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Asymptotic properties of a stochastic EM algorithm for mixtures with censored data
Umeå University, Faculty of Social Sciences, Department of Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2010 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 140, no 1, 111-127 p.Article in journal (Refereed) Published
Abstract [en]

Weak consistency and asymptotic normality is shown for a stochastic EM algorithm for censored data from a mixture of distributions under lognormal assumptions. The asymptotic properties hold for all parameters of the distributions, including the mixing parameter. In order to make parameter estimation meaningful it is necessary to know that the censored mixture distribution is identifiable. General conditions under which this is the case are given. The stochastic EM algorithm addressed in this paper is used for estimation of wood fibre length distributions based on optically measured data from cylindric wood samples (increment cores).

Place, publisher, year, edition, pages
Elsevier , 2010. Vol. 140, no 1, 111-127 p.
Keyword [en]
Censoring; Fibre length distribution; Identifiability; Increment core; Mixture; Stochastic EM algorithm
National Category
Mathematics Probability Theory and Statistics
Research subject
Mathematical Statistics; Statistics
Identifiers
URN: urn:nbn:se:umu:diva-29776DOI: 10.1016/j.jspi.2009.06.014ISI: 000271354300010OAI: oai:DiVA.org:umu-29776DiVA: diva2:278112
Available from: 2009-11-24 Created: 2009-11-23 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Estimation of wood fibre length distributions from censored mixture data
Open this publication in new window or tab >>Estimation of wood fibre length distributions from censored mixture data
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The motivating forestry background for this thesis is the need for fast, non-destructive, and cost-efficient methods to estimate fibre length distributions in standing trees in order to evaluate the effect of silvicultural methods and breeding programs on fibre length. The usage of increment cores is a commonly used non-destructive sampling method in forestry. An increment core is a cylindrical wood sample taken with a special borer, and the methods proposed in this thesis are especially developed for data from increment cores. Nevertheless the methods can be used for data from other sampling frames as well, for example for sticks with the shape of an elongated rectangular box.

This thesis proposes methods to estimate fibre length distributions based on censored mixture data from wood samples. Due to sampling procedures, wood samples contain cut (censored) and uncut observations. Moreover the samples consist not only of the fibres of interest but of other cells (fines) as well. When the cell lengths are determined by an automatic optical fibre-analyser, there is no practical possibility to distinguish between cut and uncut cells or between fines and fibres. Thus the resulting data come from a censored version of a mixture of the fine and fibre length distributions in the tree. The methods proposed in this thesis can handle this lack of information.

Two parametric methods are proposed to estimate the fine and fibre length distributions in a tree. The first method is based on grouped data. The probabilities that the length of a cell from the sample falls into different length classes are derived, the censoring caused by the sampling frame taken into account. These probabilities are functions of the unknown parameters, and ML estimates are found from the corresponding multinomial model.

The second method is a stochastic version of the EM algorithm based on the individual length measurements. The method is developed for the case where the distributions of the true lengths of the cells at least partially appearing in the sample belong to exponential families. The cell length distribution in the sample and the conditional distribution of the true length of a cell at least partially appearing in the sample given the length in the sample are derived. Both these distributions are necessary in order to use the stochastic EM algorithm. Consistency and asymptotic normality of the stochastic EM estimates is proved.

The methods are applied to real data from increment cores taken from Scots pine trees (Pinus sylvestris L.) in Northern Sweden and further evaluated through simulation studies. Both methods work well for sample sizes commonly obtained in practice.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik, 2007. 24 p.
Keyword
censoring, fibre length distribution, identifiability, increment core, length bias, mixture, stochastic EM algorithm
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-1094 (URN)978-91-7264-300-0 (ISBN)
Public defence
2007-05-16, MA121, MIT, 901 87, Umeå, 13:15
Opponent
Supervisors
Available from: 2007-04-24 Created: 2007-04-24 Last updated: 2012-08-24Bibliographically approved

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Svensson, IngridSjöstedt-de Luna, Sara
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