* Garel (2007) Garel, B. (2007). Recent asymptotic results in testing for mixtures. Computational Statistics & Data analysis, 51, 5295–5304.
  * Ghosh and Sen (1985) Ghosh, J. K. and Sen, K. P. (1985). On the asymptotic performance of the log likelihood ratio statistic for the mixture model and related results. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (edited by L. LeCam, R. A. Olshen and C.-S. Cheng), Vol. II, pp. 789–806. Wadsworth Advanced Books & Software, Monterey.
  * Godambe (1991) Godambe, V. P. (1991). Estimating Functions. Oxford University Press, Oxford.
  * Goffinet et al. (1992) Goffinet, B., Loisel, P. and Laurent, B. (1992). Testing in normal mixture models when the proportions are known. Biometrika, 79, 842–846.
  * Groeneboom and Jongbloed (2018) Groeneboom, P. and Jongbloed, G. (2018). Some Developments in the Theory of Shape Constrained Inference. Statistical Science, 33, 473–492.
  * Groeneboom and Wellner (2001) Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. Journal of Computational and Graphical Statistics, 10, 388–400.
  * Han et al. (2022) Han, Q., Sen, B. and Shen, Y. (2022). High-dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints. The Annals of Statistics, 50, 376–406.
  * Hartigan (1977) Hartigan, J. A. (1977). Distribution problems in clustering. In Classification and Clustering (edited by J. van Ryzin), pp. 45-72. Academic Press, New York.
  * Hartigan (1985) Hartigan, J. A. (1985). A failure of likelihood asymptotics for normal mixtures. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (edited by L. LeCam, R. A. Olshen and C.-S. Cheng), Vol. II, pp. 807–810. Wadsworth Advanced Books & Software, Monterey.
  * Hathaway (1985) Hathaway, R. J. (1985). A constrained formulation of maximum-likelihood estimation for normal mixture distributions. The Annals of Statistics, 13, 795–800.
  * Haughton (1997) Haughton, D. (1997). Packages for estimating finite mixtures: a review. The American Statistician, 51, 194–205.
  * Hawkins (1977) Hawkins, D. M. (1977). Testing a sequence of observations for a shift in location. Journal of the American Statistical Association, 72, 180–186.
  * Hirano and Porter (2003) Hirano, K. and Porter, J. R. (2003). Asymptotic efficiency in parametric structural models with parameter-dependent support. Econometrica, 71, 1307–1338.
  * Hogg, McKean and Craig (2019) Hogg, R. V., McKean, J. W. and Craig, A. T. (2019). Introduction to Mathematical Statistics (8th ed.). Pearson, Boston.
  * Horváth and Rice (2014a) Horváth, L. and Rice, G. (2014a). Extensions of some classical methods in change point analysis (with Discussion). TEST, 23, 219–255.
  * Horváth and Rice (2014b) Horváth, L. and Rice, G. (2014b). Rejoinder on: Extensions of some classical methods in change point analysis. TEST, 23, 287–290.
  * Huber and Ronchetti (2009) Huber, P. J. and Ronchetti, E. M. (2009). Robust Statistics (2nd ed.). John Wiley & Sons, Hoboken, New Jersey.
  * Hušková and Kirch (2012) Hušková, M. and Kirch, C. (2012). Bootstrapping sequential change-point tests for linear regression. Metrika, 75, 673–7088.
  * Huzurbazar (1948) Huzurbazar, V. S. (1948). The likelihood equation, consistency and the maxima of the likelihood function. The Annals of Eugenics, 14, 185–200.
  * Irvine (1986) Irvine, J. M. (1986). The asymptotic distribution of the likelihood ratio test for a change in the mean. Statistical Research Division Report Series, CENSUS/SRD/RR-86/10, Bureau of the Census, Washington, D.C. 20233.
  * Jarušková (1997) Jarušková, D.(1997). Some Problems with application of change-point detection methods to environmental data. Environmetrics, 8, 469–483.
  * Kasahara and Shimotsu (2015) Kasahara, H., and Shimotsu, K. (2015). Testing the number of components in normal mixture regression models. Journal of the American Statistical Association, 110, 1632–1645.
  * Khodadadi and Asgharian (2008) Khodadadi, A. and Asgharian, M. (2008). Change-point problem and regression: an annotated bibliography. COBRA Preprint Series, Working Paper 44. https://biostats.bepress.com/cobra/art44
  * Killick and Eckley (2014) Killick, R. and Eckley, I.A. (2014). changepoint: An R Package for Changepoint Analysis. Journal of Statistical Software, 58, 1–19.
  * Kim and Siegmund (1989) Kim, H. J. and Siegmund, D. (1989). The likelihood ratio test for a change-point in simple linear regression. Biometrika, 76, 409–423.
  * Kirch and Steinebach (2006) Kirch, C. and Steinebach, J. (2006). Permutation principles for the change analysis of stochastic processes under strong invariance. Journal of Computational and Applied Mathematics, 186, 64–88.
  * Kirch (2008) Kirch, C. (2008). Bootstrapping sequential change-point tests. Sequential Analysis, 27, 330–349.
  * Koenker et al. (2017) Koenker, R., Chernozhukov, V., He, X. and Peng, L. (2017). Handbook of Quantile Regression. Chapman and Hall/CRC, Boca Raton, FL.
  * Kopylev (2012) Kopylev, L. (2012). Constrained parameters in applications: Review of issues and approaches. International Scholarly Research Network ISRN Biomathematics.
  * Kopylev and Sinha (2011) Kopylev, L. and Sinha, B. (2011). On the asymptotic distribution of likelihood ratio test when parameters lie on the boundary. Sankhyā B, 73, 20–41.
  * Krishnaiah and Miao (1988) Krishnaiah, P. R. and Miao, B. Q. (1988). Review about estimation of change points. In Handbook of Statistics (edited by P. R. Krishnaiah and C. R. Rao), Vol. 7, pp. 375–402. Elsevier, Amsterdam.
  * Kudô (1963) Kudô, A. (1963). A multivariate analogue of the one-sided test. Biometrika, 50, 404–418.
  * Lancaster (2000) Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95, 391–413.
  * LeCam (1970) LeCam, L. (1970). On the assumptions used to prove asymptotic normality of maximum likelihood estimates. The Annals of Mathematical Statistics, 41, 802–828.
  * LeCam and Yang (1970) LeCam, L. and  Yang, G. L. (1990). Asymptotics in Statistics: Some Basic Concepts, Springer-Verlag, New York.
  * Lee (2010) Lee, T.-S. (2010). Change-point problems: bibliography and review. Journal of Statistical Theory and Practice, 4, 643–662.
  * Lehmann and Romano (2005) Lehman, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses (3rd ed.). Springer-Verlag, New York.
  * Lemdani and Pons (1997) Lemdani, M. and Pons, O. (1997). Likelihood ratio tests for genetic linkage. Statistics and Probability Letters, 33, 15–22.
  * Lemdani and Pons (1999) Lemdani, M. and Pons, O. (1999). Likelihood ratio tests in contamination models. Bernoulli, 5, 705–719.
  * Li et al. (2016) Li, S., Chen, J. and Li, P. (2016). MixtureInf: Inference for Finite Mixture Models. R package version 1.1. https://CRAN.R-project.org/package=MixtureInf
  * Li et al. (2009) Li, P., Chen, J. and Marriot, P. (2009). Non-finite Fisher information and homogeneity: an EM approach. Biometrika, 96, 411–426.
  * Lindsay (1995) Lindsay, B. G. (1995). Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics, Hayward.
  * Liu and Shao (2003) Liu, X. and Shao, Y. (2003). Asymptotics for likelihood ratio tests under loss of identifiability. The Annals of Statistics, 31, 807–832.
  * Lo (2008) Lo, Y. (2008). A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture. Statistics and Computing, 18, 233–240.
  * McLachlan (1987) McLachlan, G. J. (1987). On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Applied Statistics, 36, 318–324.
  * McLachlan et al. (2019) McLachlan, G., Lee, S. X. and Rathnayake, S. (2019). Finite mixture models. Annual Review of Statistics and Its Application, 6, 355–378.
  * McLachlan and Peel (2000) McLachlan, G. and Peel, D. (2000). Finite Mixture Models. John Wiley & Sons, New York.
  * Muggeo (2008b) Muggeo, V. M. R. (2008b). segmented: an R Package to Fit Regression Models with Broken-Line Relationships. R News, 8/1, 20–25. https://cran.r-project.org/doc/Rnews/
  * Muggeo et al. (2014) Muggeo, V. M. R., Atkins, D. C., Gallop, R. J. and Dimidjian, S. (2014). Segmented mixed models with random changepoints: a maximum likelihood approach with application to treatment for depression study. Statistical Modelling, 14, 293–313.
  * Murphy and van der Vaart (1997) Murphy, S. and Van der Vaart, A. (1997). Semiparametric likelihood ratio inference. The Annals of Statistics, 25, 1471–1509.
  * Murphy and van der Vaart (2000) Murphy, S. and Van der Vaart, A. (2000). On Profile Likelihood. Journal of the American Statistical Association, 95, 449–465.
  * Neyman and Scott (1948) Neyman, J. and Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 11–32.
  * Niu et al. (2016) Niu, Y.S., Hao, N. and Zhang, H. (2016). Multiple change-point detection: A selective overview. Statistical Science, 31, 611–623.
