# Applications with Scale Parameters

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This note contains several papers related to scale parameter.

## log-logistic distribution

This section is for Lesitha, G., & Yageen Thomas, P. (2013). Estimation of the scale parameter of a log-logistic distribution. Metrika, 76(3), 427–448.

apply the ranked set sampling technique to estimate the scale parameter $\alpha$ of a log-logistic distribution under a situation where the units in a sample can be ordered by judgement method without any error.

A random variable $X$ is said to have a log-logistic distribution with the scale parameter $\alpha$ and the shape parameter $\beta$ if its distribution function is given by

\[F(x;\alpha,\beta) = \frac{x^\beta}{\alpha^\beta + x^\beta}, x>0, \alpha>0, \beta>1\]The scale parameter is also equal to its median.

The applications of log-logistic distribution are well known in survival analysis of data sets such as survival times of cancer patients.

## Multinomial Logit models

This section is for Swait, J., & Louviere, J. (1993). The Role of the Scale Parameter in the Estimation and Comparison of Multinomial Logit Models. Journal of Marketing Research, 30(3), 305–314.

Multinomial logit (MNL) models are widely used in marketing research to analyze choice data, but it is not generally recognized that the unit of the utility scale in a MNL model is inversely related to the error variance. This means that, for instance, parameters of two identical utility specifications estimated from different data sources with unequal variances will necessarily differ in magnitude, even if the true model parameters that generated the utilities are identical in both sets.

Define the utility of an alternative to be $U_{in} = V_{in} + e_{in}$, where $V_{in}$ is the systematic (or explainable) proportion of the utility function and $e_{in}$ is an error term associated with joint random variation across both individuals and alternatives.

The cdf of an individual error term is

\[F(e_{in}) = \exp[-\exp(-\mu e_{in})], \mu > 0 \text{ a scalar}\]Suppose we have two samples and wish to test whether they share the same population parameters, then wish to test the hypothesis,

\[H_1: \beta_1 = \beta_2 \text{ and } \mu_1 = \mu_2\,.\]## Multi-resolution image segmentation of remotely sensed data

present a technique for estimating the scale parameter in image segmentation of remotely sensed data. The degree of heterogeneity within an image-object is controlled by a subjective measure called the “scale parameter”.

The estimation of scale parameter that builds on the idea of local variance (LV) of object heterogeneity within a scene.

Segmentation is the process of dividing remotely sensed images into discrete regions or objects that are homogeneous with regard to spatial or spectral characteristics.

These scale parameters correctly delineated three levels of image-objects representative of this subset:

- individual buildings: scale parameter 14
- blocks of buildings: scale parameter 45
- broadest land cover classes as depicted by their heights (scale parameter 82)