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Data are referred to as length-biased when the probability of inclusion of the observation in the sample is proportional to its magnitude. It has been documented in a number of research articles that the presence of length-biasedness in data arising in fields such as ecology, economics, geology, bioinformatics and medical science. Some examples of such data sets can be found in Ratnaparkhi and Naik-Nimbalkar (2012). The applications of concepts of differential geometry, such as the curvature tensor of the statistical manifolds, have been investigated with reference to statistical inference (Amari, 1985). Recently, Ratnaparkhi and Naik-Nimalkar (2012) observed that certain structural properties of parameter estimators of the length-biased lognormal distribution are significantly different from the corresponding properties of the original lognormal distribution. This article

A Note on α-Curvature of the Manifolds of the Length-biased Lognormal and Gamma Distributions in View of Related Applications in Data Analysis