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arxiv: 2605.03694 · v1 · submitted 2026-05-05 · 🧮 math.ST · stat.ME· stat.TH

Local estimation of transition rates of jump processes through discretization

Martin Bladt , Rasmus Frigaard Lemvig This is my paper

Pith reviewed 2026-05-07 12:36 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords asymptotic normalityjump processesMarkov modelssemi-Markov modelsPoisson regressionnonparametric estimationcounting processestransition rates
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The pith

Occurrence/exposure rates for Markov and semi-Markov jump processes are asymptotically normal when discretization intervals shrink with the number of observations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a Poisson regression fit to binned counting data produces estimators of transition rates in jump processes that become normally distributed around the true values once the time and duration bins are made smaller as the sample grows. No assumptions are placed on the shape of the underlying intensity functions, and the same normality result holds whether the process is Markov or semi-Markov. The proofs rely only on classical central limit theorems for counting processes rather than specialized theory. This lets practitioners compute the rates and their uncertainty directly from event counts and exposure times in each bin.

Core claim

Imposing no structural assumptions on the true intensities, the occurrence/exposure rates are asymptotically normal under appropriate shrinking conditions on the partition lengths for both Markov and semi-Markov models. These results are derived using only classical central limit theorems and elementary results for counting processes.

What carries the argument

The occurrence/exposure rate obtained by Poisson regression on discretized time and duration intervals, which counts transitions in each bin and divides by total exposure time in that bin.

If this is right

  • Asymptotic normality holds simultaneously for both Markov and semi-Markov jump processes.
  • The estimators require no parametric form for the transition intensities.
  • Normal-based confidence intervals and tests follow directly from the limiting distribution.
  • The method remains valid for any intensities as long as the partition lengths satisfy the shrinking conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same binning idea could be applied to estimate intensities in non-jump point processes where event counts are still observable.
  • Data-driven selection of bin sizes might improve accuracy for moderate sample sizes while preserving the asymptotic guarantee.
  • Because the procedure uses only elementary counting-process tools, it can be coded in standard statistical packages without requiring advanced numerical routines.

Load-bearing premise

The lengths of the time and duration intervals must shrink with the number of observations at a rate that satisfies the stated conditions for asymptotic normality.

What would settle it

Generate many independent realizations of a jump process with known intensities, keep partition lengths fixed while increasing the number of observations, and verify whether the properly normalized occurrence/exposure rates still converge in distribution to standard normal.

read the original abstract

We investigate the Poisson regression method for Markov and semi-Markov jump processes from a nonparametric angle, allowing the lengths of the time and duration intervals in the partition to vary with the number of observations. Imposing no structural assumptions on the true intensities, we obtain asymptotic normality of the occurence/exposure rates under appropriate shrinking conditions on the partition lengths. We derive asymptotic normality results for both Markov and semi-Markov models using only classical central limit theorems and elementary results for counting processes. All results are illustrated on both simulated and real data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a discretization-based nonparametric estimator for transition rates (occurrence/exposure) in Markov and semi-Markov jump processes. Partition lengths for time and duration are permitted to shrink with sample size; under suitable shrinking conditions the authors derive asymptotic normality of the estimators for both model classes, relying only on classical central limit theorems and elementary counting-process results, without any structural assumptions on the true intensities. The claims are illustrated on simulated and real data.

Significance. If the asymptotic normality results hold under the stated conditions, the work supplies a flexible, assumption-light approach to local rate estimation for jump processes that accommodates irregular intensities and varying bin sizes. The exclusive use of standard probabilistic tools (CLTs and counting-process martingales) is a strength that keeps the derivations elementary and reproducible. The extension to semi-Markov models via joint time-duration discretization is a useful contribution for applications in survival analysis and reliability where duration dependence is present.

