Expectation-Based Poisson Statistic

• Expectation-based Poisson statistic is a method that leverages a Poisson variable's mean to rigorously calibrate tests and inferential procedures.
• It employs cumulative mass function calculations and asymptotic approximations to assess properties like goodness-of-fit and extremal behavior.
• Applications include hypothesis testing, scan statistics for outbreak detection, and robust model assessment in analyses of count data.

An expectation-based Poisson statistic is any inferential quantity or test that leverages the explicit value of the expectation (mean) or expected structure under a Poisson model as its central benchmark or calibration principle. The concept encompasses a range of statistical quantities, goodness-of-fit tests, confidence procedures, and theoretical frameworks wherein the expected value (either of the data or of a function of the data) under the Poisson law is the pivot of the method. Expectation-based Poisson statistics are particularly significant in settings involving count data with Poisson or near-Poisson properties, and play a central role in both classical hypothesis testing, regularization, uncertainty quantification, and model assessment.

1. Foundational Principles and Motivation

Expectation-based Poisson statistics arise from the fact that for a Poisson random variable $X \sim \operatorname{Poi}(\lambda)$, the expectation $E[X] = \lambda$ is both the mean and variance, and the distribution is uniquely determined by this parameter. Many natural statistical events and inferences can be cast in terms of how the observed data behave with respect to this expectation, or in how one quantifies deviation from the model's theoretical mean structure.

A prototypical example is the focus on the event $\{ X \leq E[X] \}$, which motivates the paper of the probability $P(X \leq E[X]) = P( X \leq [\lambda])$ where $[\lambda]$ denotes the integer part of $\lambda$ (Li et al., 2022). Such probabilistic events embody the expectation-based paradigm: statistics and test functions are framed explicitly in terms of Poisson expectations.

Expectation-based methods contrast with population-based or conditional approaches that condition on the total observed count, and are stronger in situations where the total itself is random or is fundamentally modeled as Poissonian.

2. Properties of $P(X \leq E[X])$ and Extremal Problems

A core expectation-based Poisson statistic is the cumulative probability

$$P(X \leq E[X]) = P(X \leq [\lambda]) = \sum_{k=0}^{[\lambda]} \frac{\lambda^k}{k!} e^{-\lambda},$$

where $X \sim \operatorname{Poi}(\lambda)$ (Li et al., 2022). This event represents the probability that the observed count does not exceed its expectation.

One central problem is to characterize or bound the minimal value of $P(X \leq E[X])$ as $\lambda$ varies. The main result is that the infimum of this probability over all positive $\lambda$ is $1$, i.e.,

$$\inf_{\lambda > 0} P(X \leq E[X]) = \lim_{\lambda \to 0} e^{-\lambda} = 1,$$

but this infimum is not attained for any finite $\lambda$; it is a limit [(Li et al., 2022), Proposition 2.1]. When $0 < \lambda < 1$, $P(X \leq E[X]) = e^{-\lambda} \to 1$ as $\lambda \to 0$. For larger $\lambda \geq 1$, the cutoff $[\lambda]$ increases and the cumulative probability becomes a sum over more terms, but it remains strictly below 1.

No value of $\lambda > 0$ attains a minimum; instead, the greatest lower bound is reached as $\lambda \to 0$. A probabilistic implication is that for small means, the event of observing a count not exceeding the expectation is overwhelmingly likely.

This expectation-based extremal probability problem generalizes earlier binomial analogues (such as those rooted in Chvátal’s conjecture), and its Poisson analysis carefuly tracks the phase transitions between regimes $0 < \lambda < 1$ and $\lambda \geq 1$ using analytic and central limit arguments.

3. Theoretical Methods and Analytical Tools

Expectation-based Poisson statistics engage analytic methods that explicitly feature the expectation. For the infimum problem above, this involves:

• Direct sum manipulations of the Poisson mass function up to $[\lambda]$.
• Analysis of small $\lambda$ regime, where $P(X \leq E[X]) = e^{-\lambda} \to 1$ trivially as $\lambda \to 0$.
• Asymptotic (central limit) arguments: for large $\lambda$, approximating $P(X \leq E[X])$ using normal approximation and examining the convergence of tail probabilities.
• The observation that the function $\lambda \mapsto P(X \leq E[X])$ is continuous and strictly decreases for small $\lambda$, but never attains its infimum at a positive point.