  * Owen (1990) Owen, A. B. (1990). Empirical likelihood confidence regions. The Annals of Statistics, 18, 90–120.
  * Owen (1991) Owen, A. B. (1991). Empirical likelihood for linear models. The Annals of Statistics, 19, 1725–1747.
  * Page (1955) Page, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika,  42, 523–527.
  * Page (1957) Page, E. S. (1957). On problems in which a change in a parameter occurs at an unknown point. Biometrika,  44, 248–252.
  * Paulino and Pereira (1994) Paulino, C. D. M. and Pereira, C. A. B. (1994). On identifiability of parametric statistical models. Journal of the Italian Statistical Society, 1, 125–151.
  * Pfanzagl (2017) Pfanzagl, J. (2017). Mathematical Statistics: Essays on History and Methodology. Springer-Verlag, Berlin Heidelberg.
  * Piegorsch and Bailer (1997) Piegorsch, W. W. and Bailer, A. J. (1997). Statistics for Environmental Biology and Toxicology. Chapman & Hall, London.
  * Prakasa Rao (1992) Prakasa Rao, B. L. S. (1992). Identifiability in Stochastic Models: Characterization of Probability Distributions. Academic Press, London.
  * Quandt (1958) Quandt, R. E. (1958). The estimation of the parameters of a linear regression system obeying two separate regimes. Journal of the American Statistical Association, 53, 873–880.
  * Quandt (1960) Quandt, R. E. (1960). Tests of the hypothesis that a linear regression system obeys two separate regimes. Journal of the American Statistical Association, 55, 324–330.
  * R Core Team (2020) R Core Team (2020). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
  * Robertson et al. (1988) Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Inference. John Wiley & Sons, New York.
  * Rothenberg (1971) Rothenberg, T. J. (1971). Identification in parametric models. Econometrica, 39, 577–591.
  * Rotnitzky et al. (2000) Rotnitzky, A., Cox, D. R., Bottai, M. and Robins, J. (2000). Likelihood-based inference with singular information matrix. Bernoulli, 6, 243–284.
  * Samworth (2018) Samworth, R. J. (2018). Recent progress in log-concave density estimation. Statistical Science 33, 493–509.
  * Samworth and Bodhisattva (2018) Samworth, R. J. and Bodhisattva, S. (Eds.) (2018). Special Issue on Nonparametric Inference Under Shape Constraints. Statistical Science, 33.
  * Scheipl et al. (2008) Scheipl, F., Greven, S. and Küchenhoff, H. (2008). Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models. Computational Statistics & Data Analysis, 52, 3283–3299.
  * Scrucca et al. (2016) Scrucca, L., Fop, M., Murphy, T. B. and Raftery, A. E. (2016). mclust 5: clustering, classification and density estimation using Gaussian finite mixture models. The R Journal, 8, 289–317.
  * Self and Liang (1987) Self, S. G. and Liang, K. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under non standard conditions. Journal of the American Statistical Association, 82, 605–610.
  * Sen and Silvapulle (2002) Sen, P. K. and Silvapulle, M. J. (2002). An appraisal of some aspects of statistical inference under inequality constraints. Journal of Statistical Planning and Inference, 107, 3–43.
  * Serfling (1980) Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
  * Severini (2000) Severini, T. (2000). Likelihood Methods in Statistics. Oxford University Press, Oxford.
  * Severini (2004) Severini, T. (2004). A modified likelihood ratio statistic for some nonregular models. Biometrika, 91, 603–612.
  * Shaban (1980) Shaban, S. A. (1980). Change point problems and two-phase regression: an annotated bibliography. International Statistical Review, 48, 83–93.
  * Shapiro (1985) Shapiro, A. (1985). Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika, 72, 133–144.
  * Shapiro (1988) Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review, 56, 49–62.
  * Silvapulle and Sen (2005) Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference. John Wiley & Sons, New York.
  * Silvey (1959) Silvey, S. D. (1959). The Lagragian multiplier test. The Annals of Mathematical Statistics, 30, 382–407.
  * Sinha et al. (2012) Sinha, B., Kopylev, L. and Fox, J. (2012). Some new aspects of dose-response multi-stage models with applications. Pakistan Journal of Statistics and Operation Research, 8, 441–478.
  * Smith (1985) Smith, R. L. (1985). Maximum likelihood estimation in a class of non regular cases. Biometrika, 72, 67–92.
  * Smith (1989) Smith, R. L. (1989). A survey of non-regular problems. Proceedings of the 47th Session of the International Statistical Institute, Paris, August 29 – September 6, 1989, pp. 353–372.
  * Smith and Cook (1980) Smith, A. M. F. and Cook, D. G. (1980). Straight lines with a change point: A Bayesian analysis of some renal transplant data. Applied Statistics, 29, 180–189.
  * Sørensen (2008) Sørensen, H. (2008). Small sample distribution of the likelihood ratio effects in the random effects model. Journal of Statistical Planning and Inference, 138, 1605–1614.
  * Stram and Lee (1994) Stram, D. O. and Lee, J. W. (1994). Variance components testing in the longitudinal mixed rffects model. Biometrics, 50, 1171–1177.
  * Sun (1988) Sun, H.-J. (1988). A FORTRAN subroutine for computing normal orthant probabilities of dimensions up to nine. Communication in Statistics – Computation and Simulation, 17, 1097–1111.
  * Sofronov et al. (2020) Sofronov, G., Wendler, M. and Liebscher, V. (Eds.) (2020). Part 1: Special issue on change point detection (first 10 articles). Statistical Papers, 61, 1347–1588.
  * Thode et al. (1988) Thode, H. C. Jr., Finch, S. J. and Mendell, N. R. (1988). Simulated percentage points for the null distribution of the likelihood ratio test for a mixture of two normals. Biometrics, 44, 1195–1201.
  * Titterington (1990) Titterington, D. M. (1990). Some recent research in the analysis of mixture distributions. Statistics, 21, 619–641.
  * Ulm (1991) Ulm, K. W. (1991). A statistical method for assessing a threshold in epidemiological studies. Statistics in Medicine, 10, 341–349.
  * van der Vaart (2000) van der Vaart, A. W. (2000). Asymtptotic Statistics. Cambridge University Press, New York.
  * Wald (1949) Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Annals of Mathematical Statistics, 20, 595–601.
  * Wichitchan et al. (2019) Wichitchan, S., Yao, W., and Yang, G. (2019). Hypothesis testing for finite mixture models. Computational Statistics & Data Analysis, 132, 180–189.
  * Wilks (1938) Wilks, S. S. (1938). The large sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics, 1, 60–62.
  * Wolak (1987) Wolak, F. A. (1987). An exact test for multiple inequality and equality constraints in the linear regression model. Journal of the American Statistical Association, 82, 782–793.
  * Worsley (1979) Worsley, K. J. (1979). On the likelihood ratio test for a shift in location of normal populations. Journal of the American Statistical Association, 74, 365–367.
  * Worsley (1983) Worsley, K. J. (1983). The power of likelihood ratio and cumulative sum tests for a change in a binomial probability. Biometrika, 70, 455–464.
  * Worsley (1986) Worsley, K. J. (1986). Confidence regions and test for a change point in a sequence of exponential family random variables. Biometrika, 73, 91–104.
  * Yao and Davis (1986) Yao, Y.-C. and Davis, R.A. (1986). The asymptotic behavior of the likelihood ratio statistic for testing a shift in mean in a sequence of independent normal variates. Sankhyā A, 48, 339–353.
  * Yau and Zhao (2016) Yau, Y. and Zhao, Z. (2016). Inference for multiple change points in time series via likelihood ratio scan statistics. Journal of the Royal Statistical Society Series B (Methodological), 78, 895–916.
  * Yu (2018) Yu, Y. (2018). mixR: Finite Mixture Modeling for Raw and Binned Data. R package version 0.1.1. https://CRAN.R-project.org/package=mixR
  * Yu and Chen (2022) Yu, M. and Chen, X. (2022). A robust bootstrap change point test for high-dimensional location parameter. Electronic Journal of Statistics, 16, 1096–1152.
  * Zeileis (2006) Zeileis, A. (2006). Implementing a class of structural change tests: an econometric computing approach. Computational Statistics & Data Analysis, 50, 2987–3008
  * Zeileis et al. (2002) Zeileis, A., Leisch, F., Hornik, K. and Kleiber, C. (2002). strucchange: An R package for testing for structural change in linear regression models. Journal of Statistical Software, 7, 1–38.
  * Zhang et al. (2016) Zhang, Y., Staicu, A.-M., and Maity, A. (2016). Testing for additivity in non-parametric regression. Canadian Journal of Statistics, 44, 445–462.
  * Zhang (2018) Zhang, Y. (2018). lmeVarComp: Testing for a Subset of Variance Components in Linear Mixed Models. R package version 1.1. https://CRAN.R-project.org/package=lmeVarComp