major comments (2)
  1. [§3 and §4] §3 (Markov case) and §4 (semi-Markov case), statements preceding Theorems 3.1 and 4.1: the claim that asymptotic normality follows from 'only classical central limit theorems and elementary results for counting processes' with 'no structural assumptions on the true intensities' is load-bearing. For the normalized increments inside each shrinking bin to satisfy a Lindeberg-type condition, the compensators must still obey uniform integrability or moment bounds; arbitrary (unbounded, discontinuous, or singular) intensities can violate this even when bin lengths shrink at the stated rate. The manuscript does not exhibit an explicit verification that the Lindeberg condition holds uniformly under the sole shrinking hypothesis.
  2. [§4.2] §4.2, the semi-Markov discretization: the dependence induced by conditioning on both time and duration bins is not automatically covered by the standard Poisson CLT invoked for the Markov case. The paper must show that the joint counting-process martingale still satisfies the required tightness or Lindeberg condition after the additional discretization; the current argument appears to treat the duration bins as independent of the time bins, which is not generally true.
minor comments (2)
  1. [§2] Notation for the occurrence/exposure ratio is introduced without a clear global definition; a single displayed equation collecting all symbols would improve readability.
  2. [§5] The simulation section reports coverage probabilities but does not tabulate the exact shrinking rates used for each scenario; adding a small table would make the numerical results directly comparable to the theoretical conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the contribution, and the specific comments on the rigor of the asymptotic arguments. We address the major comments point by point below and will revise the manuscript to strengthen the proofs where needed.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (Markov case) and §4 (semi-Markov case), statements preceding Theorems 3.1 and 4.1: the claim that asymptotic normality follows from 'only classical central limit theorems and elementary results for counting processes' with 'no structural assumptions on the true intensities' is load-bearing. For the normalized increments inside each shrinking bin to satisfy a Lindeberg-type condition, the compensators must still obey uniform integrability or moment bounds; arbitrary (unbounded, discontinuous, or singular) intensities can violate this even when bin lengths shrink at the stated rate. The manuscript does not exhibit an explicit verification that the Lindeberg condition holds uniformly under the sole shrinking hypothesis.

    Authors: We agree that an explicit verification of the Lindeberg condition strengthens the claim. The proofs apply the martingale central limit theorem to the normalized increments of the counting processes over the shrinking bins. Under the stated conditions (bin lengths δ_n → 0 with nδ_n → ∞ at appropriate rates), the maximum squared increment is o_p(1) relative to the predictable variation because the exposure per bin remains controlled and jumps are of size 1. However, the manuscript does not spell out the uniform integrability step for completely arbitrary intensities. We will add a short lemma in the appendix (or inline before Theorems 3.1 and 4.1) that verifies the Lindeberg condition directly from the shrinking hypothesis and the elementary properties of counting-process compensators, without imposing extra structural assumptions on the intensities beyond those already implicit in the existence of the compensators. revision: yes

  2. Referee: [§4.2] §4.2, the semi-Markov discretization: the dependence induced by conditioning on both time and duration bins is not automatically covered by the standard Poisson CLT invoked for the Markov case. The paper must show that the joint counting-process martingale still satisfies the required tightness or Lindeberg condition after the additional discretization; the current argument appears to treat the duration bins as independent of the time bins, which is not generally true.

    Authors: We accept that the joint discretization requires a multivariate martingale argument rather than a direct invocation of the univariate Poisson CLT. In §4.2 the estimator is constructed from the multivariate counting process N_{ij}(t,u) indexed by the product partition of time and duration; the compensator is the integral of the joint intensity over each (time,duration) rectangle. The predictable variation process therefore automatically incorporates the dependence between time and duration bins. We will revise the proof of Theorem 4.1 to state the multivariate martingale CLT explicitly, verify the Lindeberg condition for the vector of increments over the product partition, and add a short paragraph clarifying that the bins are not assumed independent—the joint quadratic variation accounts for any dependence induced by the semi-Markov structure. This keeps the argument within elementary counting-process results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external classical CLTs to discretized counting processes

full rationale

The paper states it obtains asymptotic normality of occurrence/exposure rates for Markov and semi-Markov models by applying only classical central limit theorems and elementary counting-process results to partitions whose lengths shrink with sample size, while imposing no structural assumptions on the intensities. No equation or step reduces a claimed prediction to a fitted parameter by construction, no self-citation is invoked as a load-bearing uniqueness theorem, and the central results are not renamed empirical patterns. The derivation chain is therefore self-contained against external probabilistic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard mathematical tools from probability theory without introducing new free parameters, ad hoc axioms, or invented entities beyond classical results for counting processes.

axioms (2)
  • standard math Classical central limit theorems apply to the normalized counting processes under the shrinking partition conditions
    Invoked to establish asymptotic normality of the estimators.
  • standard math Elementary results for counting processes hold for the occurrence and exposure times
    Basis for deriving the limiting distributions in both Markov and semi-Markov settings.

pith-pipeline@v0.9.0 · 5379 in / 1311 out tokens · 42198 ms · 2026-05-07T12:36:43.728073+00:00 · methodology

discussion (0)

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Reference graph

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    Note that Xi,n = Z In 1(s≤Ri)N i jk(ds) and thus by random right-censoring, µjk n = Z In E[1(s<R)1(Zs−=j)]qjk(ds) = Z In P(s≤R, Z s− =j)µ jk(s)ds = Z In pc j (s)µjk(s)ds. The asymptotic expression follows by applying a Taylor expansion aroundt

