In the Diana-Perri-2 model, each respondent is asked to report the scrambled response \(z_i=W(\beta U+(1-\beta)y_i)\) where \(\beta \in [0,1)\) is a suitable constant
controlled by the researcher and \(W,U\) are scramble variables whose distribution is assumed to be known.
To estimate \(\bar{Y}\) a sample of respondents is selected according to simple random sampling with replacement.
The transformed variable is
$$r_i=\frac{z_i-\beta\mu_W\mu_U}{(1-\beta)\mu_W}$$
where \(\mu_W,\mu_U\) are the means of \(W,U\) scramble variables, respectively.
The estimated variance in this model is
$$\widehat{V}(\widehat{\bar{Y}}_R)=\frac{s_z^2}{n(1-\beta)^2\mu_W^2}$$
where \(s_z^2=\sum_{i=1}^n\frac{(z_i-\bar{z})^2}{n-1}\).
If the sample is selected by simple random sampling without replacement, the estimated variance is
$$\widehat{V}(\widehat{\bar{Y}}_R)=\frac{s_z^2}{n(1-\beta)^2\mu_W^2}\left(1-\frac{n}{N}\right)$$