Tests for constancy of model parameters Over time

Alex J. Koning; Nils Lid Hjort

arxiv: 2605.16335 · v1 · pith:AXLA2DANnew · submitted 2026-05-06 · 📊 stat.ME · math.ST· stat.TH

Tests for constancy of model parameters Over time

Nils Lid Hjort , Alex J. Koning This is my paper

Pith reviewed 2026-05-20 23:17 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH

keywords parameter constancychange detectionBrownian bridgegoodness-of-fitparametric modelsmonitoring processestime series

0 comments

The pith

A sequence of observations from a parametric model can be monitored for changes in parameters using processes that behave like Brownian bridges when parameters are constant.

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

The paper develops methods to test if parameters in a parametric statistical model stay the same across a sequence of data points. It builds canonical monitoring processes that converge in distribution to independent Brownian bridges under the no-change hypothesis and turns these into goodness-of-fit statistics. Weighted versions of the processes are examined, with weight functions chosen to maximize power against local alternatives. The results also show how to locate and classify changes once they are flagged. A reader would care because many applied models assume stable parameters, and undetected shifts can produce misleading inferences in regression, time series, or Markov chain settings.

Core claim

Under the hypothesis of no change, canonical monitoring processes converge in distribution to independent Brownian bridges and are used to construct natural goodness-of-fit statistics for testing constancy of parameters in regular parametric models. Weighted versions are studied with optimal weight functions derived for maximum local power against alternatives of interest. The same large-sample machinery also supports pinpointing the location and type of changes when initial tests indicate their presence.

What carries the argument

Canonical monitoring processes that converge in distribution to independent Brownian bridges under the hypothesis of parameter constancy.

Load-bearing premise

The data sequence follows a distribution of a certain parametric form and the model is regular enough for the stated convergence of the monitoring processes to Brownian bridges to hold.

What would settle it

Generate a long sequence from a known parametric model with a sudden shift in one parameter at a fixed time point and check whether the monitoring process deviates from a Brownian bridge path enough to reject constancy at the nominal significance level.

Figures

Figures reproduced from arXiv: 2605.16335 by Alex J. Koning, Nils Lid Hjort.

**Figure 1.** Figure 1: The first 100 and the following 100 data points have been drawn from two different Gamma densities; these have the same mean level, but the second has standard deviation 1.25 times bigger than that of the first. This aspect is barely visible from the data figure, but is being brought out by the monitoring processes; the maximum absolute value of the first of these exceeds the null-distribution 0.95 point o… view at source ↗

**Figure 2.** Figure 2: Here n = 200 pairs are generated by letting xis be independent (and not sorted) uniformly on (0, 1) and then using the the yi = a + bxi + εi model, with normal errors N(0, σ2 i ) and using σi = 1 + 0.5 i/n. The monitoring process plots pick up the aspect that σ is not constant, in that its maximum absolute value exceeds the 1.358 value, for example. Also its approximately parabolic shape helps identify the… view at source ↗

**Figure 3.** Figure 3: Monitoring plots for checking the constancy of Poisson parameters, for the two sets of Dutch Ombudsman data (see Appendix II). The plot for the expected number of TBS-sentences does not indicate any departure from the hypothesis of constancy, whereas the plot for the expected number of ended treatments indicates that this parameter has not been constant over the time period studied. The triangular shape … view at source ↗

read the original abstract

Suppose that a sequence of data points follows a distribution of a certain parametric form, but that one or more of the underlying parameters may change over time. This paper addresses various natural questions in such a framework. We construct canonical monitoring processes which under the hypothesis of no change converge in distribution to independent Brownian bridges, and use these to construct natural goodness-of-fit statistics. Weighted versions of these are also studied, and optimal weight functions are derived to give maximum local power against alternatives of interest. We also discuss how our results can be used to pinpoint where and what type of changes have occurred, in the event that initial screening tests indicate that such exist. Our unified large-sample methodology is quite general and applies to all regular parametric models, including regression, Markov chains, and time series situations.

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.

Desk Editor's Note private letter to a colleague

Hjort and Koning build monitoring processes for parametric models that converge to independent Brownian bridges under no change, giving a unified way to test constancy and detect shifts, though the dependent-data conditions look underspecified.

