Distribution Problems in Clustering

Hartigan, John

doi:10.1016/b978-0-12-714250-0.50007-3

Cited by 86 publications

(53 citation statements)

References 21 publications

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“…If the number of states in Markov-switching models is known, the EM algorithm yields consistent parameter estimates, and statistical inference proceeds via standard maximum-likelihood theory (e.g., Bickel, Ritov and Rydén 1998). If the number of states is not known, however, the likelihood ratio test to infer the true number of states breaks down because regularity conditions do not hold (see Hartigan 1977, Hansen 1992, Garcia 1998). …”

Section: Introductionmentioning

confidence: 99%

Markov-Switching Model Selection Using Kullback-Leibler Divergence

2005

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In Markov-switching regression models, we use Kullback-Leibler (KL) divergence between the true and candidate models to select the number of states and variables simultaneously. In applying Akaike information criterion (AIC), which is an estimate of KL divergence, we find that AIC retains too many states and variables in the model. Hence, we derive a new information criterion, Markov switching criterion (MSC), which yields a marked improvement in state determination and variable selection because it imposes an appropriate penalty to mitigate the over-retention of states in the Markov chain. MSC performs well in Monte Carlo studies with single and multiple states, small and large samples, and low and high noise. Furthermore, it not only applies to Markov-switching regression models, but also performs well in Markovswitching autoregression models. Finally, the usefulness of MSC is illustrated via applications to the U.S. business cycle and the effectiveness of media advertising. IntroductionEconomic systems often experience shocks that shift them from their present state into another state; for example, nations lurch into recession, government regimes change over time, and financial markets exhibit bubbles and crashes. These states tend to be stochastic and dynamic: if they occur once, they probably recur. To capture such probabilistic state transitions over time, Markov-switching models provide an analytical framework. In economics, Markov- To estimate Markov-switching models, Baum and his colleagues Petrie 1966, Baum et al. 1970) developed the forward-backward algorithm, which was extended to encompass general latent variable models under the expectation-maximization (EM) principle (see Dempster, Laird and Rubin 1977). If the number of states in Markov-switching models is known, the EM algorithm yields consistent parameter estimates, and statistical inference proceeds via standard maximum-likelihood theory (e.g., Bickel, Ritov and Rydén 1998). If the number of states is not known, however, the likelihood ratio test to infer the true number of states breaks down because regularity conditions do not hold (see Hartigan 1977, Hansen 1992, Garcia 1998 We organize this paper as follows. In Section 2, we describe the model structure and estimation algorithm for multiple state Markov-switching models. We derive the information criterion in Section 3 and investigate its properties and performance under various conditions in Section 4. Section 5 presents empirical applications to business cycles and media advertising.Section 6 concludes the paper by identifying avenues for future research. Estimating N-state Markov-switching modelsWe present the model structure, establish notation, and briefly describe the estimation of Markov-switching regressions, conditional on knowing the number of states N. Model structureConsider an N-state Markov chain. Let t s denote an N × 1 selection vector with elements s ti = 1 or 0, according to whether the Markov chain resides in the state i ( N i ,..., 1 = ). The unobserved stat...

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Section: Introductionmentioning

confidence: 99%

Markov-Switching Model Selection Using Kullback-Leibler Divergence

2005

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“…Given the low number of data available for the analysis, it was 267 considered that the loss of data through the use of firm slices was unacceptable and over-268 lapping ones were instead adopted. In addition, trials indicated that using firm time slices would Over-lapping time slices were created by using Moran's Local I test (Anselin, 1995) After data were allocated to the time slices, a partitioning clustering technique, K-means, was 290 implemented (Hartigan, 1975(Hartigan, , 1977. K-means clustering is a statistical method for grouping strength is faster and produces more discrete clusters.…”

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confidence: 99%

Sea-level change and demography during the last glacial termination and early Holocene across the Australian continent

Williams

Ulm

Sapienza

et al. 2018

Quaternary Science Reviews

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“…That is why the class of normal functions is most often chosen as an approximate model for the general set of random data; in this case, it is enough to evaluate only two parameters, and , for their sample. The mixtures of distributions on their basis, which possess the infinite differentiability, smoothness, and so forth are widely used at various simulations [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Optimal Regularities of the Normal Distribution for Estimating the Sample Statistics of the Results of a Physical Experiment

Kosobutskyy¹

2018

Ukr. J. Phys.

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Basic probabilistic principles for the formation of the normal distribution for random fluctuations of physical quantities under the action of independent random factors on the physical system have been formulated. The emphasis is made on the integrated approach to the probabilistic statistical analysis of a sample of experimental results.K e y w o r d s: normal distribution, expectation, variance, random variables.

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Distribution Problems in Clustering

Cited by 86 publications

References 21 publications

Markov-Switching Model Selection Using Kullback-Leibler Divergence

Markov-Switching Model Selection Using Kullback-Leibler Divergence

Sea-level change and demography during the last glacial termination and early Holocene across the Australian continent

Optimal Regularities of the Normal Distribution for Estimating the Sample Statistics of the Results of a Physical Experiment

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