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  • 1.
    Angelov, Angel G.
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Maximum likelihood estimation for survey data with informative interval censoring2019In: AStA Advances in Statistical Analysis, ISSN 1863-8171, E-ISSN 1863-818X, Vol. 103, no 2, p. 217-236Article in journal (Refereed)
    Abstract [en]

    Interval-censored data may arise in questionnaire surveys when, instead of being asked to provide an exact value, respondents are free to answer with any interval without having pre-specified ranges. In this context, the assumption of noninformative censoring is violated, and thus, the standard methods for interval-censored data are not appropriate. This paper explores two schemes for data collection and deals with the problem of estimation of the underlying distribution function, assuming that it belongs to a parametric family. The consistency and asymptotic normality of a proposed maximum likelihood estimator are proven. A bootstrap procedure that can be used for constructing confidence intervals is considered, and its asymptotic validity is shown. A simulation study investigates the performance of the suggested methods.

  • 2.
    Angelov, Angel G.
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Nonparametric estimation for self-selected interval data collected through a two-stage approach2017In: Metrika (Heidelberg), ISSN 0026-1335, E-ISSN 1435-926X, Vol. 80, no 4, p. 377-399Article in journal (Refereed)
    Abstract [en]

    Self-selected interval data arise in questionnaire surveys when respondents are free to answer with any interval without having pre-specified ranges. This type of data is a special case of interval-censored data in which the assumption of noninformative censoring is violated, and thus the standard methods for interval-censored data (e.g. Turnbull's estimator) are not appropriate because they can produce biased results. Based on a certain sampling scheme, this paper suggests a nonparametric maximum likelihood estimator of the underlying distribution function. The consistency of the estimator is proven under general assumptions, and an iterative procedure for finding the estimate is proposed. The performance of the method is investigated in a simulation study.

  • 3.
    Angelov, Angel G.
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Kriström, Bengt
    Umeå University, Faculty of Social Sciences, Center for Environmental and Resource Economics (CERE). Department of Forest Economics, Swedish University of Agricultural Sciences.
    Nilsson, Mats E.
    Gösta Ekman Laboratory, Department of Psychology, Stockholm University.
    Testing for stochastic dominance: Procedures with four hypothesesManuscript (preprint) (Other academic)
  • 4.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    A general central limit theorem for strong mixing sequences2014In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 94, p. 236-238Article in journal (Refereed)
    Abstract [en]

    A central limit theorem for strong mixing sequences is given that applies to both non-stationary sequences and triangular array settings. The result improves on an earlier central limit theorem for this type of dependence given by Politis, Romano and Wolf in 1997.

  • 5.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, S-901 83 Umeå, Sweden.
    Alternatives to maximum likelihood estimation based on spacings and the Kullback-Leibler divergence2008In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 138, no 6, p. 1778-1791Article in journal (Refereed)
    Abstract [en]

    An alternative to the maximum likelihood (ML) method, the maximum spacing (MSP) method, is introduced in Cheng and Amin [1983. Estimating parameters in continuous univariate distributions with a shifted origin. J. Roy. Statist. Soc. Ser. B 45, 394–403], and independently in Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112]. The method, as described by Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112], is derived from an approximation of the Kullback–Leibler divergence. Since the introduction of the MSP method, several closely related methods have been suggested. This article is a survey of such methods based on spacings and the Kullback–Leibler divergence. These estimation methods possess good properties and they work in situations where the ML method does not. Important issues such as the handling of ties and incomplete data are discussed, and it is argued that by using Moran's [1951. The random division of an interval—Part II. J. Roy. Statist. Soc. Ser. B 13, 147–150] statistic, on which the MSP method is based, we can effectively combine: (a) a test on whether an assigned model of distribution functions is correct or not, (b) an asymptotically efficient estimation of an unknown parameter θ0θ0, and (c) a computation of a confidence region for θ0.

  • 6.
    Ekström, Magnus
    Department of Forest Resource Management and Geomatics, Swedish University of Agricultural Sciences, Umeå.
    Consistency of generalized maximum spacing estimates2001In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 28, no 2, p. 343-354Article in journal (Refereed)
    Abstract [en]

    General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kullback–Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby & Ekstrom (1997) (using simple spacings) and Ekstrom (1997b) (using high order spacings) to obtain a class of methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao & Hahn (1999) for the MSP method. In particular, it will be proved that GMSP methods give consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.

  • 7.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Consistency of generalized maximum spacing estimates1997Report (Other academic)
    Abstract [en]

    General methods for the estimation of distributions can be derived from approx­imations of certain information measures. For example, both the maximum like­lihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kuliback-Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distri­butions is obtained from an approximation based on spacings of an information measure, were used by Ranneby and Ekström (1997) (using simple spacings) and Ekström (1997) (using high order spacings) to obtain a class of estimation methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao and Hahn (1996) for the MSP method. In particular, it will be shown that GMSP methods give Ll consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.

  • 8.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Generalized maximum spacing estimators1997Report (Other academic)
    Abstract [en]

    The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general method for estimating param­eters in univariate continuous distributions and is known to give consistent and asymptotically efficient estimates under general conditions. This method can be derived from an approximation based on simple spacings of the Kullback-Leibler information.