##  SUPPLEMENTARY MATERIAL
B Annotated bibliography

  * Böhning and Dietz (1995) Böhning, D. and Dietz, E. (1995). Discussion of the paper by Cheng and Traylor (1995). Journal of the Royal Statistial Society Series B (Methodological), 57, 33–34.
[§3.1] Present a counterexample for finite mixtures.

  * Chant (1974) Chant, D. (1974). On asymptotic tests of composite hypotheses in non standard conditions. Biometrika, 61, 291–298.
[§3.2] Investigates the limiting distribution of the maximum likelihood
estimator when $\theta$ lies on the boundary of a closed parameter space.

  * Chen et al. (2015) Chen, Y., Huang, J., Ning, Y., Liang, K.-Y., and Lindsay, B.G. (2015). A conditional composite likelihood ratio test with boundary constraints. Biometrika, 105, 225–232.
[§3.2] Generalize Susko’s (2013) results to composite likelihoods.

  * Chen et al. (2017) Chen, Y., Ning, J., Ning, Y., Liang, K.-Y., and Bandeen-Roche, K. (2017). On pseudolikelihood inference for semiparametric models with boundary problems. Biometrika, 104, 165–179.
[§3.2] Establish the asymptotic behaviour of the pseudo likelihood ratio
statistic under semiparametric models when testing the hypothesis that the
parameter of interest lies on the boundary of its parameter space.

  * Drton (2009) Drton, M. (2009). Likelihood ratio test and singularities. The Annals of Statistics, 37, 979–1012.
[§3.2] Uses tools from algebraic geometry to study the asymptotic distribution
of the likelihood ratio statistic when the true parameter value is a
singularity, as for instance a cusp.

  * Feder (1968) Feder, P. I. (1968). On the distribution of the log likelihood ratio test statistic when the true parameter is near the boundaries of the hypothesis regions. The Annals of Mathematical Statistics,  39, 2044–2055.
[§3.2] Extends Chernoff’s (1954) results to the case where the true parameter
value is ‘near’ the boundary of $\Theta_{0}$ and $\Theta_{1}$. Assumes that
the true parameter value $\theta^{0}_{n}=\theta^{0}+o(1)$ is a sequence of
points, not necessarily in $\Theta_{0}$ or $\Theta_{1}$, such that
$\theta^{0}_{n}$ approaches the boundary given by the intersection
$\bar{\Theta}_{0}\cap\bar{\Theta}_{1}$ of their complementary subsets
$\bar{\Theta}_{0}$ and $\bar{\Theta}_{1}$.

