Section 10.2 Estimating Parameters
Statistic is the science of data analysis. The primary goal in statistic is to draw meaningful and solid conclusions on a given phenomena on the basis of observed data. Typically, the data emerges as a sample of observations. An observation is the outcome of a measurement (or several measurements) that is taken for a subject that belongs to the sample. These observations may be used in order to investigate the phenomena of interest. The conclusions are drawn from the analysis of the observations.
A key aspect in statistical inference is the association of a probabilistic model to the observations. The basic assumption is that the observed data emerges from some distribution. Usually, the assumption is that the distribution is linked to a theoretical model, such as the Normal, Exponential, Poisson, or any other model that fits the specifications of the measurement taken.
A standard setting in statistical inference is the presence of a sequence of observations. It is presumed that all the observations emerged from a common distribution. The parameters one seeks to estimate are summaries or characteristics of that distribution.
For example, one may be interested in the distribution of price of cars. A reasonable assumption is that the distribution of the prices is the \(\mathrm{Exponential}(\lambda)\) distribution. Given an observed sample of prices one may be able to estimate the rate \(\lambda\) that specifies the distribution.
The target in statistical point estimation of a parameter is to produce the best possible guess of the value of a parameter on the basis of the available data. The statistic that tries to guess the value of the parameter is called an estimator. The estimator is a formula applied to the data that produces a number. This number is the estimate of the value of the parameter.
An important characteristic of a distribution, which is always of interest, is the expectation of the measurement, namely the central location of the distribution. A natural estimator of the expectation is the sample average. However, one may propose other estimators that make sense, such as the sample mid-range that was presented in the previous chapter. The main topic of this chapter is the identification of criteria that may help us choose which estimator to use for the estimation of which parameter.
In the next section we discuss issues associated with the estimation of the expectation of a measurement. The following section deals with the estimation of the variance and standard deviation — summaries that characterize the spread of the distribution. The last section deals with the theoretical models of distribution that were introduced in the first part of the book. It discusses ways by which one may estimate the parameters that characterize these distributions.