    In the present paper, we introduce a class of estimation methods, derived from approximations based on mth order spacings of certain information measures, i.e. the ^-divergences introduced by Csiszâr (1963). The introduced class of methods includes the MSP method as a special case. A subclass of these methods was considered earlier in Ranneby and Ekström (1997), i.e. those based on first order spacings. Here it is found that such methods can be improved by using high order spacings. We also show that the suggested methods give consistent estimates under general conditions. [1]

    [1]Research was supported by The Bank of Sweden Tercentenary Foundation.

  • 9.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Maximum product of spacings estimation1997In: Encyclopedia of Statistical Sciences, Update Volume 3, John Wiley & Sons, 1997, p. 443-447Chapter in book (Other academic)
  • 10.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Maximum spacing methods and limit theorems for statistics based on spacings1997Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general estimation method for continuous univariate distributions. The MSP method, which is closely related to the maximum likelihood (ML) method, can be derived from an approximation based on simple spacings of the Kullback-Leibler information. It is known to give consistent and asymptotically efficient estimates under general conditions and works also in situations where the ML method fails, e.g. for the three parameter Weibull model.

    In this thesis it is proved under general conditions that MSP estimates of parameters in the Euclidian metric are strongly consistent. The ideas behind the MSP method are extended and a class of estimation methods is introduced. These methods, called generalized MSP methods, are derived from approxima­tions based on sum-functions of rath order spacings of certain information mea­sures, i.e. the ^-divergences introduced by Csiszår (1963). It is shown under general conditions that generalized MSP methods give consistent estimates. In particular, it is proved that generalized MSP methods give L1 consistent esti­mates in any family of distributions with unimodal densities, without any further conditions on the distributions. Other properties such as distributional robust­ness are also discussed. Several limit theorems for sum-functions of rath order spacings are given, for ra fixed as well as for the case when ra is allowed to in­crease to infinity with the sample size. These results provide a strongly consistent nonparametric estimator of entropy, as well as a characterization of the uniform distribution. Further, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives.

  • 11.
    Ekström, Magnus
    Department of Forest Resource Management and Geomatics, Swedish University of Agricultural Sciences, Umeå.
    Nonparametric estimation of the variance of sample means based on nonstationary spatial data2002In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 31, no 10, p. 1743-1775Article in journal (Refereed)
    Abstract [en]

    In Politis and Romano (Politis, D.N.; Romano, J.P. Nonparametric Resampling for Homogeneous Strong Mixing Random Fields. Journal of Multivariate Analysis 1993, 47, 301–328.), different block resampling estimators of variance of general linear statistics, e.g., a sample mean, were proposed under the assumption of stationarity. In the present paper such estimators of variance of sample means, computed from nonstationary spatially indexed data , where is a finite subset of the integer lattice , are studied. Consistency of estimators of variance will be shown for the following kind of data: Observations taken from different lattice points are allowed to come from different distributions, and the dependence structure is allowed to differ over the lattice. We assume that all observed values are from distributions with the same expected value, or with expected values that decompose additively into directional components. Furthermore, it will be assumed that observations separated by a certain distance are independent.

  • 12.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    On the consistency of the maximum spacing method1998In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 70, no 2, p. 209-224Article in journal (Refereed)
    Abstract [en]

    The main result of this paper is a consistency theorem for the maximum spacing method, a general method of estimating parameters in continuous univariate distributions, introduced by Cheng and Amin (J. Roy. Statist. Soc. Ser. A45 (1983) 394–403) and independently by Ranneby (Scand. J. Statist.11 (1984) 93–112). This main result generalizes a theorem of Ranneby (Scand. J. Statist.11 (1984) 93–112). Also, some examples are given, which shows that this estimation method works also in cases where the maximum likelihood method breaks down.

  • 13.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Powerful parametric tests based on sum-functions of spacings2013In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 40, no 4, p. 886-898Article in journal (Refereed)
    Abstract [en]

    Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this paper, tests based on sum-functions of sample spacings are proposed, and large sample theory of the tests are presented under simple null hypotheses as well as under close alternatives. Tests, which are optimal within this class, are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test in regular parametric models. Some examples are given, which show that the proposed tests work also in situations where the likelihood ratio test breaks down. Extensions to more general hypotheses are discussed.

  • 14.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå.
    Quantifying spatial patterns of landscapes2003In: Ambio, ISSN 0044-7447, E-ISSN 1654-7209, Vol. 32, no 8, p. 573-576Article in journal (Refereed)
    Abstract [en]

    In this paper we discuss information theoretical landscape indices based on data from digitized maps in grid format: measures based on Shannon's entropy, e.g. the measures of diversity and contagion; and measures based on conditional entropy, e.g. a new index which can be seen as an alternative to the measure of contagion that does not have the disadvantage of being highly correlated to the measure of diversity. We also introduce a measurement on how much information Is contained in a coarse-scale map about a fine-scale map.