  * Moran (1971) Moran, P. A. P. (1971). Maximum likelihood estimation in non standard conditions. Mathematical Proceedings of the Cambridge Philosophical Society, 70, 441–450.
[§3.2] Investigates the limiting distribution of the maximum likelihood
estimator when $\theta$ lies on the boundary of a closed parameter space.

  * Susko (2013) Susko, E. (2013). Likelihood ratio tests with boundary constraints using data-dependent degrees of freedom. Biometrika, 100, 1019–1023.
[§3.2] Make the same assumptions than Self and Liang (1987), but require that
all nuisance parameters are interior points of $\Theta$. Propose a data-
dependent solution which avoids the calculation of the mixing weights of the
chi-bar squared limit distribution and performs well in terms of power and
type I error.

  * Vu and Zhou (1997) Vu, H. T. V. and Zhou, S. (1997). Generalization of likelihood ratio tests under nonstandard conditions. The Annals of Statitics, 25, 897–916.
[§3.2] Derive the large-sample distribution of estimators obtained from
estimating functions for models involving covariates. The non-standard
asymptotic distribution of the likelihood ratio statistic for the two-way
nested variance components model is derived as an example.

  * Wood (2013) Wood, S. N. (2013). A simple test for random effects in regression models. Biometrika, 100, 1005–1010.
[§3.3] Extends (2004) to generalized linear mixed models with multiple
variance components. Exploits the link between random effects and penalized
regression to develop a simple simulation-free test for a null variance
component based on the restricted likelihood ratio, which under the null
hypothesis follows a weighted sum of squared independent standard normal
random variables.

  * Zhang and Lin (2008) Zhang, D. and Lin, X. (2008). Variance component testing in generalized linear mixed models for longitudinal/clustered data and other related topics. In Random Effect and Latent Variable Model Selection (D. B. Dunson Editor), p. 19-36, Lecture Notes in Statistics, Springer-Verlag, New York.
[§3.3] Extend Stram and Lee (1994) ideas to generalized linear mixed effects
models to test if between-subject variation is absent.

  * Claeskens et al. (2008) Claeskens, G., Nguti, R. and Janssen, P. One-sided tests in shared frailty models. Test, 17, 69–82.
[§3.4] Propose likelihood ratio tests in frailty models. Extend Maller and
Zhou (2003) to allow for the presence of covariates. Consider also the
limiting null distribution of the score statistic.

  * Maller and Zhou (2003) Maller, R.A. and Zhou, X. (2003). Testing for individual heterogeneity in parametric models for event-history data. Mathematical Methods of Statistics, 12, 276–304.
[§3.4] Show that under minimal conditions on the censoring mechanism, the
likelihood ratio statistic for homogeneity testing asymptotically distributes
as a $\bar{\chi}^{2}(\omega,1)$ with $\omega=(0.5,0.5)$.

  * Molenberghs and Verbeke (2007) Molenberghs, G. and Verbeke, G. (2007). Likelihood ratio, score and Wald tests in a constrained parameter set. The American statistician, 61, 22–27.
[§3.4] Compare the performance of the Wald, score and likelihood ratio
statistics in multivariate one-sided testing. Suggest to consider the
likelihood ratio as the default choice.

  * Andrews and Ploberger (1994) Andrews, D. W. K. and Ploberger W. (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica, 62, 1383–1414.
[§4.2] Discuss asymptotically optimal tests for the linear model with unknown
error variance. Use their results to test for a one-time structural change
with unknown change-point and discuss several other examples.

  * Bowden (1973) Bowden, R. (1973). The theory of parametric identification. Econometrica, 41, 1069–1074.
[§4.2] Sets out a general criterion for the identifiability of a statistical
system based on Kullback’s information integral.

  * Davies (1977) Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64, 247–254.
[§4.2] Investigates the construction of optimal likelihood-based tests under
loss of identifiability for a two-parameter model when the test statistic
follows a normal distribution.

  * Davies (1987) Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 64, 33–43.
[§4.2] Extends Davies (1977) to the case when the model distribution is chi-
squared.

  * Davies (2002) Davies, R. B. (2002). Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case. Biometrika, 89, 484–489.
[§4.2] Extends Davies (1977) and Davies (1987) to the linear model with
unknown error variance.

  * Fortunati et al. (2012) Fortunati, S., Gini, F., Greco, M., Farina, A., Graziano, A. and Giompapa, S. (2012). An identifiability criterion in the presence of random nuisance parameters. Proceedings of the 20th European Signal Processing Conference (EUSIPCO 2012). Bucharest, August 27–31, 2012.
[§4.2] Extend Bowden’s (1973) result which connects parameter identifiability
to non-singularity of the information matrix to the nuisance parameter case.

  * Ritz and Skovgaard (2005) Ritz, C. and Skovgaard, Ib M. (2005). Likelihood ratio tests in curved exponential families with nuisance parameters present only under the alternative. Biometrika, 92, 507–517.
[§4.2] Derive the asymptotic distribution of the likelihood ratio and of the
related score statistic for a general curved exponential family for which some
nuisance parameters vanish under the null hypothesis. Their results are
derived without the need to assume compactness of the parameter space, a
condition which is generally required when some parameters vanish under the
null hypothesis.

  * Song et al. (2009) Song, R., Kosorok, M. R. and Fine, J. P. (2009). On asymptotically optimal tests under loss of identifiability in semiparametrics models. The Annals of Statistics, 37, 2409–2444.
[§4.2] Consider tests of hypothesis when the parameters are not identifiable
under the null hypothesis in the context of semiparametric models.

  * Aitchison and Silvey (1960) Aitchison, J. and Silvey, S. D. (1960). Maximum likelihood procedures and associated tests of significance. Journal of the Royal Statistical Society, Series B (Methodological), 22, 154-171.
[§4.3] Address the problem of singular information matrix when the null
hypothesis is specified by constraints on the parameters and the outcome of
the test dictates whether it is necessary to provide estimates of these
parameters.

  * Barnabani (2002) Barnabani, M. (2006). Inference in the indeterminate problem. STATISTICA, 1, 59–75.
[§4.3] Proposes to maximise a suitably penalized log-likelihood function which
guarantees that the corresponding estimator of the parameter is consistent and
asymptotically normal. Allows one to construct a Wald type test statistic
which has a limiting chi-squared distribution both under null and alternative
hypotheses.

  * Ekvall and Bottai (2022) Ekvall, K. O. and Bottai, M. (2022). Confidence regions near singular information and boundary points with applications to mixed models. The Annals of Statistics, 50, 1806–1832.
[§4.3] Propose confidence regions with asymptotically correct uniform coverage
probability of parameters whose Fisher information matrix can be singular at
important points of the parameter set.

