The DOI matches its Crossref record.
The Use of Maximum Likelihood Estimates in $\chi^2$ Tests for Goodness of Fit
Open accessAbstract
The usual test that a sample comes from a distribution of given form is performed by counting the number of observations falling into specified cells and applying the $\chi^2$ test to these frequencies. In estimating the parameters for this test, one may use the maximum likelihood (or equivalent) estimate based (1) on the cell frequencies, or (2) on the original observations. This paper shows that in (2), unlike the well known result for (1), the test statistic does not have a limiting $\chi^2$-distribution, but that it is stochastically larger than would be expected under the $\chi^2$ theory. The limiting distribution is obtained and some examples are computed. These indicate that the error is not serious in the case of fitting a Poisson distribution, but may be so for the fitting of a normal.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cited by
Econometric Models for Count Data with an Application to the Patents-R & D Relationship
Hausman, J., Hall, B. H., & Griliches, Z. (1984) · Econometrica · doi:10.2307/1911191 · Cited by 3,393
On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown
Lilliefors, H. W. (1967) · Journal of the American Statistical Association · doi:10.2307/2283970 · Cited by 4,530
Musolino, A., Naldi, N., Bortesi, B., Pezzuolo, D., Capelletti, M., Missale, G., Laccabue, D., Zerbini, A., Camisa, R., Bisagni, G., et al. (2008) · Journal of Clinical Oncology · doi:10.1200/jco.2007.14.8957 · Cited by 981
Fried, L. P., Xue, Q.-L., Cappola, A. R., Ferrucci, L., Chaves, P., Varadhan, R., Guralnik, J. M., Leng, S. X., Semba, R. D., Walston, J. D., et al. (2009) · The Journals of Gerontology Series A: Biological Sciences and Medical Sciences · doi:10.1093/gerona/glp076 · Cited by 539
Ng, L., Pitt, V., Huckvale, K., Clavisi, O., Turner, T., Gruen, R., & Elliott, J. H. (2014) · Systematic Reviews · doi:10.1186/2046-4053-3-121 · Cited by 38
Metadata
- Authors
- Chernoff, H., & Lehmann, E. L.
- Venue
- The Annals of Mathematical Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1954-09
- Volume
- 25
- Issue
- 3
- Pages
- 579-586
- ISSN
- 0003-4851 (ISSN-L)
- OpenAlex
- W2151385137
- Cited by
- 559
- References
- 0
- FWCI
- 4.52
- Type
- journal article
- Language
- en
- Open access
- bronze
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: The Use of Maximum Likelihood Estimates in $\chi^2$ Tests for Goodness of Fit. https://app.cochat.ai/papers/10.1214/aoms/1177728726
- The paper
- Chernoff, H., & Lehmann, E. L. (1954). The Use of Maximum Likelihood Estimates in $\chi^2$ Tests for Goodness of Fit. The Annals of Mathematical Statistics. https://doi.org/10.1214/aoms/1177728726
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
