J. A. Nelder, Daryl Pregibon · Biometrika · 1987
Journal Article An extended quasi-likelihood function Get access J. A. NELDER, J. A. NELDER Department of Mathematics, Imperial CollegeLondon SW7 2BZ, U.K. Search for other works by this author on: Oxford Academic Google Scholar D. PREGIBON D. PREGIBON Statistics and Data Analysis Research Department, AT&T Bell LaboratoriesMurray Hill, New Jersey 07974, U.S.A. Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 74, Issue 2, June 1987, Pages 221–232, https://doi.org/10.1093/biomet/74.2.221 Published: 01 June 1987 Article history Received: 01 November 1984 Revision received: 01 August 1986 Published: 01 June 1987
Cited by 391
W. M. Patefield · 2016
Suppose xx,x2, ...,xn is a sample from a probability distribution depending on a set of r parameters, 6X, d2, ..., 0r, in which we are interested and a set of q v 2, ..., (?)q, in which we are not interested. The term nuisance parameters has been used by Kalbfleisch and Sprott (1970), Edwards (1972) and others. If the joint density of x is/(x; G, 0), where the functional form of/ is known, then the log-likelihood (or support) function of 8 and <p given the sample x is i(e,0) = log/(x;M).
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Peng Yu, Chad A. Shaw · 2014
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Katie Grantham Lough, Robert B. Stone, Irem Y. Tumer · 2005
Risk assessments are necessary to anticipate and prevent accidents from occurring or repeating. This paper focuses specifically on the relationship between function and risk by presenting a mathematical mapping from product function to risk likelihood. This type of mapping will aid designers by removing the subjectivity of the likelihood value from a risk element, provide three key risk element properties (design parameter, failure mode, and likelihood) for numerous risk elements with one simple mathematical calculation, and provide a means for inexperienced designers to effectively address risk in the conceptual design phase. In addition, the level of detail a functional model should be written in to produce adequate likelihood risk assessments is examined. Three case studies are used to validate the proposed mapping: A spacecraft orientation subsystem, a subsystem to guide science instruments on an extraterrestrial, and subsystems from a Bell 206 rotorcraft. The work described in this paper continues is part of ongoing research relating function to failure.
Cited by 27
S. R. Majumdar · Research in Statistics · 2024
The fact that the graph of the exponential function exp is always at or above the straight line through the origin with slope exp(1) is well-known and can be easily proved using differential calculus. We provide a simple algebraic proof of that fact and use that fact to construct a template for maximizing the likelihood function that avoids the log-likelihood function and differentiation in a number of examples, including the two-parameter normal distribution family.
Cited by 1Open access
W. M. Patefield · Biometrika · 1985
A method is described for determining, from the maximized likelihood function, the asymptotic variance-covariance matrix of a subset of parameters. The technique is applied to the slope of a bivariate linear structural relationship.
Cited by 19
Djallel Bouneffouf · HAL (Le Centre pour la Communication Scientifique Directe) · 2019
Dirichlet-multinomial (DMN) distribution is commonly used to model over-dispersion in count data. Precise and fast numerical computation of the DMN log-likelihood function is important for performing statistical inference using this distribution, and remains a challenge. To address this, we use mathematical properties of the gamma function to derive a closed form expression for the DMN log-likelihood function. Compared to existing methods, calculation of the closed form has a lower computational complexity, hence is much faster without comprimising computational accuracy.
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F. C. Porter · Nuclear Instruments and Methods in Physics Research Section A Accelerators Spectrometers Detectors and Associated Equipment · 1996
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Thorn Thaler <thorn.thaler@thothal.com> · 2009
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Marc Mangel (39580) · Figshare · 2023
<p>Shown is the log-likelihood function (the logarithm of the right side of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0281710#pone.0281710.e031" target="_blank">Eq 6</a>) as the positivity rate declines when <i>p</i><sub><i>FN</i></sub> = 0.25, <i>p</i><sub><i>FP</i></sub> = 0.05 and <i>T</i> = 100 tests are administered for positivity (A) 0.075, (B) 0.06, (C) 0.0525, and (D) 0.04.</p>
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John E. Piper · IEEE Signal Processing Letters · 1994
The maximum likelihood method is designed to yield high resolution estimates in a multiple source environment. The article derive a simplified representation of the maximum likelihood function for the two source case. This case is instructive to understand when and why the full power of the maximum likelihood method should be used. This approach extends the results of a previous paper (see J. Piper, IEEE Trans. Signal Processing, vol.42, no.2, p.412, 1994) that showed how the maximum likelihood function calculation can be transformed from a matrix problem to a vector product. The maximum likelihood function for spectral estimation is given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
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Michael Evans · Statistics & Probability Letters · 1989
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Katalin Hajdú-Szücs (5007434), Nóra Fenyvesi (5007431), József Stéger (5007428) et al. · Figshare · 2018
<p>The ball hit the front wall. can be evaluated in space given the positions of the sensors (marked by white disks) to find its minimum, which indicates where the event took place. (0.5 m from the right corner and 3 m above the floor, marked by a blue disk).</p>
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Flávio B. Gonçalves, Pedro Franklin · arXiv (Cornell University) · 2019
We discuss a general definition of likelihood function in terms of Radon-Nikod\'{y}m derivatives. The definition is validated by the Likelihood Principle once we establish a result regarding the proportionality of likelihood functions under different dominating measures. This general framework is particularly useful when there exists no or more than one obvious choice for a dominating measure as in some infinite-dimensional models. We discuss the importance of considering continuous versions of densities and how these are related to the Likelihood Principle and the basic concept of likelihood. We also discuss the use of the predictive measure as a dominating measure in the Bayesian approach. Finally, some examples illustrate the general definition of likelihood function and the importance of choosing particular dominating measures in some cases.