  * El-Helbawy and Hassan (1994) El-Helbawy, A. T. and Hassan, T. (1994). On the Wald, Lagrangian multiplier and likelihood ratio tests when the information matrix is singular. Journal of the Italian Statistical Society, 1, 51–60.
[§4.3] Build upon Silvey (1959) and develop modified formulae for the Wald,
score and likelihood ratio statistics which, under standard regularity
conditions, asymptotically follow a chi-squared distribution with degrees of
freedom specified by the number of constraints.

  * Jin and Lee (2018) Jin, F. and Lee, L.-F. (2018). Lasso maximum likelihood estimation of parametric models with singular information matrices. Econometrics, 6, 8.
[§4.3] Propose to fit the parameters of models with singular information
matrix by adaptive lasso while allowing the true parameter vector to lie on
the boundary of the parameter space.

  * Azaïs et al. (2006) Azaïs, J.-M., Gassiat, É. and Mercadier, C. (2006). Asymptotic distribution and local power of the log-likelihood ratio test for mixtures: bounded and unbounded cases. Bernoulli, 12, 775–799.
[§5.2] Provide the asymptotic distribution of the likelihood ratio statistic
under the null hypothesis of a $K_{0}$-component model and under contiguous
alternatives for a general mixture of parametric populations for a bounded
parameter space.

  * Azaïs et al. (2009) Azaïs, J.-M., Gassiat, É. and Mercadier, C. (2009). The likelihood ratio test for general mixture models with or without a structural parameter. ESAIM: Probability and Statistics, 13, 301–327.
[§5.2] Consider likelihood ratio for testing homogeneity in the general
$K$-component model, with application to Gaussian, Poisson and binomial
distributions, and testing for the number of components of a finite mixture
with or without a nuisance parameter. A number of conditions need be imposed
to avoid divergence of the limiting distribution of the likelihood ratio test.

  * Chen and Kalbfleisch (2005) Chen, J. and Kalbfleisch, J. D. (2005). Modified likelihood ratio test in finite mixture models with a structural parameter. Journal of Statistical Planning and Inference, 129, 93–107.
[§5.2] Study a modification of the likelihood ratio statistic similar to that
proposed by Chen et al. (2001) to verify the hypothesis of a homogeneous model
against the alternative of a Gaussian mixture of two or more components with a
common and unknown variance. Show, in particular, that the $\chi^{2}_{2}$
distribution represents a stochastic upper bound to the limiting null
distribution of the test statistic.

  * Chen et al. (2008) Chen, J., Li, P. and Fu, Y. (2008). Testing homogeneity in a mixture of von Mises distributions with a structural parameter. The Canadian Journal of Statistics, 36, 129–142. [§5.2] Derive the asymptotic null distribution of the modified likelihood ratio test introduced in Chen et al. (2001) and of a further modification, called the iterative modified likelihood ratio test, for testing model homogeneity against the alternative that the model is a two-component von Mises mixture with unknown mean directions without and with nuisance parameters.
  * Jeffries (2003) Jeffries, N. O. (2003). A note on “Testing the number of components in a normal mixture”, Biometrika, 90, 991–994.
[§5.2] Disproves Lo et al. (2001) result based on the fact that it requires
conditions on the structure of the parameter space that are generally not met
when the null hypothesis of a $K_{0}$-component model holds, with $K_{0}\geq
1$.

  * Lo et al. (2001) Lo, Y., Mendell, N. R. and Rubin, D. B. (2001). Testing the number of components in a normal mixture. Biometrika, 88, 767–778.
[§5.2] Prove that in the Gaussian case the distribution of the likelihood
ratio statistic based on the Kullback-Leibler information criterion converges
under the null hypothesis to a weighted sum of $\chi^{2}_{1}$ distributions.

  * Garel (2001) Garel, B. (2001). Likelihood ratio test for univariate Gaussian mixture. Journal of Statistical Planning and Inference, 96, 325–350.
[§5.2.2] Discusses seven distinct cases of homogeneity testing using the
likelihood ratio for the general two-component Gaussian mixture model by
imposing different restrictions on the means and the variances.

  * Lei and Qin (2009) Lei, Q. L. and Qin, Y. S. (2009). A modified likelihood ratio test for homogeneity in bivariate normal mixtures of two samples. Journal of Systems Science and Complexity, 22, 460–468.
[§5.2.2] Extend Qin and Smith (2006) to the two-sample problem by using the
modified likelihood statistic.

  * Liu and Shao (2004) Liu, X. and Shao, Y. (2004). Asymptotics for likelihood ratio test in a two-component normal mixture model. Journal of Statistical Planning and Inference, 123, 61–81.
[§5.2.2] Derive the asymptotic distribution of the likelihood ratio statistic
for a two-component mixture model. Furthermore, show that the likelihood ratio
diverges to infinity in probability at rate $O\\{\log(\log n)\\}$ if the mean
parameters are unbounded, in accordance with Hartigan (1985) and Chen and Chen
(2001, Theorem 2).

  * Polymenis and Titterington (1999) Polymenis, A. and Titterington, D. M. (1999). A note on the distribution of the likelihood ratio statistic for normal mixture with known proportions. Journal of Statistical Computation and Simulation, 64, 167–175.
[§5.2.2] Analyse empirically the $d=1$ scenarios treated by Goffinet et al.
(1992) and give an heuristic explanation for the slow convergence. Propose to
refer to the $\bar{\chi}^{2}(\tilde{\omega},1)$ distribution with suitably
defined mixing proportions $\tilde{\omega}$ instead of the theoretical value
$\omega=(0.5,0.5)$ to improve the approximation in finite samples.

  * Qin and Smith (2004) Qin, Y. S. and Smith, B. (2004). Likelihood ratio test for homogeneity in normal mixtures in the presence of a structural parameter. Statistica Sinica, 14, 1165–1177.
[§5.2.2] Consider a mixture of two normal distributions as in (2003) with the
restrictions on the mean parameters given by Chen and Chen (2001).
Identifiability is guaranteed by setting $\pi\leq 0.5$. In addition the mixing
proportion need to satisfy $\min(\pi,1-\pi)\geq\epsilon$ for some positive
$\epsilon<1/2$. The likelihood ratio asymptotically follows a fifty-fifty
mixture of a $\chi^{2}_{1}$ and a $\chi^{2}_{2}$ distribution under the
hypothesis of homogeneity.

  * Qin and Smith (2006) Qin, Y. S. and Smith, B. (2006). The likelihood ratio test for homogeneity in bivariate normal mixtures. Journal of Multivariate Analysis, 97, 474–491.
[§5.2.2] Generalize Qin and Smith (2004) to a bivariate normal mixture model
with known covariance matrix under the condition $\min(\pi,1-\pi)\geq\epsilon$
for some positive $\epsilon<1/2$ (Theorem 1). In practice, the limiting
distribution must be found numerically, though an approximation is provided in
their Section 4.

