On the use of the Box-Cox transformation in censored and truncated regression models

2020 ◽  
Vol 15 (3) ◽  
pp. 197-214
Author(s):  
Fabrizio Carlevaro ◽  
Yves Croissant
Keyword(s):  
Density Function ◽  
Regression Models ◽  
Scale Parameter ◽  
Random Variable ◽  
Normal Random Variable ◽  
Truncated Regression ◽  
Tobit Models ◽  
Box Cox Transformation

In this paper we revisit some issues related to the use of the Box-Cox transformation in censored and truncated regression models, which have been overlooked by the econometric and statistical literature. We first analyze the shape of the density function of the random variable which, rescaled by a Box-Cox transformation, leads to a normal random variable. Then, we identify the value ranges of the Box-Cox scale parameter for which a regular expectation of the derived random variable does not exist. This result calls for an extension of the concept of expectation, which can be computed regardless of the value of the scale parameter. For this purpose, we extend the concept of mean of a rescaled series of observations to the case of a random variable. Finally, we run estimates of censored and truncated Box-Cox standard Tobit models to determine the range of the scale parameter most relevant for empirical demand analyzes. These estimates highlight significant deviations from the assumption of normality of the dependent variable towards highly right skewed and leptokurtic distributions with no expectation.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

On application of a certain formula

1971 ◽  
Vol 70 (1) ◽  
pp. 57-59
Author(s):  
J. K. Wani
Keyword(s):  
Density Function ◽  
Divided Difference ◽  
Random Variable ◽  
Multiple Integral

In this paper we first demonstrate how a certain formula, which expresses (n − 1 )th divided difference in the form of a multiple integral, may be used to obtain the density function of a suitable random variable and then apply this to obtain the density of a useful variate.

A renewal density theorem in the multi-dimensional case

1967 ◽  
Vol 4 (1) ◽  
pp. 62-76 ◽  
Author(s):  
Charles J. Mode
Keyword(s):  
Density Function ◽  
Covariance Matrix ◽  
Branching Processes ◽  
Random Variable ◽  
Density Theorem ◽  
Positive Definite ◽  
Dimensional Case ◽  
Age Dependent ◽  
Image Position ◽  
Mean Vector

SummaryIn this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(x) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn(x) be the n-fold convolution of f(x) with itself, and set Then for arbitrary choice of integers k1, …, kp–1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x1, …, xp) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.

Improved approximate confidence intervals for the mean of a log-normal random variable

2002 ◽  
Vol 21 (10) ◽  
pp. 1443-1459 ◽  
Author(s):  
Douglas J. Taylor ◽  
Lawrence L. Kupper ◽  
Keith E. Muller
Keyword(s):  
Confidence Intervals ◽  
Random Variable ◽  
Normal Random Variable ◽  
The Mean ◽  
Approximate Confidence Intervals ◽  
Log Normal

scholarly journals Optimal Detection of Bilinear Dependence in Short Panels of Regression Data

2020 ◽  
Vol 43 (2) ◽  
pp. 143-171
Author(s):  
Aziz Lmakri ◽  
Abdelhadi Akharif ◽  
Amal Mellouk
Keyword(s):  
Null Hypothesis ◽  
Regression Models ◽  
Scale Parameter ◽  
Regression Coefficient ◽  
Monte Carlo Study ◽  
Test Statistics ◽  
Bilinear Model ◽  
Large N ◽  
Optimal Tests ◽  
Error Terms

In this paper, we propose parametric and nonparametric locally andasymptotically optimal tests for regression models with superdiagonal bilinear time series errors in short panel data (large n, small T). We establish a local asymptotic normality property– with respect to intercept μ, regression coefficient β, the scale parameter σ of the error, and the parameter b of panel superdiagonal bilinear model (which is the parameter of interest)– for a given density f1 of the error terms. Rank-based versions of optimal parametric tests are provided. This result, which allows, by Hájek’s representation theorem, the construction of locally asymptotically optimal rank-based tests for the null hypothesis b = 0 (absence of panel superdiagonal bilinear model). These tests –at specified innovation densities f1– are optimal (most stringent), but remain valid under any actual underlying density. From contiguity, we obtain the limiting distribution of our test statistics under the null and local sequences of alternatives. The asymptotic relative efficiencies, with respect to the pseudo-Gaussian parametric tests, are derived. A Monte Carlo study confirms the good performance of the proposed tests.

scholarly journals Occupation times of alternating renewal processes with Lévy applications

2018 ◽  
Vol 55 (4) ◽  
pp. 1287-1308 ◽  
Author(s):  
Nicos Starreveld ◽  
Réne Bekker ◽  
Michel Mandjes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variable ◽  
Levy Processes ◽  
Normal Random Variable ◽  
Occupation Time ◽  
Renewal Processes ◽  
Tail Asymptotics ◽  
Occupation Times

AbstractIn this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

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