  • 15.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Strong consistency of the maximum spacing estimate1996In: Theory of Probability and Mathematical Statistics, ISSN 0094-9000, Vol. 55, p. 55-72Article in journal (Refereed)
  • 16.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Strong limit theorems for sums of logarithms of high order spacings1999In: Statistics (Berlin), ISSN 0233-1888, E-ISSN 1029-4910, Vol. 33, no 2, p. 153-169Article in journal (Refereed)
    Abstract [en]

    Several strong limit theorems are proved for sums of logarithms of mth order spacings from general distributions. In all given results, the order mof the spacings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well as a characterization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives.

  • 17.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Strong limit theorems for sums of logarithms of mth order spacings1997Report (Other academic)
    Abstract [en]

    Several strong limit theorems axe proved for sums of logarithms of mth order spacings from general distributions. In all given results, the order of the spac­ings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well as a charac­terization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives. [1]

    [1] Research was supported by The Bank of Sweden Tercentenary Foundation.

  • 18.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå.
    Subsampling variance estimation for nonstationary spatial lattice data2008In: Scandinavian Journal of Statistics, ISSN 0303-6898, E-ISSN 1467-9469, Vol. 35, no 1, p. 38-63Article in journal (Refereed)
    Abstract [en]

    Most proposed subsampling and resampling methods in the literature assume stationary data. In many empirical applications, however, the hypothesis of stationarity can easily be rejected. In this paper, we demonstrate that moment and variance estimators based on the subsampling methodology can also be employed for different types of non-stationarity data. Consistency of estimators are demonstrated under mild moment and mixing conditions. Rates of convergence are provided, giving guidance for the appropriate choice of subshape size. Results from a small simulation study on finite-sample properties are also reported.

  • 19.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå.
    Sum-functions of spacings of increasing order2006In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 136, no 8, p. 2535-2546Article in journal (Refereed)
    Abstract [en]

    We discuss a class of statistics based on spacings of increasing order and show that these statistics are almost surely consistent. Special attention is devoted to estimation of φφ-divergences and to tests for uniformity on the unit interval. It is shown that tests for uniformity, based on sum-functions of spacings, are strongly consistent against all absolutely continuous alternatives having support [0,1].

  • 20.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Tests based on sum-functions of spacings2012Conference paper (Other academic)
    Abstract [en]

    Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this talk I present large sample theory of tests based on symmetric functions of sample spacings under simple null hypotheses as well as under close alternatives.  Tests which are optimal within this class are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test. Some examples are given, which show that the new tests work also in situations where the likelihood ratio test breaks down.

  • 21.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Tests based on sum-functions of spacings2012Conference paper (Other academic)
    Abstract [en]

    Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this talk I present large sample theory of tests based on symmetric functions of sample spacings under simple null hypotheses as well as under close alternatives.  Tests which are optimal within this class are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test. Some examples are given, which show that the new tests work also in situations where the likelihood ratio test breaks down.

  • 22.
    Ekström, Magnus
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Esseen, Per-Anders
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Westerlund, B.
    Grafström, A.
    Jonsson, B. G.
    Ståhl, G.
    Logistic regression for clustered data from environmental monitoring programs2018In: Ecological Informatics, ISSN 1574-9541, E-ISSN 1878-0512, Vol. 43, p. 165-173Article in journal (Refereed)
    Abstract [en]

    Large-scale surveys, such as national forest inventories and vegetation monitoring programs, usually have complex sampling designs that include geographical stratification and units organized in clusters. When models are developed using data from such programs, a key question is whether or not to utilize design information when analyzing the relationship between a response variable and a set of covariates. Standard statistical regression methods often fail to account for complex sampling designs, which may lead to severely biased estimators of model coefficients. Furthermore, ignoring that data are spatially correlated within clusters may underestimate the standard errors of regression coefficient estimates, with a risk for drawing wrong conclusions. We first review general approaches that account for complex sampling designs, e.g. methods using probability weighting, and stress the need to explore the effects of the sampling design when applying logistic regression models. We then use Monte Carlo simulation to compare the performance of the standard logistic regression model with two approaches to model correlated binary responses, i.e. cluster-specific and population-averaged logistic regression models. As an example, we analyze the occurrence of epiphytic hair lichens in the genus Bryoria; an indicator of forest ecosystem integrity. Based on data from the National Forest Inventory (NFI) for the period 1993-2014 we generated a data set on hair lichen occurrence on > 100,000 Picea abies trees distributed throughout Sweden. The NFI data included ten covariates representing forest structure and climate variables potentially affecting lichen occurrence. Our analyses show the importance of taking complex sampling designs and correlated binary responses into account in logistic regression modeling to avoid the risk of obtaining notably biased parameter estimators and standard errors, and erroneous interpretations about factors affecting e.g. hair lichen occurrence. We recommend comparisons of unweighted and weighted logistic regression analyses as an essential step in development of models based on data from large-scale surveys.

  • 23.
    Ekström, Magnus
    et al.
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå.
    Jammalamadaka, S. R.
    UCSB, Santa Barbara, USA.
    A general measure of skewness2012In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 82, no 8, p. 1559-1568Article in journal (Refereed)
    Abstract [en]

    A very general measure of skewness based on the quantiles is introduced, which includes several well-known measures as special cases. Sample versions of our measure may be used as test statistics for testing the hypothesis of symmetry about an unknown value. We provide large sample theory for such a statistic and discuss the asymptotic relative efficiencies of this against some competing test statistics for symmetry.

