A Characterization of the Normal Distribution

Lukács, Eugene

doi:10.1214/aoms/1177731647

Cited by 108 publications

(48 citation statements)

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“…where u i1 is the random intercept and u i2 is the random slope for time; with u i ∼ M V N (0, V ) and with V a two by two variance-covariance matrix of the u i , with diagonal terms var(u i1 ) = v 11 , and var(u i2 ) = v 22 , and an off-diagonal covariance cov(u i1 , u i2 ) = v 12 = v 21 . Hence, the two random terms have a correlation of v 12…”

Section: Mixed-effects Modellingmentioning

confidence: 99%

Modelling escalation in crime seriousness: a latent variable approach

Francis

Liu

2015

METRON

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This paper investigates the use of latent variable models in assessing escalation in crime seriousness. It has two aims. The first is to contrast a mixed-effects approach to modelling crime escalation with a latent variable approach. The paper therefore examines whether there are specific subgroups of offenders with distinct seriousness trajectory shapes. The second is methodological-to compare mixed-effects modelling used in previous work on escalation with group-based trajectory modelling and growth mixture modelling (mixture of mixed-effects models). The availability of software is an issue, and comparisons of fit across software packages is not straightforward. We suggest that mixture models are necessary in modelling crime seriousness, that growth mixture models rather than group-based trajectory models provide the best fit to the data, and that R gives the best software environment for comparing models. Substantively, we identify three latent groups, with the largest group showing crime seriousness increases with criminal justice experience (measured through number of conviction occasions) and decreases with increasing age. The other two groups show more dramatic non-linear effects with age, and non-significant effects of criminal justice experience. Policy considerations of these results are briefly discussed.

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Section: Mixed-effects Modellingmentioning

confidence: 99%

Modelling escalation in crime seriousness: a latent variable approach

Francis

Liu

2015

METRON

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“…The normal (Gaussian) distribution is a very common continuous probability distribution and it is important in statistics because it is often used in natural and social sciences to represent real-valued random variables whose distributions are not known [1,4,5]. The normal distribution is also useful due to its relation to the central limit theorem.…”

Section: Introductionmentioning

confidence: 99%

MATLAB-Based Stochastic Modeling: Distribution Analysis of Commercial Fishery Length-Frequency Samples Taken From the Black Sea (Bulgaria)

Zlateva¹,

Alexandrova²,

Nikolov³

et al. 2017

IJOER

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“…In a famous paper Lukacs [9] showed, using characteristic functions and dispensing with the excessive moment conditions providing only E{X\) < oo, that independence of X and S 2 is enough to characterize the normal distribution. Kawata and Sakamoto [6] in a paper submitted in 1944 but whose publication was delayed due to World War II, showed (understandably unaware of Lukacs [9]) also using characteristic function technology, that the condition E(X 2 ) < oo could be removed entirely.…”

Section: Introductionmentioning

confidence: 99%

“…Kawata and Sakamoto [6] in a paper submitted in 1944 but whose publication was delayed due to World War II, showed (understandably unaware of Lukacs [9]) also using characteristic function technology, that the condition E(X 2 ) < oo could be removed entirely.…”

Section: Introductionmentioning

confidence: 99%

Normal characterization by zero correlations

Seneta

Székely

2006

J. Aust. Math. Soc.

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Suppose Xj,i = 1,..., n are independent and identically distributed with E\X { \ r < oo, r = 1, 2 If Cov ((X -/*)', S 2 ) = 0 for r = 1, 2 , . . . , where /x = EX,, S 2 = £" = 1 (X, -X) 2 /(n -1), and X = Y11=i Xi/ n ' t n e n w e show that X, ~ ,yK(^,<7 2 ), where a 2 = Var(X,). This covariance zero condition characterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of X and S 2 using semiinvariants. 2000 Mathematics subject classification: primary 60E05, 62E10.

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A Characterization of the Normal Distribution

Cited by 108 publications

References 0 publications

Modelling escalation in crime seriousness: a latent variable approach

Modelling escalation in crime seriousness: a latent variable approach

MATLAB-Based Stochastic Modeling: Distribution Analysis of Commercial Fishery Length-Frequency Samples Taken From the Black Sea (Bulgaria)

Normal characterization by zero correlations

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