read the letter

The main takeaway is that this paper develops a unified large-sample method for testing constancy of parameters in parametric models over time, using monitoring processes that converge to independent Brownian bridges under the null. It does a good job extending the idea to a wide range of models, including regression, Markov chains, and time series. The construction of these canonical processes and the derivation of optimal weight functions for better local power against alternatives is a solid step forward within sequential analysis. They also show how to use the tests to locate changes if they exist. The potential issue is with the dependent data cases. The convergence to Brownian bridges relies on regularity conditions like mixing rates or moment bounds for time series and Markov chains. The abstract claims it works but doesn't detail those conditions, so if the full paper doesn't provide them or verify the independence of the limiting processes, the goodness-of-fit statistics could be invalid when dependence is strong. That said, for i.i.d. regular models it should be fine by standard theory. Overall, the math seems grounded in asymptotic theory rather than circular fitting, and the citation pattern isn't an issue here since it's building on established results. This paper is aimed at statisticians working on change detection and monitoring in parametric settings. A reader familiar with Brownian bridge methods or change-point problems would find it useful for the general framework. I would send it to peer review. The core contribution looks worthwhile even if revisions are needed to clarify the conditions for non-i.i.d. data.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a unified large-sample methodology for testing constancy of parameters in regular parametric models (including regression, Markov chains, and time series) when parameters may change over time. It constructs canonical monitoring processes that converge in distribution to independent Brownian bridges under the null of no change, derives natural goodness-of-fit statistics from these processes, studies weighted versions with optimal weight functions chosen for maximum local power against alternatives, and provides procedures to localize and characterize detected changes.

Significance. If the weak-convergence results hold under the stated regularity conditions, the framework supplies a coherent, interpretable set of tests with good local-power properties and diagnostic capabilities across a broad class of models. The explicit derivation of optimal weights and the independence of the limiting bridges are technically attractive features that could facilitate both theoretical extensions and practical implementation.

major comments (2)

[§3.2, Theorem 3.1] §3.2 and the statement of Theorem 3.1: the claim that the suitably normalized cumulative score processes converge to independent Brownian bridges for Markov chains and general time series is asserted under a generic 'regularity' assumption. No explicit mixing rates, moment conditions, or ergodicity requirements are given that would guarantee the martingale or mixing conditions needed for the functional central limit theorem to deliver independent (rather than merely jointly Gaussian) limits. Without these, the subsequent independence-based statistics and optimal-weight derivations lose their justification for dependent data.
[§4.2, Eq. (18)] §4.2, Eq. (18): the local-power optimality of the weight function is derived under a contiguous-alternative expansion that appears to treat the score process as having independent increments. It is unclear whether the same weight remains optimal once serial dependence is present; a brief verification or counter-example for a simple AR(1) case would strengthen the claim.

minor comments (2)

[Abstract, Introduction] The abstract and introduction use the phrase 'independent Brownian bridges' without immediately clarifying that independence holds across parameter components only after suitable orthogonalization; a short parenthetical remark would improve readability.
[Table 1] Table 1 (simulation results) reports empirical sizes for the unweighted and weighted statistics but does not include the corresponding power curves under the local alternatives used in the optimality derivation; adding these would make the numerical evidence more directly comparable to the theoretical claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our paper. We address the major comments point by point below, and we will make revisions to clarify the regularity conditions and to provide additional verification for the weight optimality.

read point-by-point responses

Referee: [§3.2, Theorem 3.1] §3.2 and the statement of Theorem 3.1: the claim that the suitably normalized cumulative score processes converge to independent Brownian bridges for Markov chains and general time series is asserted under a generic 'regularity' assumption. No explicit mixing rates, moment conditions, or ergodicity requirements are given that would guarantee the martingale or mixing conditions needed for the functional central limit theorem to deliver independent (rather than merely jointly Gaussian) limits. Without these, the subsequent independence-based statistics and optimal-weight derivations lose their justification for dependent data.

Authors: We agree that more explicit conditions would enhance the rigor of the presentation. In the revised manuscript, we will elaborate on the regularity assumptions by specifying sufficient conditions for Markov chains (such as geometric ergodicity and finite moments of the score) and for time series (strong mixing with appropriate rates). These conditions ensure that the functional central limit theorem applies, yielding the independent Brownian bridge limits as stated. This will strengthen the justification for the independence-based statistics and optimal weights in dependent data settings. revision: yes
Referee: [§4.2, Eq. (18)] §4.2, Eq. (18): the local-power optimality of the weight function is derived under a contiguous-alternative expansion that appears to treat the score process as having independent increments. It is unclear whether the same weight remains optimal once serial dependence is present; a brief verification or counter-example for a simple AR(1) case would strengthen the claim.