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April Kerby, David B. Marx, Ashok Samal et al. · 2007
Clustering has been widely used as a tool to group multivariate observations that have similar characteristics. However, there have been few attempts at formulating a method to group similar multivariate observations while taking into account their spatial location (R.T. Ng and J. Han, 1994), (J. Cuzick and R. Edwards, 1990), (G.C. Simbahan and A. Dobermann, 2006). This paper proposes a method to spatially cluster similar observations based on their likelihoods. The geographic or spatial location of the observations can be incorporated into the likelihood of the multivariate normal distribution through the variance-covariance matrix. The variance-covariance matrix can be computed using any specific spatial covariance structure. Therefore, observations within a cluster which are spatially close to one another will have a larger likelihood than those observations which are not close to one another. This results in spatially close observations being placed into the same cluster.
Cited by 3
Emine Ecem Özer · Marmara University Open Access System · 2023
Sağkalım analizi; tedavi sürecinde bir hastalığa neden olan sebepleri ya da bir konuda risk faktörlerini belirlemede büyük önem taşıyan ve sağlık alanında, hastaların yaşam sürelerine etki eden faktörleri belirlemede yaygın olarak kullanılan bir yöntemdir. Sağkalım analizinde, en çok karşılaşılan problem sağkalım süresinin net olarak belirlenememesidir. Böyle bir durumda sansürlü veriler oluşur. Üç çeşit sansürlü veri vardır. Bunlar soldan, sağdan ve aralıklı sansürlü veridir. Tahminler oluşturularak ve test istatistikleri kullanılarak değişkenlerin sağkalım analizi yapılır. Yapılan bir çalışmada örneklemin alındığı verinin dağılımı ile ilgili kesin ve yeterli bilgi olmadığı durumlarda klasik olabilirlik yöntemi kullanmak doğru olmamaktadır. Dağılımla ilgili yeterli bilgi olmadığı durumlarda deneysel olabilirlik yöntemi kullanılır. Bu çalışmada bu yöntem kullanılacaktır. İkinci bölümde sağkalım analizi, veri türleri, yöntemler, kullanılan fonksiyonlar ve parametrelerin tahmin yöntemleri, deneysel olabilirlik yöntemi, tehlikenin terimlerinde olabilirlik ve birinci tıp kısıtlama altında maksimumunun bulunması, asimptotik özellikleri, olabilirliğin iki versiyonunun kıyası hakkında bilgi verilmiştir. Üçüncü bölümde iki gerçek veri üzerinde deneysel olabilirlik yöntemi incelenmiştir. Dördüncü bölümde simülasyon çalışması yapılmıştır. Beşinci bölümde sonuçlar paylaşılmıştır. Bu tez çalışmasının amacı, sağlık çalışmalarında yaygın olarak kullanılan sağkalım fonksiyon parametrelerinin, parametrik olmayan deneysel olabilirlik yöntemi ile tahmin edilmesidir. Gerçek veri ve simülasyon çalışmasından üretilen verilerin kapsama olasılıkları kriteri dikkate alınarak karşılaştırma yapılmıştır.
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Jan F. Bjørnstad · Journal of the American Statistical Association · 1996
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Jan F. Bjørnstad · Journal of the American Statistical Association · 1996
Abstract The parametric likelihood and likelihood principle (LP) play a central role in parametric methodology and in the foundations of statistics. The main purpose of this article is to extend the concepts of likelihood and LP to general inferential aims and models covering, for example, prediction and empirical Bayes models. The likelihood function is the joint distribution of the data and the unobserved variables of inferential interest, considered as a function of the parameters and these inferential variables. LP is based on this likelihood, and the principles of sufficiency (SP) and conditionality (CP) are modified such that the equivalence SP and CP ⇔ LP is valid, generalizing Birnbaum's theorem.
Cited by 86
Tapan K. Nayak, Subrata Kundu · 2003
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Benjamin Crysup, August E. Woerner · Forensic Science International Genetics · 2022
Cited by 4