  * (40) Chen, H. and Chen, J. (2001b). The likelihood ratio test for homogeneity in finite mixture models. The Canadian Journal of Statistics, 29, 201–215.
[§5.3] Consider the same setting than Böhning et al. (1994), though the
component distributions are allowed to belong to a generic parametric family.
They show that under suitable conditions, which guarantee identifiability of
the mixture and regularity of the component distributions, the limiting
distribution of the likelihood ratio is the distribution of the squared
supremum of a left-truncated standard Gaussian process, whose autocorrelation
function is explicitly presented.

  * Garel (2007) Garel, B. (2007). Recent asymptotic results in testing for mixtures. Computational Statistics & Data analysis, 51, 5295–5304.
[§5.3] Provides an overview of asymptotic results for testing homogeneity for
the general two-component mixture model (5.1) and illustrates some new results
such as the calculation of tail probabilities and asymptotic power for both
bounded and unbounded parameter spaces.

  * Liu et al. (2003) Liu, X., Pasarica, C. and Shao, Y. (2003). Testing homogeneity in gamma mixture models. Scandinavian Journal of Statistics, 30, 227–239.
[§5.3] Characterise the asymptotic behaviour of the likelihood ratio for
testing model homogeneity against a two-component gamma mixture with known
shape and unknown rate parameters. Show that under the null hypothesis the
asymptotic distribution agrees with the distribution of the square of Davies’s
(1977) statistic. Further show that if the unknown rate parameter belongs to
an unbounded set, the likelihood ratio diverges to infinity in probability at
rate $O\\{\log(\log n)\\}$, in accordance with Hartigan (1985).

  * Mendell et al. (1991) Mendell, N. R., Thode, H. C. Jr. and Finch, S. J. (1991). The likelihood ratio test for the two-component normal mixture problem. Biometrics, 47, 1143–1148.
[§5.3] Investigate numerically the distribution of the likelihood ratio under
the alternative hypothesis of a two-component Gaussian mixture with unequal
means and common variances for a wide range of mixing proportions. The authors
conjecture that the limiting distribution is a non-central $\chi^{2}_{2}$
distribution.

  * Gombay and Horváth (1994b) Gombay, E. and Horváth, L. (1994b). An application of the maximum likelihood test to the change-point problem. Stochastic Processes and their Applications, 50, 161–171.
[§6.1] Prove that a suitably centered and rescaled likelihood ratio statistics
converges to a Gumbel distribution under the null hypothesis of no change.

  * Gombay and Horváth (1996) Gombay, E. and Horváth, L. (1996). On the rate of approximations for maximum likelihood tests in change-point models. Journal of Multivariate Analysis, 56, 120–152.
[§6.1] Generalise Gombay and Horváth (1994b) to the case where a fixed
nuisance parameter is present.

  * Gombay and Horváth (1999) Gombay, E. and Horváth, L. (1999). Change-points and bootstrap. Environmetrics, 10, 725–736.
[§6.1] Detect possible changes in the distribution of random vector by using
the weighted bootstrap.

  * Haccou et al. (1987) Haccou, P., Meelis, E. and van de Geer, S. (1987). The likelihood ratio test for the change point problem for exponentially distributed random variables. Stochastic Processes and their Applications, 27, 121–139.
[§6.1] Show that under the null hypothesis of no change in the rate parameter
of an exponential distribution, the distribution of the likelihood ratio
statistic converges to an extreme value distribution. The limiting
distribution of the likelihood ratio statistic is obtained by using the theory
of uniform quantile process.

  * Henderson (1990) Henderson, R. (1990). A problem with the likelihood ratio test for a change-point hazard rate model. Biometrika, 77, 835–843.
[§6.1] Considers modified likelihood ratio tests for survival data as in
Worsley (1988).

  * Hinkley (1970) Hinkley, D. V. (1970). Inference about the change-point in a sequence of random variables. Biometrika, 57, 1–17.
[§6.1] Derives limit theorems for the likelihood ratio for a change in a
sequence of binomial distributions.

  * Horváth (1989) Horváth, L. (1989). The limit distributions of the likelihood ratio and cumulative sum tests for a change in binomial probability. Journal of Multivariate Analysis, 31, 148–159.
[§6.1] Derives limit theorems for the likelihood ratio for a change in a
sequence of binomial distributions.

  * Loader (1992) Loader, C. R. (1992). A log-linear Model for a Poisson process change point. The Annals of Statistics, 20, 1391–1411.
[§6.1] Tests for the presence of a change point in a non-homogeneous Poisson
process. Large deviation techniques are used to approximate the significance
level, and approximations for the power function are provided. A coal mining
accident data set is used to illustrate the methodology.

  * Robbins et al. (2016) Robbins, M. W., Gallagher, C. M. and Lund, R. B. (2016). A general regression changepoint test for time series data. Journal of the American Statistical Association, 111, 670–683.
[§6.1] Develop likelihood ratio tests for the detection of a single change-
point in time series regression, considering both the time-varying situation
of Section 6.3 and multi-phase regression as in Section 6.3.

  * Robbins et al. (2011) Robbins, M., Gallagher, C., Lund, R. and Aue, A. (2011). Mean shift testing in correlated data. Journal of Time Series Analysis, 32, 498–511.
[§6.1] Address the problem of change-point detection in time series. Point out
that the likelihood ratio statistic converges to infinity if there is no
continuity constraint on the regression function at the change-point.

  * Sadooghi-Alvandi et al. (2011) Sadooghi-Alvandi, S. M., Nematollahi, A. R. and Habibi, R. (2011). Test procedures for change point in a general class of distributions, Journal of Data Science, 9, 111–126.
[§6.1] Consider change point detection for a general class of distributions.
Derive the exact and asymptotic null distributions of the quasi-Bayes and
likelihood ratio statistics using results from the theory of Brownian motion
and bridge processes.

  * Said et al. (2017) Said, K.K, Ning, W. and Tian, Y. (2017). Likelihood procedure for testing changes in skew normal model with applications to stock returns. Communications in Statistics – Simulation and Computation, 46, 6790–6802.
[§6.1] Using the results of (1997), investigated a trimmed version of the
likelihood ratio statistic for detecting changes in parameters of the skew
normal distribution. The asymptotic distribution of the test statistics is
again a Gumbel distribution.

  * Siegmund (1988) Siegmund, D. (1988). Confidence sets in change-point problems. International Statistical Review, 56, 31–48.
[§6.1] Discusses several methods, based on the likelihood ratio, for the
construction of a confidence interval for the change-point in a sequence of
independent observations from completely specified distributions. The results
are generalised to the construction of confidence regions for the change-point
and the parameters which index the exponential family from which the
independent observations are drawn.