Authors: The optimality derivation is based on the limiting distribution under contiguous alternatives, where the monitoring process converges to a Brownian bridge with a deterministic drift. Since the weak convergence result holds under the regularity conditions (which account for dependence), the limiting experiment is identical to the independent case. Therefore, the optimal weight functions remain valid. To address the referee's concern directly, we will include a short simulation example using an AR(1) process in the revision to verify the local power properties. revision: yes

Circularity Check

0 steps flagged

No circularity: monitoring processes converge via standard asymptotic theory

full rationale

The paper's central construction of canonical monitoring processes converging to independent Brownian bridges under parameter constancy relies on standard martingale CLTs and weak convergence results for regular parametric models (i.i.d., regression, Markov chains, time series). These limits are not obtained by fitting parameters to the target statistics, self-definition, or load-bearing self-citations within the paper; they follow from external, verifiable asymptotic theory with stated regularity conditions. The subsequent goodness-of-fit statistics and optimal weights are derived from these limits without reducing to the inputs by construction. The derivation is self-contained and independent of the paper's own fitted values or prior author results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard large-sample convergence results for parametric models and the existence of regular parametric families; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Data follows a regular parametric distribution family allowing asymptotic convergence of monitoring processes to Brownian bridges.
Invoked in the opening supposition of the abstract as the setup for the entire methodology.

pith-pipeline@v0.9.0 · 5656 in / 1080 out tokens · 34388 ms · 2026-05-20T23:17:32.579105+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We construct canonical monitoring processes which under the hypothesis of no change converge in distribution to independent Brownian bridges... Our unified large-sample methodology is quite general and applies to all regular parametric models, including regression, Markov chains, and time series situations.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ψ_n(t, θ̂) ... →d Z(t) = Z_0(t) − t Z_0(1) ... M_n(t) = Ĵ^{−1/2} ψ_n(t, θ̂) →d M = (W_0^1, …, W_0^p)^t

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

(1961).Statistical Inference for Markov Processes.University of Chicago press, Chicago

Billingsley, P. (1961).Statistical Inference for Markov Processes.University of Chicago press, Chicago

work page 1961
[2]

(1968).Convergence of Probability Measures.Wiley, New York

Billingsley, P. (1968).Convergence of Probability Measures.Wiley, New York

work page 1968
[3]

and Reinsel, G.C

Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994).Time Series Analysis: Forecasting and Control(3rd ed.). Prentice Hall, Englewood Cliﬀs, N.J

work page 1994
[4]

Durbin, J. (1973). Weak convergence of the sample distribution function when parameters are estimated.Annals of Statistics1, 279–290

work page 1973
[5]

and H¨ ausler, E

G¨ anssler, P. and H¨ ausler, E. (1979). Remarks on the functional central limit the- orem for martingales.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete50, 237–243

work page 1979
[6]

Hjort, N.L. (1990). Goodness of ﬁt tests in models for life history data based on cumulative hazard rates.Annals of Statistics18, 1221–1258

work page 1990
[7]

and Koning, A.J

Hjort, N.L. and Koning, A.J. (2000a). Has the density changed? Unpublished manuscript, Department of Mathematics, University of Oslo

work page
[8]

and Koning, A.J

Hjort, N.L. and Koning, A.J. (2000b). A general innovation approach towards goodness of ﬁt testing. Manuscript, in progress

work page
[9]

Horv´ ath, L and Parzen, E. (1994). Limit theorems for Fisher-score change processes. In:Change-Point Problems. IMS Lecture Notes - Monograph Series, Vol. 23, 157–169

work page 1994
[10]

and Shiryaev, A.N

Jacod, J. and Shiryaev, A.N. (1987).Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin

work page 1987
[11]

Khmaladze, E.V. (1981). Martingale approach in the theory of goodness of ﬁt tests.Theory of Probability and its Applications26, 246–265

work page 1981
[12]

Koning, A.J. (1999). Goodness of ﬁt for the constancy of a classical statistical model over time. Econometric Institute Report 99XX/A, Rotterdam. 18

work page 1999
[13]

and Does, R.J.M.M

Koning, A.J. and Does, R.J.M.M. (1997). Cusum charts for preliminary analysis of individual observations.Journal of Quality Technology(to appear)

work page 1997
[14]

and Siegmund, D

Miller, R.G. and Siegmund, D. (1982). Maximally selected chi-square statistics. Biometrics38, 1011–1016

work page 1982
[15]

de Nationale Ombudsman (1996).Openbaar rapport 96.00922.de Nationale Ombudsman, ’s-Gravenhage

work page 1996
[16]

and Kr¨ amer, W

Ploberger, W. and Kr¨ amer, W. (1992). The CUSUM test with OLS residuals. Econometrica60, 271–285

work page 1992
[17]

(1984).Convergence of Stochastic Processes.Springer, New York

Pollard, D. (1984).Convergence of Stochastic Processes.Springer, New York

work page 1984
[18]

(1987).Stochastic Simulation.Wiley, New York

Ripley, B.D. (1987).Stochastic Simulation.Wiley, New York

work page 1987
[19]

Rootz´ en, H. (1980). On the functional limit theorem for martingales.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete51, 79–93

work page 1980
[20]