  • 24.
    Ekström, Magnus
    et al.
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå.
    Jammalamadaka, S. R.
    UCSB, Santa Barbara, USA.
    Addendum to “An asymptotically distribution-free test of symmetry” [Journal of Statistical Planning and Inference 137 (2007) 799–810]2009In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 139, no 4, p. 1569-1571Article in journal (Refereed)
  • 25.
    Ekström, Magnus
    et al.
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå, Sweden.
    Jammalamadaka, S. Rao
    UCSB, USA.
    An asymptotically distribution-free test of symmetry2007In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 137, no 3, p. 799-810Article in journal (Refereed)
    Abstract [en]

    A procedure, based on sample spacings, is proposed for testing whether a univariate distribution is symmetric about some unknown value. The proposed test is a modification of a sign test suggested by Antille and Kersting [1977. Tests for symmetry. Z. Wahrscheinlichkeitstheorie verw. Gebiete 39, 235-255], but unlike Antille and Kersting's test, our modified test is asymptotically distribution-free and is usable in practice. A simulation study indicates that the proposed test maintains the nominal level of significance, alpha fairly accurately even for samples of size as small as 20, and a comparison with the classical test based on sample coefficient of skewness, shows that our test has good power for detecting different asymmetric distributions.

  • 26.
    Ekström, Magnus
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Mirakhmedov, S. M.
    Jammalamadaka, S. R.
    A class of asymptotically efficient estimators based on sample spacings2018In: The 27th Nordic Conference in Mathematical Statistics: Abstracts, 2018Conference paper (Other academic)
    Abstract [en]

    We consider general classes of estimators based on higher-order sample spacings,called Generalized Spacings Estimators (GSEs). Such classes of estimators areobtained by minimizing approximations of Csisz´ar divergences between distributionsin the model and the true underlying distribution; maximum likelihood estimators(MLEs) may be derived in a similar way using the Kullback-Leibler divergence. Ourresults generalize several earlier studies on spacings-based estimation, by utilizingnon-overlapping spacings that are of an order which increases with the sample size.The GSEs are shown to be consistent as well as asymptotically normal under a fairlygeneral set of regularity conditions. When both the order of the spacings and thenumber of spacings grow with the sample size, an asymptotically efficient class ofestimators, called the “Minimum Power Divergence Estimators,” are shown to exist.Simulation studies give further support to the performance of these asymptoticallyefficient estimators in finite samples, and compare well relative to the MLEs as wellas corresponding estimators based on “overlapping” higher order spacings. Unlikethe MLEs, some of these estimators are also shown to be quite robust under heavycontamination.

  • 27.
    Ekström, Magnus
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. Department of Forest Resource Management, Swedish University of Agricultural Sciences, Umeå, Sweden.
    Mirakhmedov, S. M.
    Jammalamadaka, S. Rao
    A class of asymptotically efficient estimators based on sample spacings2019In: Test (Madrid), ISSN 1133-0686, E-ISSN 1863-8260Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider general classes of estimators based on higher-order sample spacings, called the Generalized Spacings Estimators. Such classes of estimators are obtained by minimizing the Csiszár divergence between the empirical and true distributions for various convex functions, include the “maximum spacing estimators” as well as the maximum likelihood estimators (MLEs) as special cases, and are especially useful when the latter do not exist. These results generalize several earlier studies on spacings-based estimation, by utilizing non-overlapping spacings that are of an order which increases with the sample size. These estimators are shown to be consistent as well as asymptotically normal under a fairly general set of regularity conditions. When the step size and the number of spacings grow with the sample size, an asymptotically efficient class of estimators, called the “Minimum Power Divergence Estimators,” are shown to exist. Simulation studies give further support to the performance of these asymptotically efficient estimators in finite samples and compare well relative to the MLEs. Unlike the MLEs, some of these estimators are also shown to be quite robust under heavy contamination.

  • 28.
    Ekström, Magnus
    et al.
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå.
    Sjöstedt-de Luna, Sara
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Subsampling methods to estimate the variance of sample means based on nonstationary spatial data with varying expected values2004In: Journal of the American Statistical Association, ISSN 0162-1459, E-ISSN 1537-274X, Vol. 99, no 465, p. 82-95Article in journal (Refereed)
    Abstract [en]

    Subsampling and block resampling methods have been suggested in the literature to nonparametrically estimate the variance of statistics computed from spatial data. Usually stationary data are required. However, in empirical applications, the assumption of stationarity often must be rejected. This article proposes nonparametric methods to estimate the variance of (functions of) sample means based on nonstationary spatial data using subsampling. We assume that data are observed on a lattice in some region of R-2. In the data that we consider, the information in the different picture elements (pixels) of the lattice are allowed to come from different distributions, with smoothly varying expected values, or with expected values decomposed additively into directional components. Furthermore, pixels are assumed to be locally dependent, and the dependence structure is allowed to differ over the lattice. Consistent variance estimators for (functions of) sample means, together with convergence rates in mean square, are provided under these assumptions. An example with applications to forestry, using satellite data, is discussed.

