A Bayesian model for local smoothing in kernel density by Brewer M. J.

By Brewer M. J.

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We are surrounded by data that shows an inherent variability. The issue addressed in this chapter is how we may describe this variability. Traditional statistics focuses on the proportions of observations lying in particular sections of the range of possible values. Corresponding plots and measurements focus on this theme, for example, the classical histogram. 1 say, it tells us very little on its own. For it to be meaningful, we need to know how it relates to the rest of the data. If we are told that all the data lie between 230 and 240 and that the observation is 12th out of 40 in increasing order, then we begin to get more of a picture of the situation.

10. 5 MODELLING WITH QUANTILE FUNCTIONS 21 is totally flat, so consider modelling the distribution by adding a multiple, K, of the uniform to the logistic: S(p) = ln[p/(1 – p)] + Kp. 11 shows the shape of the PDF for this model and it is seen that it suitably combines the properties of the uniform and the logistic distributions, havin g a flattened logistic shape. 1) indicating the shift in position of the distribution due to the added uniform distribution. 15 the exponential and the reflected exponential distributions were combined.

75. 1 illustrates. , the frequency, of observations lying in specified ranges called class intervals. Notice that the interval given by, say, 11+ to 13, includes the exact value 13 but starts just beyond the value of 11, which would be classified in the previous interval. The sum of the frequencies gives the total number of observations. To evaluate the medians and quartiles in these circumstances requires some manipulation of the table. As the use of frequency tables usually implies large data sets, we will not make the ˜ o (p) and Q ˜ (p), which becomes type of distinction we made between Q negligible for large n.

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