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Rayner, G.D.; MacGillivray, Helen

doi:10.1023/a:1013120305780

Cited by 91 publications

(17 citation statements)

References 17 publications

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“…Rayner and MacGillivray [13] examined the effect of non-normality on the distribution of (numerical) maximum likelihood estimators. The g-and-k distribution [12] can be defined in terms of its quantile function as:…”

Section: The G-and-k Distributionmentioning

confidence: 99%

Power of <i>t</i>-test for Simple Linear Regression Model with Non-normal Error Distribution: A Quantile Function Distribution Approach

Jahan

Khan

2012

J. Sci. Res.

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One of the major assumptions of the regression analysis is the normality assumption of the model error. We generally assume that the error term of the simple linear regression model is normally distributed. But in this paper g-and-k distribution is used as the underlying assumption for the distribution of error in simple linear regression model and a numerical study is conducted to see what extent of the deviation from normality causes what extent of effect on the size and power of t-test for simple linear regression model with the deviation being measured by a set a of skewness and kurtosis parameters. The strength of t-test is evaluated by observing the power function of t-test. The simulation result shows that, the performance of the t-test for simple linear regression model with g-and-k error distribution is seen to be vastly affected in presence of excess kurtosis and small samples (i.e. n<100).t-test is size robust under normal situation. Skewness and kurtosis parameter has a very little effect on the size of the t-test.

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Section: The G-and-k Distributionmentioning

confidence: 99%

Power of <i>t</i>-test for Simple Linear Regression Model with Non-normal Error Distribution: A Quantile Function Distribution Approach

Jahan

Khan

2012

J. Sci. Res.

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“…where F and G are cumulative distribution functions (CDFs). In addition, there are numerous authors who have studied the generalized properties of quantile-based functionals of asymmetry and kurtosis see (Balanda and MacGillivray 1990;Rayner and MacGillivray 2002).…”

Section: Examples Of Linear Parametric Quantile Time Series With Distmentioning

confidence: 99%

“…One of the most well known of these classes of transformation is the g-and-h transformations which involve a skewness transformation of type g and a kurtosis transformation of type h. If one replaces the kurtosis transformation of the type h with the type k, one obtains the g-and-k family of distributions discussed by Rayner and MacGillivray (2002). If the type h transformation is replaced by the type j transformation, one obtains the g-and-j transformations of Fischer and Klein (2004).…”

Section: Tukey Class Of Elongation Mapsmentioning

confidence: 99%

“…Furthermore, using these moment identities, one can easily then find the skew, kurtosis, and coefficient of variations for model families such as the g-and-h, the g-distributions and h-distributions. In addition to these simple population summaries of the g-and-h model, one could also consider other generalized properties of quantile-based functionals of asymmetry and kurtosis (see Balanda and MacGillivray 1990;Rayner and MacGillivray 2002;Balanda and MacGillivray 1988).…”

Section: Properties Of the G-and-h Quantile Error Familymentioning

confidence: 99%

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General Quantile Time Series Regressions for Applications in Population Demographics

Peters

2018

Risks

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The paper addresses three objectives: the first is a presentation and overview of some important developments in quantile times series approaches relevant to demographic applications—secondly, development of a general framework to represent quantile regression models in a unifying manner, which can further enhance practical extensions and assist in formation of connections between existing models for practitioners. In this regard, the core theme of the paper is to provide perspectives to a general audience of core components that go into construction of a quantile time series model. The third objective is to compare and discuss the application of the different quantile time series models on several sets of interesting demographic and mortality related time series data sets. This has relevance to life insurance analysis and the resulting exploration undertaken includes applications in mortality, fertility, births and morbidity data for several countries, with a more detailed analysis of regional data in England, Wales and Scotland.

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“…These methods can now be replaced by the, fast and usually efficient, maximum likelihood estimator (MLE). Rayner and MacGillivray [43] introduce a numerical MLE procedure based on quantile functions, but they conclude that “sample sizes significantly larger than 100 should be used to obtain reliable estimates.” Simulations in Section 5 show that the MLE using the closed form Lambert W × F X distribution converges quickly and is accurate even for sample sizes as small as N = 10.…”

Section: Parameter Estimationmentioning

confidence: 99%

The Lambert Way to Gaussianize Heavy‐Tailed Data with the Inverse of Tukey’s h Transformation as a Special Case

Goerg

2015

The Scientific World Journal

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I present a parametric, bijective transformation to generate heavy tail versions of arbitrary random variables. The tail behavior of this heavy tail Lambert W × F X random variable depends on a tail parameter δ ≥ 0: for δ = 0, Y ≡ X, for δ > 0 Y has heavier tails than X. For X being Gaussian it reduces to Tukey's h distribution. The Lambert W function provides an explicit inverse transformation, which can thus remove heavy tails from observed data. It also provides closed-form expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield analytic expression for Tukey's h pdf and cdf. Parameters can be estimated by maximum likelihood and applications to S&P 500 log-returns demonstrate the usefulness of the presented methodology. The R package implements most of the introduced methodology and is publicly available on

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Untitled

Cited by 91 publications

References 17 publications

Power of <i>t</i>-test for Simple Linear Regression Model with Non-normal Error Distribution: A Quantile Function Distribution Approach

Power of <i>t</i>-test for Simple Linear Regression Model with Non-normal Error Distribution: A Quantile Function Distribution Approach

General Quantile Time Series Regressions for Applications in Population Demographics

The Lambert Way to Gaussianize Heavy‐Tailed Data with the Inverse of Tukey’s h Transformation as a Special Case

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