  • 29.
    Eriksson, Amanda
    et al.
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Gauslaa, Yngvar
    Palmqvist, Kristin
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Esseen, Per-Anders
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Morphology drives water storage traits in the globally widespreadlichen genus Usnea2018In: Fungal ecology, ISSN 1754-5048, E-ISSN 1878-0083, Vol. 35, p. 51-61Article in journal (Refereed)
    Abstract [en]

    Links between lichen morphology, internal/external water storage and distribution patterns are poorly known. We compared mass- (WC, % H2O) and area-based (WHC, mg H2O cm−2) hydration traits in seven pendent or shrubby Usnea species from oceanic to continental climates. All species held more external than internal water. Internal WHC and WC increased with specific thallus mass (STM, mg cm−2), while external WC decreased. Shrubby species had higher STM and total WHC than pendent ones. The continental Usnea hirta (shrubby) had the highest total and external storage; the suboceanic Usnea longissima (pendent) had the lowest internal storage. Morphology drives hydration traits and explains distributions of some Usnea species, but such traits did not distinguish oceanic from widespread species. Shrubby species maximize water storage and thus prolong hydration after rainfall events and/or hydration by dew. The low internal WHC in pendent species is likely an adaptation to frequent hydration in humid air.

  • 30.
    Esseen, Per-Anders
    et al.
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Westerlund, Bertil
    Palmqvist, Kristin
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Jonsson, Bengt Gunnar
    Grafström, Anton
    Ståhl, Göran
    Broad-scale distribution of epiphytic hair lichens correlates more with climate and nitrogen deposition than with forest structure2016In: Canadian Journal of Forest Research, ISSN 0045-5067, E-ISSN 1208-6037, Vol. 46, no 11, p. 1348-1358Article in journal (Refereed)
    Abstract [en]

    Hair lichens are strongly influenced by forest structure at local scales, but their broad-scale distributions are less understood. We compared the occurrence and length of Alectoria sarmentosa (Ach.) Ach., Bryoria spp., and Usnea spp. in the lower canopy of > 5000 Picea abies (L.) Karst. trees within the National Forest Inventory across all productive forest in Sweden. We used logistic regression to analyse how climate, nitrogen deposition, and forest variables influence lichen occurrence. Distributions overlapped, but the distribution of Bryoria was more northern and that of Usnea was more southern, with Alectoria's distribution being intermediate. Lichen length increased towards northern regions, indicating better conditions for biomass accumulation. Logistic regression models had the highest pseudo R-2 value for Bryoria, followed by Alectoria. Temperature and nitrogen deposition had higher explanatory power than precipitation and forest variables. Multiple logistic regressions suggest that lichen genera respond differently to increases in several variables. Warming decreased the odds for Bryoria occurrence at all temperatures. Corresponding odds for Alectoria and Usnea decreased in warmer climates, but in colder climates, they increased. Nitrogen addition decreased the odds for Alectoria and Usnea occurrence under high deposition, but under low deposition, the odds increased. Our analyses suggest major shifts in the broad-scale distribution of hair lichens with changes in climate, nitrogen deposition, and forest management.

  • 31. Grafström, Anton
    et al.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics. Department of Forest Resource Management, Swedish University of Agricultural Sciences, SE-90183 Umeå, Sweden.
    Jonsson, Bengt Gunnar
    Esseen, Per-Anders
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Ståhl, Göran
    On combining independent probability samples2019In: Survey Methodology, ISSN 0714-0045, E-ISSN 1492-0921, Vol. 45, no 2, p. 349-364Article in journal (Refereed)
    Abstract [en]

    Merging available sources of information is becoming increasingly important for improving estimates of population characteristics in a variety of fields. In presence of several independent probability samples from a finite population we investigate options for a combined estimator of the population total, based on either a linear combination of the separate estimators or on the combined sample approach. A linear combination estimator based on estimated variances can be biased as the separate estimators of the population total can be highly correlated to their respective variance estimators. We illustrate the possibility to use the combined sample to estimate the variances of the separate estimators, which results in general pooled variance estimators. These pooled variance estimators use all available information and have potential to significantly reduce bias of a linear combination of separate estimators.

  • 32. Jonsson, Bengt Gunnar
    et al.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Esseen, Per-Anders
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Grafström, Anton
    Ståhl, Göran
    Westerlund, Bertil
    Dead wood availability in managed Swedish forests: policy outcomes and implications for biodiversity2016In: Forest Ecology and Management, ISSN 0378-1127, E-ISSN 1872-7042, Vol. 376, p. 174-182Article in journal (Refereed)
    Abstract [en]

    Dead wood is a critical resource for forest biodiversity and widely used as an indicator for sustainable forest management. Based on data from the Swedish National Forest Inventory we provide baseline information and analyze trends in volume and distribution of dead wood in Swedish managed forests during 15 years. The data are based on ≈30,000 sample plots inventoried during three periods (1994–1998; 2003–2007 and 2008–2012). The forest policy has since 1994 emphasized the need to increase the amount of dead wood in Swedish forests. The average volume of dead wood in Sweden has increased by 25% (from 6.1 to 7.6 m3 ha−1) since the mid-1990s, but patterns differed among regions and tree species. The volume of conifer dead wood (mainly from Picea abies) has increased in the southern part of the country, but remained stable or decreased in the northern part. Heterogeneity of dead wood types was low in terms of species, diameter and decay classes, potentially negatively impacting on biodiversity. Overall, we found only minor effects of the current forest policy since most of the increase can be attributed to storm events creating a pulse of hard dead wood. Therefore, the implementation of established policy instruments (e.g. legislation and voluntary certification schemes) need to be revisited. In addition to the retention of dead trees during forestry operations, policy makers should consider calling for more large-scale targeted creation of dead trees and management methods with longer rotation cycles.