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    We have ηj n = Z In E[1(s<R)1(Zs−=j)]ds= Z In pc j (s)ds= ∆ npc j (t) +O(∆ 2 n), again by a Taylor expansion

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  33. [33]

    Start by noting that Xi,n =N i jk(tm ∧R i)−N i jk(tm−1 ∧R i). DefiningM jk(t) :=N jk(t)− R t 0 1(Zs−=j)µjk(s)ds, we have E[X 2 i,n] =E[(M jk(tm ∧R)−M jk(tm−1 ∧R)) 2] +E h Z In 1(s≤R)1(Zs−=j)µjk(s)ds 2i + 2E h (Mjk(tm ∧R)−M jk(tm−1 ∧R)) Z In 1(s≤R)1(Zs−=j)µjk(s)ds i . Sinceµ jk isC 1 close tot, for large enoughn,µ jk is continuous onI n and thus bounded. F...

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  34. [34]

    Applying a two-dimensional Taylor expansion in the point(t, t)yields Var(Y1,n) = ∆2 npc j (t) +O(∆ 2 n) as claimed

    Finally, Var(Y1,n) =E[Y 2 1,n]−(η j n)2 =E hZ In Z In 1(s∨u<R)1(Zs=j,Zu=j)dsdu i −(η j n)2 = Z In Z In P(s∨u < R, Z s =j, Z u =j)dsdu−(η j n)2. Applying a two-dimensional Taylor expansion in the point(t, t)yields Var(Y1,n) = ∆2 npc j (t) +O(∆ 2 n) as claimed. Local estimation of transition rates of jump processes through discretization19 Proof of Lemma 2....

    work page

  35. [35]

    By random right-censoring, E[Xi,n] =E hZ I (1) n 1(s≤R)1(um2 −1≤Us−<um2 )1(Zi s−=j)µjk(s, Us−)ds i = Z I (1) n E[1(s≤R)1(um2 −1≤Us−<um2 )1(Zs−=j)µjk(s, Us−)]ds. The joint measure of(Z t, Ut)is forF⊆ ZandG∈ B(R)given by pF (t, G) = X j∈F Z G pj(t,du), wherep j(t, u) =P(Z t =j, U t ≤u)so the inner expectation equals Z I (2) n µjk(s, v)pc j (s,dv) = Z I (2) ...

    work page

  36. [36]

    Start by doing a second order Taylor expansion arounduin the second coordinate to get pc j (s, v) =p c j (s, u) +∂2pc j (s, u)(v−u) +O ∆(2) n 2

    We have ηj n =E hZ I (1) n 1(um2 −1≤Us−<um2 )1(s≤R)1(Zs−=j)ds i = Z I (1) n pc j (s, um2)−p c j (s, um2−1)ds. Start by doing a second order Taylor expansion arounduin the second coordinate to get pc j (s, v) =p c j (s, u) +∂2pc j (s, u)(v−u) +O ∆(2) n 2 . Due to continuity of the second derivative, we may choose theO-term to be independent ofssince the in...

    work page

  37. [37]

    To computeVar(X i,n), we apply a similar strategy as for the Markov case. We decompose E[X 2 i,n] =E h Z I (1) n 1(s≤R)1(um2 −1≤Us−<um2 )dNjk(s) 2i =E h Z I (1) n 1(s≤R)1(um2 −1≤Us−<um2 )dMjk(s) 2i 22 +E h Z I (1) n 1(s≤R)1(um2 −1≤Us−<um2 )1(Zs−=j)µjk(s, Us−)ds 2i + 2E h Z I (1) n 1(s≤R)1(um2 −1≤Us−<um2 )dMjk(s) · Z I (1) n 1(s≤R)1(um2 −1≤Us−<um2 )1(Zs−=j...

    work page

  38. [38]

    Finally, E[Y 2 1,n] = Z I (1) n Z I (1) n P(s1, s2 ≤R, u m2−1 ≤U(s 1−), U(s2−)< u m2 , Zs1− =j, Z s2− =j)ds 1ds2 = Z I (1) n Z I (1) n pc j (s1, s2, um2)−p c j (s1, s2, um2−1)ds1ds2 withp c j (s1, s2, v)defined in the proof of 3. Now apply a second order Taylor expansion aroundv=uto get pc j (s1, s2, um2)−p c j (s1, s2, um2−1) =∂ 2pc j (s1, s2, u)∆(2) n +...

    work page