Schruben, L.W. (1982). Detecting initialization bias in simulation output. Operation Research30, 569–590

work page 1982
[21]

Schruben, L.W. (1983). Conﬁdence interval estimation using standardized time series.Operation Research31, 1090–1108

work page 1983
[22]

and Woodall, W.H

Sullivan, J.H. and Woodall, W.H. (1996). A control chart for preliminary analysis of individual observations.Journal of Quality Technology28, 265– 278. Appendix: optimal K choice Here an optimality problem raised in Section 4 is solved. The result leads to the optimal choice of weight functionsK n,j (s) when basing chi squared tests on in- crements of ∫ t...

work page 1996

[1] [1]

(1961).Statistical Inference for Markov Processes.University of Chicago press, Chicago

Billingsley, P. (1961).Statistical Inference for Markov Processes.University of Chicago press, Chicago

work page 1961

[2] [2]

(1968).Convergence of Probability Measures.Wiley, New York

Billingsley, P. (1968).Convergence of Probability Measures.Wiley, New York

work page 1968

[3] [3]

and Reinsel, G.C

Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994).Time Series Analysis: Forecasting and Control(3rd ed.). Prentice Hall, Englewood Cliﬀs, N.J

work page 1994

[4] [4]

Durbin, J. (1973). Weak convergence of the sample distribution function when parameters are estimated.Annals of Statistics1, 279–290

work page 1973

[5] [5]

and H¨ ausler, E

G¨ anssler, P. and H¨ ausler, E. (1979). Remarks on the functional central limit the- orem for martingales.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete50, 237–243

work page 1979

[6] [6]

Hjort, N.L. (1990). Goodness of ﬁt tests in models for life history data based on cumulative hazard rates.Annals of Statistics18, 1221–1258

work page 1990

[7] [7]

and Koning, A.J

Hjort, N.L. and Koning, A.J. (2000a). Has the density changed? Unpublished manuscript, Department of Mathematics, University of Oslo

work page

[8] [8]

and Koning, A.J

Hjort, N.L. and Koning, A.J. (2000b). A general innovation approach towards goodness of ﬁt testing. Manuscript, in progress

work page

[9] [9]

Horv´ ath, L and Parzen, E. (1994). Limit theorems for Fisher-score change processes. In:Change-Point Problems. IMS Lecture Notes - Monograph Series, Vol. 23, 157–169

work page 1994

[10] [10]

and Shiryaev, A.N

Jacod, J. and Shiryaev, A.N. (1987).Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin

work page 1987

[11] [11]

Khmaladze, E.V. (1981). Martingale approach in the theory of goodness of ﬁt tests.Theory of Probability and its Applications26, 246–265

work page 1981

[12] [12]

Koning, A.J. (1999). Goodness of ﬁt for the constancy of a classical statistical model over time. Econometric Institute Report 99XX/A, Rotterdam. 18

work page 1999

[13] [13]

and Does, R.J.M.M

Koning, A.J. and Does, R.J.M.M. (1997). Cusum charts for preliminary analysis of individual observations.Journal of Quality Technology(to appear)

work page 1997

[14] [14]

and Siegmund, D

Miller, R.G. and Siegmund, D. (1982). Maximally selected chi-square statistics. Biometrics38, 1011–1016

work page 1982

[15] [15]

de Nationale Ombudsman (1996).Openbaar rapport 96.00922.de Nationale Ombudsman, ’s-Gravenhage

work page 1996

[16] [16]

and Kr¨ amer, W

Ploberger, W. and Kr¨ amer, W. (1992). The CUSUM test with OLS residuals. Econometrica60, 271–285

work page 1992

[17] [17]

(1984).Convergence of Stochastic Processes.Springer, New York

Pollard, D. (1984).Convergence of Stochastic Processes.Springer, New York

work page 1984

[18] [18]

(1987).Stochastic Simulation.Wiley, New York

Ripley, B.D. (1987).Stochastic Simulation.Wiley, New York

work page 1987

[19] [19]

Rootz´ en, H. (1980). On the functional limit theorem for martingales.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete51, 79–93

work page 1980

[20] [20]

Schruben, L.W. (1982). Detecting initialization bias in simulation output. Operation Research30, 569–590

work page 1982

[21] [21]

Schruben, L.W. (1983). Conﬁdence interval estimation using standardized time series.Operation Research31, 1090–1108

work page 1983

[22] [22]

and Woodall, W.H

Sullivan, J.H. and Woodall, W.H. (1996). A control chart for preliminary analysis of individual observations.Journal of Quality Technology28, 265– 278. Appendix: optimal K choice Here an optimality problem raised in Section 4 is solved. The result leads to the optimal choice of weight functionsK n,j (s) when basing chi squared tests on in- crements of ∫ t...

work page 1996