  • 33.
    Lidestav, Gun
    et al.
    Swedish University of Agricultural Sciences, Umeå.
    Ekström, Magnus
    Department of Forest Resource Management and Geomatics, Swedish University of Agricultural Sciences, Umeå.
    Introducing gender in studies on management behaviour among non-industrial private forest owners2000In: Scandinavian Journal of Forest Research, ISSN 0282-7581, E-ISSN 1651-1891, Vol. 15, no 3, p. 378-386Article in journal (Refereed)
    Abstract [en]

    The genderization of non-industrial private forest (NIPF) ownership creates different conditions for male and female forest owners. To compare male and female forest management behaviour and to examine whether observed differences should be understood in terms of gender, data from the Swedish National Board of Forestry's interview inquiry were used. Differences in frequency of harvesting and silvicultural operations were examined by binary logistic regression and differences in harvesting volume were tested by a multiple regression analysis. In all analyses sex of the owner was introduced as one of the explanatory variables. The sex of the owner was found to have a significant effect on the frequency of harvesting, cleaning and supplementary planting, but not on planting and mechanized scarification. In the cases when the sex of the owner was a significant factor, the degree of activity among the female owners was found to be lower. Results regarding harvested volumes did not expose ally significant differences in harvesting management strategies between male and female owners.

  • 34.
    Löfgren, Stefan
    et al.
    Institutionen för vatten och miljö, SLU.
    Ranneby, Bo
    Centre of Biostochastics, SLU.
    Ekström, Magnus
    Centre of Biostochastics, SLU.
    Yu, Jun
    Centre of Biostochastics, SLU.
    Naturliga bakgrundshalter av bly, zink och arsenik i Svenska ytvatten baserat på metallernas haltvariation i morän och sedimentära jordar: implikationer för EU:s ramdirektiv för vatten och nationella miljömål2007Report (Other academic)
    Abstract [en]

    In the autumn 2006, lakes and running waters should be classified according to the EU water framework directive (2000/60/EU), which successively is implemented into the Swedish Environmental Act. Surface water monitoring should be in operation and environmental action plans, based on environmental assessments, should be in force at catchment level. The aim is to achieve “Good ecological status” in 2015. For each lake and running water, type specific reference conditions should be determined with regard to hydromorphological, physical-chemical and biological parameters. The reference status should represent conditions without human influences.

    The aim of this project was to develop models for estimating reference levels on the Pb, Zn and As concentrations in Swedish lakes based on the natural prerequisites regarding geology, vegetation, hydrology and surface water chemistry. Environmental authorities should be able to use the models in assessments demanded by the EU water framework directive and coupled to the national goals “Living lakes and streams” and “Non-toxic environment”.

    Based on the results from the year 2000 Swedish national lake survey, the geochemical and biogeochemical surveys of the Swedish Geological Survey, official statistics of climate, topography, land use etc., statistical models have been developed in order to estimate the natural background levels of Pb, Zn and As in lakes with catchments dominated by forests, mires and/or alpine ecosystems. The models are based on data from 644 lakes and approximately 28 000 geochemical observations. Multiple linear regressions were used for creating the models. For Pb and Zn, the least squares estimation method was used, while a robust estimation method was used for As (e.g. the robust fit is minimally influenced by outliers in the independent variables space, in the response space, or in both). Detailed analyses of the statistical prerequisites were performed and the established models fulfil the theoretical criterions necessary for the chosen techniques. The variations in Pb, Zn and As concentrations can to a large extent, 71%, 65% and 54%, respectively, be explained by lake internal factors with coupling to the humus levels (TOC) and acidity status (pH). In addition, the lake metals concentrations seem to be influenced by the accumulated atmospheric deposition of metals and acidifying compounds.

    By creating a second type of models, where geochemical and biogeochemical parameters were forced into the models, the degrees of explanation increased by 3, 7 and 3 percentage units for Pb, Zn and As, respectively. This marginal explanation increase indicates that the national lake survey is a weak tool for analysing the geochemical influences on the metal concentrations in surface waters. The lakes are randomly selected and it is only by chance some few lakes, with high metal concentrations in the water due to elevated metal concentrations in the soils, are sampled. In most of the objects, the humus and pH levels overshadow the geochemistry influence since metal concentrations in soils are generally low.

    Locally and in areas with elevated metal concentrations in the soils, other studies have demonstrated a correlation between the geochemistry and the surface water concentrations of metals. Hence, there is still a need for assessments studying at what metal levels in the soils the geochemistry is of importance for the metal concentrations in surface waters. Therefore, we suggest the Swedish Geological Survey to initiate such a project focusing on studies in areas with naturally occurring high metal concentrations in the soils and where small headwater streams are the aquatic response systems.

