The standard logistic distribution has important uses in describing growl distri

Question

The standard logistic distribution has important uses in describing growl distribution. It has also attracted interesting applications in the modeling of uhen and as a substitute for the normal issues, hemolytic uremic syndrome data for children, physicochemical phenomen distribution the standard vival time of diagnosed leukemia patients, and weight gain data. A generalized eim variables asi sting tive respiratory disease prevalence on smoking and age, degrees of pneumoconiosis osycholoicalroposed e of chronic of pneumoconiosis in coal miners, geological psychological issues, sur- based on the fact that the difference of two independent Gumbel distributed logistic distribution is proposed, logistic distribution. The generalized logistic distribution (GLD) has received s of the probability sribution de- random variables has the standard additional attention in estimating its parameters for practical usage see for example Asgharzadeh (1S). The form tion (pdf) and cumulative distribution function (odf) of the two parameter noted by GLD(2,0 are given, respectively, by generalized logistic distribution de- F(x)-(1+exp(-6x)", -“ 0. Here and 0 are the shape and scale parameters, respectively, the above GLD was a generalization of the logistic distribution by Johnson et al. 1 161. For - 1, the GLD gistic and it is symmetric. The pdf in (1) has been obtained by compounding an extree a gamma distribution, different estimation procedures can be found in Chen and Balakrishma becomes the standard lo- extreme value distribution with The rest of the paper is organized as follows. In Section 2, we derive point confidence interval based on maximum likelihood estimation. The parametric discussed in Section 3. Section 4 describes Bayes estimates and construction o bootstrap confidence intervals are f credible intervals using the MCMC Section 5 contains the analysis of a numerical example to illustrate our proposed methods. A simula- tion studies are reported in order to give an assessment of the performance of the different estimation methods in Section 6. Finally we conclude with some comments in Section 7 2. Maximum Likelihood Estimation Suppose thatxu),x49, ,xu") be the lower record values of size n from the generalized logistic distri- bution GLD(3,0). The likelihood function for observed record was given by see Arnold et al. 17) where () and F() are given respectively, by (1) and (2), the likelihood function can be obtained by subs- tituting from (1) and (2) in (3) and written as e(a.elr).ro.(1+exp(«r.)Thiapp The natural logarithm of the likelihood function (4) is given by and and equating the results to zero, we obtain the likel to tions for the parameters and as -log 1+expl-a aL(a.ea)A rLi0) exp(-x

Explanation / Answer

1. Mean

mean of n observations is affected by both location and scale change of the observations.

mean is based on the all the observations of the set.

2.Median

median is the positional mean

it is affected by the location and scale change.

we dont need to know all the observations of the set to compute median just we need their order.

3.Mode

mode is affected by location and scale change.

we need to know only the modal class to compute mode not all the observations.

4. Variance

it is not affected by location change but by scale change.

it measures the avg deviation of the observations from mean value.

5.skewness

not affected by location and scale change.

used to know the shape of the distribution.

6.kurtosis

not affected by location and scale change.

use to know the size of the distribution.

7.Moments about 0

affected by both location and scale change.

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