# Estimation and Modelling

• N. Unnikrishnan Nair
• P. G. Sankaran
• N. Balakrishnan
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

## Abstract

Earlier in Chaps. 3 and 7, several types of models for lifetime data were discussed through their quantile functions. These will be candidate distributions in specific situations. The selection of one of them or a new one is dictated by how well it can justify the data generating mechanisms and satisfy well other criteria like goodness of fit. Once the question of an initial choice of the model is resolved, the problem is then to test its adequacy against the observed data. This is accomplished by first estimating the parameters of the model and then carrying out a goodness-of-fit test. This chapter addresses the problem of estimation as well as some other modelling aspects.

In choosing the estimates, our basic objective is to get estimated values that are as close as possible to the true values of the model parameters. One method is to seek estimate that match the basic characteristics of the model with those in the sample. This includes the method of percentiles and the method of moments that involve the conventional moments, L-moments and probability weighted moments. These methods of estimation are explained along with a discussion of the properties of these estimates. In the quantile form of analysis, the method of maximum likelihood can also be employed. The approach of this method, when there is no tractable distribution function, is described. Many functions required in reliability analysis are estimated by nonparametric methods. These include the quantile function itself and other functions such as quantile density function, hazard quantile function and percentile residual quantile function. We review some important results in these cases that furnish the asymptotic distribution of the estimates and the proximity of the proposed estimates to the true values.

## Keywords

Quantile Function Life Distribution Sample Quantile Probability Weighted Moment Percentile Point

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