioned has significantly and directly contributed intellectually to the project and has given their approval for its publication.

M.A (Yunusa) conceptualization, methodology, writing-orginal draft.

A. (Awwal) data collection analysis, writing-review and editing.

References

1. Adejumobi A., Yunusa M. A., Erinola Y. A. and Abubakar K. An efficient logarithmic ratio type estimator of finite population mean under simple random sampling. International Journal of Engineering and Applied Physics. Vol. 3(2), pp. 700-705, (2023).
2. Azeem, M., Shabbir, J., Salahuddin, N., Hussain, S. and Salam, A. (2023) A comparative study of randomized response techniques using separate and combined metrics of efficiency and privacy. PLOS ONE, 19(1), 24-42. (2023).
3. Audu A., Gidado A., Dauran N. S., Buda S., Abdulazeez S. A. and Yunusa M. A. On the efficiency of modified regression-type estimators using robust regression and non-conventional measures of dispersion. Asian research journal of Mathematics, 18 (2), 1-26. (2022).
   CrossRef
4. Audu A., Gidado A., Dauran N.S., Buda S., Abdulazeez S. A., Yunusa M.A. and Abubakar I. Modified robust regression type estimators with multi-auxiliary variables using Non-conventional measures of dispersion. Nigerian Journal of Basic and Applied Sciences, 31 (1), 08-25. (2023).
   CrossRef
5. Bahl S. and Tuteja R.K. Ratio and product type exponential estimators. Information and Optimization Sciences, 12 (1), 159-163. (1991)
   CrossRef
6. Bar-Lev S.K., Bobovitch E. and Bouka B. A note on randomized response models for quantitative data, Metrika, 60, 255-260. (2004).
   CrossRef
7. Cochran W.G. The estimation of the yields of cereal experiments by sampling for the ratio grain to total produce. Journal of Agriculture Society, 30, 262-275. (1940).
   CrossRef
8. Diana G. and Perri P.F. New scrambled response models for estimating the mean of a sensitive quantitative character. Journal of Applied Statistics, 37(11), 1875-1890. (2010).
   CrossRef
9. Eichhorn B.H. and Hayre L.S. Scrambled randomized response methods for obtaining sensitive quantitative data. Journal of Statistical Planning and inference, 7(4), 307-316. (1983).
   CrossRef
10. Gjestvang C.R. and Singh S. An improved randomized response model: Estimation of mean. Journal of Applied Statistics, 36(12), 1361-1367. (2009)
    CrossRef
11. Grover L.K. and Kaur P. An improved exponential type estimator of population mean of sensitive variable using optional randomized response technique. Pakistan journal of Statistics and operation research, 15(1), 49-59. (2019).
    CrossRef
12. Grover L.K. and Kaur P. An improved estimator of the finite population mean in simple random sampling. Models Assisted Statistics and Application, 6(1), 47-55. (2011).
    CrossRef
13. Gupta S., Gupta B. and Singh S. Estimation of sensitivity level of personal interview survey questions. Journal of Statistical Planning and Inference, 100(2), 239-247. (2002).
    CrossRef
14. Gupta S., Thornton B., Shabbir J. and Singhal S. A comparison of multiplicative and additive optional RRT models. Journal Statistical theory and application, 5(3), 226-239. (2006).
15. Gupta S., Shabbir J. and Sehra S. Mean and sensitivity estimation in optional randomized response models. Journal of Statistical Planning and inference, 14(110), 2870-2874. (2010).
    CrossRef
16. Huang K.C. Unbiased estimators of mean, variance and sensitivity level for quantitative characteristics in finite population sampling, Metrika, 71, 341-352. (2010).
    CrossRef
17. Hussain Z. Improvement of the Gupta and Thornton scrambling model through double use of randomization device. International journal Academic Research Business and Social Sciences, 291-297. (2012).
18. Kadilar C. and Cingi H. Ratio estimators in simple random sampling. Applied Mathematics and Computation, 151, 893-902. (2006).
    CrossRef
19. Koyuncu N., Gupta S. and Sousa R. Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communication in Statistics-Simulation and Computation, 43(7), 1583-1594. (2010).
    CrossRef
20. Mehta S., Dass B.K., Shabbir J. and Gupta S. A third stage optional randomized response model. Journal of Statistical theory and Practice, 6(3), 412-427. (2012).
    CrossRef
21. Murthy M.N. Product method of estimation. The Indian journal of Statistics, 26(1), 69-74. (1964).
22. Narjis G. and Shabbir J. An efficient new scrambled response model for estimating sensitive population mean in successive sampling. Communications in Statistics-Simulation and Computation, 1-18. (2021).
    CrossRef
23. Patidar P. and Singh H.P. An improved class of estimators of population mean of sensitive variable using optional randomized response technique. Pakistan journal of Statistics and operation research, 19(3), 537-550. (2023).
    CrossRef
24. Pollock K.H. and Bek Y. A comparison of three randomized response models for quantitative data. Journal of American Statistical Application, 71, 884-886. (1976).
    CrossRef
25. Sousa R., Shabbir J., Real P.C. and Gupta S. Ratio estimation of the mean of a sensitive variable in the presence of auxiliary information. Journal of Statistics theory and Practice, 4(3), 495-507. (2010).
    CrossRef
26. Singh H.P. and Mathur N. Estimation of population mean when coefficient of variation is known using scrambled response technique. Journal of Statistical Planning and inference, 131,144. (2005).
    CrossRef
27. Singh H.P. and Tailor R. Use of known correlation coefficient in estimating the finite population mean. Statistics in Transition, 6, 555-560. (2003).
28. Sahai A. A simple randomized response technique in complex surveys, Metron, 65, 59-66. (2007).
29. Sisodia B.V.S. and Dwivedi V.K. A modified ratio estimator using coefficient of variation of auxiliary variable. Journal of Indian Society Agricultural Statistics, 33, 13-18. (1981).
30. Shahzad U., Hanif M., Koyuncu N. and Luengo A.V.G. A new family of estimators for mean estimation alongside the sensitivity issues. Journal of reliability and Statistical Studies, 10(2), 43-63. (2017).
31. Tarray T.A. and Singh H.P. A simple way of improving the Bar-Lev, Bobovitch and Boukai randomized response model. Kuwait Journal of Science, 44(4), 83-90. (2017).
32. Tiwari K.K., Bhougal S., Kumar S. and Rather K.U. Using randomized response to estimate the population mean of a sensitive variable under measurement error. Journal of Statistical Theory and Practice, 16-28. (2022).
    CrossRef
33. Upadhyaya L.N. and Singh H.P. Use of transformed auxiliary variable in estimating the finite population mean. Biometrical journal, 41, 627-636. (1991).
    CrossRef
34. Warner S.L. Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association, 60(309), 63-69. (1965).
    CrossRef
35. Warner S.L. The linear randomized response model. Journal of the American Statistical Association, 66(336), 884-888. (1971).
    CrossRef
36. Yunusa M.A., Audu A., Ishaq O.O. and Beki D.O. An efficient exponential type estimator for estimating finite population mean under simple random sampling. Annals Computer Science Series, 19 (1), 46-51. (2021).