Such explicit calculations are representative of expectation-based approaches: the core feature is the direct use of $E[X]$ to define test boundaries, integration domains, or indices of functions.

Further, expectation-based Poisson statistics are often involved in other areas: for example, tail inequalities for the probability that $X$ exceeds its mean (or some threshold function of its mean) are commonly controlled via expectation-based tail bounds (Pelekis, 2016), with explicit connections made between upper bounds for tail conditional expectations and lower bounds for tail probabilities.

4. Connections and Implications in Statistical Inference

Expectation-based Poisson statistics have direct implications for:

• Goodness-of-fit testing: Procedures that compare observed cumulative distribution functions (up to $[\lambda]$) to the expected Poisson cumulative function.
• Scan statistics and outbreak detection: As in scan statistics for count data, expectation-based procedures calibrate the test using expected Poisson counts rather than totals or alternative parameterizations (Allévius et al., 2017). This improves sensitivity, especially in heterogeneous or zero-inflated data.
• Model comparison: In likelihood ratio and power-divergence test statistics (Daly, 2023), expectation-based constructions provide the asymptotic distributions by matching expectations of Poisson distributions with theoretical statistics.
• Non-asymptotic probability bounds: Expectation-based Poisson statistics lead naturally to bounds and inequalities—such as extremal probabilities and tail expectations—directly interpretable in terms of deviations from the mean rather than concentration inequalities around quantiles.

A key insight is that conditioning on the expectation, rather than on a fixed total or using percentiles, often yields sharper and more interpretable inferential statements in stochastic models with Poisson structure.

5. Relationship to Related Distributions and Generalizations

The expectation-based minimum probability problem has natural analogues for geometric and Pascal distributions (also addressed in (Li et al., 2022)). This highlights the universality of conditioning on, or formulating statistics via, expectation, regardless of the underlying discrete law.

Additionally, in sum-of-event scenarios or compound Poisson phenomena, expectation-based approaches can systematically quantify error, bias, and uncertainty by comparing observed or simulated statistics to their Poisson-expectation analogues. This underpins refined inference in Monte-Carlo and reweighting schemes, local limit approximations, and regularization approaches.

Although the particular result for Poisson variables is that the minimal $P(X \leq E[X])$ equals 1 in the limit as $\lambda \to 0$ (and is not achieved otherwise), analogous problems for other discrete laws display distinct and sometimes richer behaviors, reinforcing the structural singularity of the Poisson distribution regarding expectation-based events.

6. Summary Table: $P(X \leq E[X])$ for Poisson Distribution

Parameter range	Value of $P(X \leq E[X])$	Limiting behavior
$0 < \lambda < 1$	$e^{-\lambda}$	$\to 1$ as $\lambda \to 0$
$\lambda \geq 1$	$$\sum_{k = 0}^{[\lambda]} \frac{\lambda^k}{k!}e^{-\lambda}$$	$<$ 1
All $\lambda > 0$	No minimum attained; $\inf = 1$

This tabulation expresses the main finding: the probability that a Poisson variable does not exceed its mean cannot be made arbitrarily small; its minimal possible value is 1, achieved only in the limit of vanishing mean.

7. Broader Impact and Future Directions

Expectation-based Poisson statistics provide a conceptual pivot for the development of hypothesis tests, goodness-of-fit procedures, risk bounds, and theoretical analysis in Poisson and related count models. They clarify the nontrivial behaviors of tail and central probabilities as functions of the Poisson mean, inform the design of robust statistical tools (especially when total counts themselves are random), and guide analytic approximations for both direct probabilities and functionals of Poisson data.

Their generalization to other discrete distributions is an area of ongoing research, as expectation-based extremal probability and inference problems expose deep structural differences among discrete laws, with consequences in combinatorics, stochastic process theory, and applied probability.