  • 35. Mirakhmedov, Sherzod M.
    et al.
    Jammalamadaka, Sreenivasa R.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Edgeworth expansions for two-stage sampling with applications to stratified and cluster sampling2015In: Canadian Journal of Statistics / Revue Canadienne de Statistique, ISSN 0319-5724, Vol. 43, no 4, p. 578-599Article in journal (Refereed)
    Abstract [en]

    A two-term Edgeworth expansion for the standardized version of the sample total in a two-stagesampling design is derived. In particular, for the commonly used stratified and cluster sampling schemes,formal two-term asymptotic expansions are obtained for the Studentized versions of the sample total. Theseresults are applied in conjunction with the bootstrap to construct more accurate confidence intervals for theunknown population total in such sampling schemes.

  • 36.
    Olsson, David
    et al.
    Umeå University, Faculty of Medicine, Department of Public Health and Clinical Medicine, Occupational and Environmental Medicine.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, 901 83 Umeå, Sweden.
    Forsberg, Bertil
    Umeå University, Faculty of Medicine, Department of Public Health and Clinical Medicine, Occupational and Environmental Medicine.
    Temporal variation in air pollution concentrations and preterm birth: a population based epidemiological study2012In: International Journal of Environmental Research and Public Health, ISSN 1661-7827, E-ISSN 1660-4601, Vol. 9, no 1, p. 272-285Article in journal (Refereed)
    Abstract [en]

    There is growing evidence of adverse birth outcomes due to exposure to air pollution during gestation. However, recent negative studies are also reported. The aim of this study was to assess the effect of ozone and vehicle exhaust exposure (NO2) on the length of the gestational period and risk of preterm delivery. We used data from the Swedish Medical Birth Registry on all vaginally delivered singleton births in the Greater Stockholm area who were conceived during 1987–1995 (n = 115,588). Daily average levels of NO2 (from three measuring stations) and ozone (two stations) were used to estimate trimester and last week of gestation average exposures. Linear regression models were used to assess the association between the two air pollutants and three exposure windows, while logistic regression models were used when analyzing associations with preterm delivery ( < 37 weeks gestation). Five percent were born preterm. The median gestational period was 40 weeks. Higher levels of ozone during the first trimester were associated with shorter gestation as well as with an elevated risk of preterm delivery, the odds ratio from the most complex model was 1.06 (95% CI: 1.00–1.13) per 10 μg/m3 increase in the mean daily 8-h maximum concentration. Higher levels of ozone during the second trimester were associated with shorter gestation but the elevated risk of preterm delivery was not statistically significant. Higher levels of ozone and NO2 during the last week of gestation were associated with a shorter duration of gestation and NO2 also with preterm delivery. There were no significant associations between first and second trimester NO2 exposure estimates and studied outcomes. The effect of first trimester ozone exposure, known to cause oxidative stress, was smallest among women who conceived during autumn when vitamin D status, important for fetal health, in Scandinavian women is the highest.Keywords: pregnancy; preterm birth; vehicle emissions; ozone; nitrogen dioxide

  • 37.
    Ranneby, Bo
    et al.
    Department of Forest Resource Management and Geomatics Swedish University of Agricultural Sciences, Umeå, Sweden.
    Ekström, Magnus
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Maximum spacing estimates based on different metrics1997Report (Other academic)
    Abstract [en]

    The maximum spacing (MSP) method, introduced by Cheng and Amin (1983) and independently by Ranneby (1984), is a general method for estimating param­eters in univariate continuous distributions and is known to give consistent and asymptotically efficient estimates under general conditions. This method, which is closely related to the maximum likelihood (ML) method, can be derived from an approximation based on simple spacings of the Kullback-Leibler information. In the present paper, the ideas behind the MSP metod axe extended and a class of estimation methods is derived from approximations of certain information mea­sures, i.e. the ^-divergences introduced by Csiszâr (1963). We call these methods generalized maximum spacing (GMSP) methods, and it will be shown under gen­eral conditions that they give consistent estimates. GMSP methods have the advantage that they work also in situations where the ML method breaks down, e.g. due to an unbounded likelihood function. Other properties, such as asymp­totic normality and the behaviour of the estimates when the assigned model is only approximately true, will be discussed.[1]

    [1] Research was supported by MISTRA, the Foundation for Strategic Environmental Research.

