---
title: Peláez–Lamata’s CI in Poisson-ICT Models
url: https://www.emergentmind.com/topics/pelaez-lamata-s-ci
type: topic
---

# Peláez–Lamata’s CI in Poisson-ICT Models

The Peláez–Lamata confidence interval (CI), as presented in the context of item count techniques (ICT) for estimating the prevalence of sensitive traits, refers to an exact finite-sample confidence interval for the sensitive fraction $\pi$ in the Poisson-ICT model. In this paradigm, responses are structured to protect respondent privacy when estimating the proportion of a population possessing a sensitive attribute, by leveraging a mixture of Bernoulli and Poisson variables and a randomized sampling framework. The construction, as fully detailed by Jaworski & Zieliński, achieves nominal frequentist coverage for all $\pi$ and is governed by the interplay between sample size, a privacy-tuning parameter, and observed data [2410.14301].

## 1. Poisson-ICT Model Setup and Notation

The ICT model is formalized by randomly splitting a population sample of size $n$ into two subsamples, $n_1$ and $n_2$ ($n = n_1 + n_2$). Each participant is characterized by:

- $Z_i \sim \mathrm{Bernoulli}(\pi)$: the sensitive binary trait, with prevalence $\pi$.
- $X_i \sim \mathrm{Poisson}(\lambda)$: a “neutral” count variable, independent of $Z_i$.

Individual responses are not directly observed. Instead:
- Group 1 ($n_1$ respondents): $Y_{1i} = X_{1i} - Z_{1i}$
- Group 2 ($n_2$ respondents): $Y_{2j} = X_{2j} + Z_{2j}$

The model relies on the no-design-effect and no-liars assumptions as per Blair & Imai (2012), and $\lambda > 0$ is a design parameter selected for privacy control.

The principal observed statistic is:
$$
T = \sum_{j=1}^{n_2} Y_{2j} - \sum_{i=1}^{n_1} Y_{1i} = \xi + \eta
$$
where
- $\eta \sim \mathrm{Binomial}(n, \pi)$,
- $\xi \sim \mathrm{Skellam}(n_2\lambda, n_1\lambda)$.

Because $\xi$ and $\eta$ are independent, the cumulative distribution function (CDF) for $T$ given $\pi$ is
$$
F_\pi(t) = P_\pi\{T \leq t\} = \sum_{k=0}^n P\{\xi \leq t-k\} \binom{n}{k} \pi^k (1-\pi)^{n-k}
$$

## 2. Construction of the Exact Confidence Interval

The CI for $\pi$ is constructed by inverting the acceptance region derived from $F_\pi(t)$, which is monotonic in $\pi$. For an observed statistic $t_{\text{obs}}$ and two-sided confidence level $\gamma$ (e.g., $0.95$), the lower and upper confidence bounds $\pi_L$ and $\pi_U$ are defined by:

\[
\begin{align*}
& F_{\pi_L}(t_{\text{obs}}) = \frac{1+\gamma}{2} \\
& F_{\pi_U}(t_{\text{obs}}^-) = \frac{1-\gamma}{2}
\end{align*}
\]

Where the CDF for $T$ at integer $t$ can be evaluated via a Skellam distribution mass function, $g(m)$, using the modified Bessel function of the first kind, $I_{|m|}$, and Marcum-Q representation. Solving (3a) and (3b) numerically (as there is no closed-form) yields the CI endpoints.

## 3. Sensitivity of Interval Length to Privacy and Sample Size

The CI’s random length, $L(T, \gamma; \lambda, n_1, n_2) = \pi_U - \pi_L$, responds to both $\lambda$ and $n$:

- $L$ increases in $\lambda$; this reflects the privacy-accuracy tradeoff (more noise widens the CI).
- $L$ decreases in $n$; sample size increases precision.
- $L$ is maximal near $\pi=0.5$ and minimized near $\pi=0$ or $1$ for fixed $n,\lambda,\gamma$.

Practical study design must balance $\lambda$ (privacy) and $n$ (efficiency) to attain a desired interval width. Sufficient $n$ for targeted CI length, given privacy $\lambda$, can be computed empirically. For example, with $\pi_0=0.10$, $\gamma=0.95$, and privacy-tuned $\lambda=0.693$, achieving an average interval length $\leq 0.06$ requires $n\approx 3100$, and $n\approx 4500$ for $\leq 0.05$ [2410.14301].

## 4. Privacy Protection and Tuning of $\lambda$

Respondent anonymity is formalized by requiring that $P\{Z=1|Y=y\} \leq \tau$ with high probability $\delta$ for all $\pi\leq \pi_0$. Conditional probabilities for $Z=1$ given an observed response are:

- Subtraction arm ($Y_1$): $P\{Z_1=1|Y_1=y_1\} = [1 + ((y_1+1)/\lambda)((1/\pi)-1)]^{-1}$, $y_1 \geq -1$
- Addition arm ($Y_2$): $P\{Z_2=1|Y_2=y_2\} = [1 + (\lambda/y_2)((1/\pi)-1)]^{-1}$, $y_2 \geq 1$, and $=0$ if $y_2=0$

Selecting $\lambda$ requires solving two numerical inequalities to ensure the privacy guarantee holds; e.g., for $\pi_0=0.3$, $\tau=0.05$, $\delta=0.95$, minimal $\lambda \approx 5.364$ [2410.14301].

## 5. Coverage Properties and Comparison with Asymptotic Methods

The exact CI derived by inversion satisfies
$$
P_\pi\left\{ \pi_L(T) \leq \pi \leq \pi_U(T) \right\} \geq \gamma\quad \forall \pi \in [0,1]
$$
guaranteeing (at least) nominal coverage for all sample sizes. This outperforms the method-of-moments (MM) Normal-based CI, which:
- Uses $\hat{\pi}_{MM} = (\bar{Y}^{(2)} - \bar{Y}^{(1)})/2$ with variance estimator
$$
\hat{V} = \frac{1}{4}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)\left(\lambda + \hat{\pi}_{MM}(1-\hat{\pi}_{MM})\right)
$$
- Under-covers, especially for moderate $n$ (actual coverage can be \textless $95\%$).

## 6. Implementation Details and Numerical Examples

Numeric computation of $F_\pi(t)$ uses standard routines for the Skellam CDF (e.g., R’s `pskellam` or Marcum-Q via `mpmath`). The binomial mixture is then summed as per equation (1). Root-finding for $\pi_L$ and $\pi_U$ on the interval $[0,1]$ can employ bisection (e.g., R's `uniroot`). $T$ provides the one-to-one mapping to confidence limits.

Illustrative example: $n_1 = n_2 = 50$, $\lambda = 0.5$, $\gamma = 0.95$, $T=4$ yields numerically $\pi_L \approx 0.04$, $\pi_U \approx 0.21$. Tabulated results further support study planning; Table 2 in the cited work gives values for the supremum expected CI length over $\pi\leq 0.1$.

## 7. Significance and Research Context

The exact finite-sample CI for $\pi$ in the Poisson-ICT model, as constructed by Jaworski & Zieliński, provides a rigorous methodological advance in privacy-preserving inference for sensitive survey items. By leveraging the monotonicity property of $F_\pi$, the method maintains at least nominal coverage for any $\pi$, irrespective of sample size or privacy parameter, and natively supports design-time privacy tuning via $\lambda$. Both theoretical and empirical comparisons establish its superiority to conventional asymptotic CIs under operational regimes relevant to indirect questioning and privacy-ensured statistics [2410.14301].

Source: https://www.emergentmind.com/topics/pelaez-lamata-s-ci