  * Visek (2003) Visek, T. (2003). The likelihood ratio method for testing changes in the parameters of double exponential observations. textitJournal of Statistical Planning and Inference, 113, 79–111.
[§6.1] Constructs procedures for testing a change in the distribution of a
sequence of independent and identically distributed random variables which
follow a double exponential law. The change can occur either in the location
of the distribution, in the scale or in both.

  * Wang et al. (2020) Wang, T., Tian, W. and Ning, W. (2020) Likelihood ratio test change-point detection in the skew slash distribution. Published online in Communications in Statistics – Simulation and Computation.
[§6.1] Using the results of (1997), recently investigated a trimmed version of
the likelihood ratio statistic for detecting changes in parameters of the skew
slash distribution. The asymptotic distribution of the test statistics is
again a Gumbel distribution.

  * Worsley (1988) Worsley, K. J. (1988). Exact percentage points of the likelihood-ratio test for a change-point hazard-rate model. Biometrics, 44, 259–263.
[§6.1] Considers test for detecting a change in the hazard function. The
likelihood ratio statistic is shown to be unbounded, but the exact null
distribution of a suitably modified likelihood ratio test is provided.

  * Chen and Gupta (1997) Chen, J. and Gupta, A. K. (1997). Testing and locating variance change points with application to stock prices. Journal of American Statistical Association,  92, 739–747.
[§6.2] Test and locate multiple change-points in the variance of a series of
independent normal observations with known mean using the Schwarz information
criterion.

  * Chen and Gupta (2004) Chen, J. and Gupta, A. K. (2004). Test and locate multiple change-points in the variance of a series of independent normal observations with known mean using the Schwarz information criterion. Statistics, 38, 17–28.
[§6.2] Generalize Chen and Gupta (1997) to the multivariate case.

  * Hawkins (1992) Hawkins, D. M. (1992). Detecting shifts in functions of multivariate location and covariance parameters. Journal of Statistical Planning and Inference, 33, 233–244.
[§6.2] Generalizes his 1977 paper to study eight procedures—which, however, do
not include the likelihood ratio—for monitoring possible shifts in the mean
vector or covariance matrix of an arbitrary multivariate random variable.

  * Horváth (1993) Horváth, L. (1993). The maximum likelihood method for testing changes in the parameters of normal observations. The Annals of Statistics, 21, 671–680.
[§6.2] Derives the asymptotic distribution of the likelihood ratio statistic
for testing whether the mean and/or the variance of a sequence of normal
observations changed over time at an unknown point $\tau$.

  * Hinkley (1970) Hinkley, D. V. (1970). Inference about the change-point in a sequence of random variables. Biometrika, 57, 1–17. [§6.2] Determine the asymptotic distribution of the maximum likelihood estimator of $\tau$ and of the likelihood ratio statistic for testing the null hypothesis $H_{0}:\tau=\tau_{0}$, that is, that the change occurred at a given time point $\tau_{0}$ in a sequence of normal variables.
  * Inclàn (1993) Inclàn, C. (1993). Detection of multiple changes of variance using posterior odds. Journal of Business and Economic Statistics, 11, 289–300.
[§6.2] Detects a single possible change-point in the variance of a sequence of
independent Gaussian random variables with known common mean using a Bayesian
approach.

  * James et al. (1987) James, B., James, K. L. and Siegmund, D. (1987). Tests for a change-point. Biometrika, 74, 71–83.
[§6.2] Compare various test statistics for detecting mean shifts in univariate
normal distributions, which also include the likelihood ratio.

  * James et al. (1992) James, B., James, K. L. and Siegmund, D. (1992). Asymptotic approximation for likelihood ratio test and confidence regions for a change point in mean of a multivariate normal distribution. Statistica Sinica, 2, 69–90.
[§6.2] Compare various test statistics for detecting mean shifts as in James
et al. (1987), but this time for the multivariate case.

  * Sen and Srivastava (1975) Sen, A. and Srivastava, M. S. (1975). Some one-sided tests for change in level. Technometrics,  17, 61–64.
[§6.2] Provide a Bayesian solution to the problem considered in Hawkins
(1977).

  * Srivastava and Worsley (1986) Srivastava, M.S. and Worsley, K.J. (1986). Likelihood ratio tests for a change in the multivariate mean. Journal of the American Statistical Association, 81, 199–204.
[§6.2] Consider change-point detection in location for the multivariate normal
distribution. The likelihood ratio statistic is shown to be equivalent to the
maximum of Hotteling’s two sample statistic and that the same statistic can be
used to test for extra-multinomial variation in a contingency table.

  * Tang and Gupta (1987) Tang, J. and Gupta, A. K.(1988). On testing homogeneity of variances for Gaussian models. Journal of Statistical Computation and Simulation, 27(2), 155–173.
[§6.2] Use Bartlett’s statistic to detect a single possible change-point in
the variance of a sequence of independent Gaussian random variables with known
common mean.

  * Andrews (1993) Andrews, D. W. K.(1993). Tests for parameter instability and structural change with unknown change point. Econometrica, 61, 821–856.
[§6.3] Considers Wald, score and likelihood ratio type tests based on the
generalized method of moments to detect a possible structural change in
multiple regression with unknown change-point. Tables of critical values are
provided. If the change-point is fixed, the test statistics follow a chi-
squared distribution with $p$ degrees of freedom.

  * Andrews et al. (1996) Andrews, D. W. K., Lee, I. and Ploberger, W. (1996). Optimal changepoint tests for normal linear regression. Journal of Econometrics, 70, 9–38.
[§6.3] Extend Andrews (1993) and determine a class of finite-sample optimal
tests for the existence of a single or multiple changes at unknown time points
in multiple linear regression with normal errors and known variance. Exact
critical values are obtained straightforwardly on a case by case basis using
simulation. Simulation is furthermore used to compare the power of the
proposed test statistics.

  * Aue et al. (2008) Aue, A., Horváth, L., Hušková, M. and Kokoszka, P. (2008). Testing for changes in polynomial regression. Bernoulli, 14, 637–660.
[§6.3] Test the null hypothesis of no change in the polynomial regression
parameters against the alternative of a break at an unknown time point. Derive
the extreme value distribution of a maximum type test statistic which is
asymptotically equivalent to the likelihood ratio. The approximation proves to
work well even for small sample sizes. Previous ground breaking work for
likelihood ratio testing in polynomial regression is reviewed in their Section
1.

  * Deutsch (1992) Deutsch, J. (1992). Linear regression under two separate regimes: An empirical distribution for Quandt’s log-likelihood ratio. Applied Economics, 24, 123–127.
[§6.3] Provides an empirical distribution for Quandt’s (1960) statistic to
verify the existence of a change point when assessing the stability of
regression relationships over time.

