PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

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Homogeneous Poisson rate MLE in a positive-volume window

Statement

Let $W$ be as in Finite-window spatial observations and configurations and suppose $v = ∣ W ∣ > 0$ . Consider the homogeneous family $λ_{ρ} (x) = ρ$ with parameter space either $[0, \infty)$ or $(0, \infty)$ . For an observed configuration with $n = η (W)$ occurrences, if $n > 0$ the unique maximum likelihood estimate is $\hat{ρ} = n / v$ . If $n = 0$ , the likelihood is maximized at $ρ = 0$ when zero is in the parameter space; on $(0, \infty)$ its supremum is the boundary value at $ρ ↓ 0$ and is not attained.

Facts & Assumptions

Given: $v > 0$ , a finite configuration $η$ with count $n = η (W)$ , and the homogeneous intensity $λ_{ρ} = ρ$ .

[F1]

For $ρ > 0$ , the log likelihood up to a parameter-independent constant is $n \log ρ - v ρ$ ; for a zero event factor the likelihood is zero. (Finite-window spatial Poisson likelihood relative to unit rate)

[F2]

$\log$ is differentiable on $(0, \infty)$ with derivative $1 / t$ , is continuous and strictly increasing, and obeys the quotient law. (The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t)

[F3]

A function continuous on a closed real interval and differentiable on its interior satisfies the mean value theorem. (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ )

Proof

Given: The hypotheses in the Statement.

1.1givenF1F2algebra

Suppose $n > 0$ and put $\hat{ρ} = n / v > 0$ . For $ρ > 0$ set $t = ρ / \hat{ρ}$ . By [F1] and the logarithm quotient law [F2], the log-likelihood difference is $ℓ (ρ) - ℓ (\hat{ρ}) = n (\log t - t + 1)$ .

1.2givenF1algebra

Suppose $n = 0$ . For $ρ > 0$ , [F1] gives log likelihood $- v ρ$ , which is strictly below $0$ and tends to $0$ as $ρ ↓ 0$ . At $ρ = 0$ , the empty product is $1$ and the density in Finite-window spatial Poisson likelihood relative to unit rate has log likelihood equal to its parameter-independent constant, so the normalized log likelihood value is $0$ . Thus zero is the maximizer when admitted and is only a supremum when the parameter space is $(0, \infty)$ .

2.1givenstep 1.1F1F2F3algebra

Let $ψ (t) = \log t - t + 1$ . For each $s > 0$ , $\frac{ψ (s + h) - ψ (s)}{h} = \frac{\log (s + h) - \log s}{h} - 1 ⟶ \frac{1}{s} - 1$ as $h \to 0$ , by [F2]; hence $ψ^{'} (s) = 1 / s - 1$ . If $t > 1$ , apply [F3] on $[1, t]$ : its intermediate point $c > 1$ has $ψ^{'} (c) < 0$ , so $ψ (t) < ψ (1) = 0$ . If $0 < t < 1$ , apply [F3] on $[t, 1]$ : its intermediate point has $ψ^{'} (c) > 0$ , so again $ψ (t) < ψ (1) = 0$ . Equality holds at $t = 1$ . Therefore [F1] and step 1.1 give the unique maximum at $ρ = \hat{ρ}$ among positive rates; at $ρ = 0$ the observed nonempty configuration has likelihood zero.

3.1step 1.1step 2.1step 1.2∎

Steps 1.1 and 2.1 prove the positive-count claim, and step 1.2 proves the zero-count boundary claim.

Depends on

Used by

Homogeneous Poisson rate estimation at positive-count and zero-count boundaries Example

Dependency tree · two levels

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Sources

Adrian Baddeley, Spatial Point Processes and their Applications, complete lecture notes (standard reference, not scraped)
Günter Last and Mathew Penrose, Lectures on the Poisson Process, author-hosted 2017 version (standard reference, not scraped)