  • 38.
    Ringvall, Anna
    et al.
    SLU.
    Snäll, Tord
    University of Colorado at Boulder, USA.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, S-901 83 Umeå, Sweden.
    Ståhl, Göran
    SLU.
    Unrestricted guided transect sampling for surveying sparse species2007In: Canadian Journal of Forest Research, ISSN 0045-5067, E-ISSN 1208-6037, Vol. 37, no 12, p. 2575-2586Article in journal (Refereed)
    Abstract [en]

    We present a modification of an earlier presented method using prior auxiliary information in the layout of survey strips. The idea is to imitate a skilled surveyor who purposively seeks the species of interest. Yet, the method “unrestricted guided transect sampling” (UGTS) is a probability sampling method. In comparison with a strip survey using no auxiliary information, UGTS gave 11%–64% lower standard errors for estimates of species population size in three simulated forest types. In a test in six stands where European aspen (Populus tremula L.) and an epiphytic moss (Orthotrichum speciosum Nees) had been mapped, UGTS gave a small improvement in some stands but considerably higher standard errors in other stands with kNN estimates of volume of deciduous trees derived from satellite images as covariate values. With covariates values simulated from aspen basal area, UGTS gave 8%–75% lower standard error than a strip survey using no auxiliary information. The study shows a gain in precision by using auxiliary information both in the design and in estimation when surveying sparse species but also that the correlation between the covariate and the variable of interest has to be relatively strong to make the method worthwhile.

  • 39.
    Ståhl, Göran
    et al.
    Department of Forest Resource Management, SwedishUniversity of Agricultural Sciences, SE 901 83 Umeå, Sweden.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Dahlgren, Jonas
    Department of Forest Resource Management, SwedishUniversity of Agricultural Sciences, SE 901 83 Umeå, Sweden.
    Esseen, Per-Anders
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Grafström, Anton
    Department of Forest Resource Management, SwedishUniversity of Agricultural Sciences, SE 901 83 Umeå, Sweden.
    Jonsson, Bengt-Gunnar
    Department of Natural Sciences, Mid-Sweden University, SE 851 70 Sundsvall, Sweden.
    Informative plot sizes in presence-absence sampling of forest floor vegetation2017In: Methods in Ecology and Evolution, ISSN 2041-210X, E-ISSN 2041-210X, Vol. 8, no 10, p. 1284-1291Article in journal (Refereed)
    Abstract [en]

    1. Plant communities are attracting increased interest in connection with forest and landscape inventories due to society’s interest in ecosystem services. However, the acquisition of accurate information about plant communities poses several methodological challenges. Here, we investigate the use of presence-absence sampling with the aim to monitor state and change in plant density. We study what plot sizes are informative, i.e. the estimators should have as high precision as possible.

    2. Plant occurrences were modelled through different Poisson processes and tests were developed for assessing the plausibility of the model assumptions. Optimum plot sizes were determined by minimizing the variance of the estimators. While state estimators of similar kind as ours have been proposed in previous studies, our tests and change estimation procedures are new.

    3. We found that the most informative plot size for state estimation is 1.6 divided by the plant density, i.e. if the true density is 1 plant per square metre the optimum plot size is 1.6 square metres. This is in accordance with previous findings. More importantly, the most informative plot size for change estimation was smaller and depended on the change patterns. We provide theoretical results as well as some empirical results based on data from the Swedish National Forest Inventory.

    4. Use of too small or too large plots resulted in poor precision of the density (and density change) estimators. As a consequence, a range of different plot sizes would be required for jointly monitoring both common and rareplants using presence-absence sampling in monitoring programmes.

  • 40.
    Yu, Jun
    et al.
    Umeå University, Faculty of Science and Technology, Mathematical statistics.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Asymptotic properties of high order spacings under dependence assumptions2000In: Mathematical Methods of Statistics, ISSN 1066-5307, E-ISSN 1934-8045, Vol. 9, no 4, p. 437-448Article in journal (Refereed)
    Abstract [en]

    Strong limit theorems are proved for sums of logarithms of spacings of increasing order when the observations satisfy a phi-mixing or an alpha-mixing condition. Applications of the results in goodness of fit and parametric estimation problems are discussed.

  • 41.
    Yu, Jun
    et al.
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå, Sweden.
    Ekström, Magnus
    Centre of Biostochastics, Swedish University of Agricultural Sciences, Umeå, Sweden.
    Multispectral image classification using wavelets: a simulation study2003In: Pattern Recognition, ISSN 0031-3203, E-ISSN 1873-5142, Vol. 36, no 4, p. 889-898Article in journal (Refereed)
    Abstract [en]

    This work presents methods for multispectral image classification using the discrete wavelet transform. Performance of some conventional classification methods is evaluated, through a Monte Carlo study, with or without using the wavelet transform. Spatial autocorrelation is present in the computer-generated data on different scenes, and the misclassification rates are compared. The results indicate that the wavelet-based method performs best among the methods under study.

  • 42.
    Yu, Jun
    et al.
    SLU, Department of Forest Resource Management and Geomatics.
    Ekström, Magnus
    Umeå University, Faculty of Social Sciences, Umeå School of Business and Economics (USBE), Statistics.
    Nilsson, Mats
    SLU, Department of Forest Resource Management and Geomatics.
    Image Classication using Wavelets with Application to Forestry2000In: Proceedings of the 5th International Conference on Methodological Issues in Official Statistics, 2000Conference paper (Refereed)
    Abstract [en]

    This work presents methods for image classication using the discrete wavelet transform. Performance of some conventional classication methods is evaluated, throughboth a Monte Carlo study and a real Landsat TM image together with the National Forest Inventory (NFI) eld data, with or without using the wavelet transform. In our evaluation on the real data, the bootstrap is applied to estimate classication errors. The results indicate that the wavelet based method performs best among the methods under study.

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