  * Feder (1975a) Feder, P. I. (1975a). On asymptotic distribution theory in segmented regression problems – identified case. Annals of Statistics, 3, 49–83.
[§6.3] Puts the bases for Feder (1975b) by developing the asymptotic
distribution theory of least squares estimators in broken line regression.

  * Feder (1975b) Feder, P. I. (1975b). The log likelihood ratio in segmented regression. The Annals of Statistics, 3, 84–97.
[§6.3] Contribution related to Hinkley (1969), which studies the asymptotic
distribution of the likelihood ratio statistic in segmented regression, which
are models where the analytical form of the regression mean changes according
to the values the covariate $x$ takes on. In particular, proves that under
suitable identifiability conditions the limiting distributions of Wilks and
Chernoff still apply.

  * Hawkins (1980) Hawkins, D. M. (1980). A note on continuous and discontinuous segmented regressions. Technometrics, 22, 443–444.
[§6.3] Distinguishes between continuous and discontinuous models. If the model
is discontinuous the distribution of the likelihood ratio for verifying the
presence of two segments diverges to infinity.

  * Hinkley (1969) Hinkley, D. V. (1969). Inference about the intersection in two-phase regression. Biometrika, 56, 495–504.
[§6.3] Considers tests on the parameter
$\gamma=\frac{\alpha_{1}-\alpha_{2}}{\beta_{2}-\beta_{1}}$, which identifies
the $x$ value at which the two straight lines cross each other. Furthermore,
investigates the asymptotic distribution of the maximum likelihood estimator
of $\gamma$, showing that the normal approximation may be poor for small
samples.

  * Hinkley (1971) Hinkley, D. V. (1971). Inference in two-phase regression. Journal of the American Statistical Association, 66, 736–743.
[§6.3] Building on the grounds of Hinkley (1969), investigates two cases: (i)
there is no change $\alpha_{1}=\alpha_{2}$ and $\beta_{1}=\beta_{2}$, and (ii)
the response variable is constant until the change-point $\tau$
($\beta_{1}=0$). On empirical grounds, suggests to approximate the finite-
sample distribution by an $F_{1,n-4}$, respectively. He furthermore develops
confidence intervals for the change-point $\tau$ and joint confidence regions
for the change-point and the model parameters.

  * Kelly (2015) Kelly, G. E. (2015). Approximations to the p-values of tests for a change-point under non-standard conditions. Journal of Statistical Computation and Simulation, 86, 1430–1449.
[§6.3] Considers three variants of the likelihood-based statistics studied by
Andrews and Ploberger (1994) for the general regression setting with time
trend regressors. Critical values are obtained via simulation.

  * Kim (1994) Kim, H.-J. (1994). Tests for a change-point in linear regression. In Change-point Problems. IMS Lecture Notes - Monograph Series (edited by E. Carlstein, H.-G. Möller and D. Siegmud), 23, pp. 170–176. IMS, Hayward.
[§6.3] Extend Kim and Siegmund (1989) to multiple linear regression.

  * Kim and Cai (1993) Kim, H. J. and Cai, L. (1993). Robustness of the likelihood ratio test for a change in simple linear regression. Journal of the American Statistical Association, 88, 864–871.
[§6.3] Study the robustness of the likelihood ratio test in simple linear
regression.

  * Knowles et al. (1991) Knowles, M., Siegmund, D. and Zhang, H. P. (1991). Confidence semilinear regression, Biometrika, 78 15–31.
[§6.3] Provide confidence intervals and joint confidence regions based on the
likelihood ratio statistic for the change-point in a broken-line regression
model with $K=1$.

  * Koul and Qian (2002) Koul, H. L. and Qian, L. (2002). Asymptotics of maximum likelihood estimator in a two-phase linear regression model. Journal of Statistical Planning and Inference, 108, 99–119.
[§6.3] Consider two-phase linear regression with arbitrary error distribution
and fixed jump in the linear predictor at the true change-point $\tau$. The
maximum likelihood estimator $\hat{\tau}$ is shown to be consistent and the
finite-sample distribution of the standardized maximum likelihood estimator to
converge weakly to the distribution of a compound Poisson process.

  * Lund and Reeves (2002) Lund, R. and Reeves, J. (2002). Detection of undocumented changepoints: A revision of the two-phase regression model. Journal of Climate, 15, 2547–2554.
[§6.3] Consider the same problem as Hinkley (1969) and conjecture that the
asymptotic approximation of the finite-sample distribution of the statistics
may involve the Gumbel distribution.

  * Luo et al. (1997) Luo, X., Turnbull, B. W. and Clark, L. C. (1997). Likelihood ratio tests for a changepoint with survival data. Biometrika, 84, 555-565.
[§6.3] Derive the asymptotic distribution of the likelihood ratio statistic to
test for a possible time-lag effect in covariates in the presence of right-
censored observations.

  * Reeves et al. (2007) Reeves, J., Chen, J., Wang, X. L., Lund, R. B. and Lu, Q. (2007). A review and comparison of changepoint detection techniques for climate data. Journal of Applied Meteorology and Climatology, 46, 900–915.
[§6.3] Give an application for the identification of a possible shift in mean
temperature values in climate data.

  * Robison (1964) Robison, D. E. (1964). Estimates for the points of intersection of two polynomial regressions. Journal of the American Statistical Association, 59, 214–224.
[§6.3] Generalize Sprent (1961) to polynomial regression.

  * Siegmund and Zhang (1994) Siegmund, D. O. and Zhang, H. (1994). Confidence regions in broken-line regression. In Change-point Problems. IMS Lecture Notes - Monograph Series (edited by E. Carlstein, H.-G. Möller and D. Siegmud), 23, pp. 292–316. IMS, Hayward. int Problems.
[§6.3] Provide as Knowles et al. (1991) confidence intervals and joint
confidence regions based on the likelihood ratio statistic for the change-
point in a broken-line regression model with $K=1$.

  * Sprent (1961) Sprent, P. (1961). Some hypotheses concerning two-phase regression lines. Biometrics, 17, 634–645.
[§6.3] For a known change-point $\tau$, he uses the likelihood ratio to test a
number of hypotheses on the relationship between the two straight lines which
form the broken-line regression model.

  * Banerjee (2008) Banerjee, M. (2008). Estimating monotone, unimodal and U-shaped failure rates using asymptotic pivots. Statistica Sinica, 18(2), 467–492.
[§7] Proposes a method, based on asymptotic pivots, for constructing
nonparametric confidence sets for a monotone failure rate, and for unimodal or
U-shaped hazards.

  * Groeneboom and Jongbloed (2015) Groeneboom, P. and Jongbloed, G. (2015). Nonparametric confidence intervals for monotone functions. The Annals of Statistics, 43(5), 2019–2054.
[§7] Obtain confidence intervals for distribution functions and monotone
densities by inverting the acceptance region of the nonparametric likelihood
ratio test.