Question

Some Elementary Statistical Inferences nple itesa estimate of -hz+h). This suggests the following estimate of f(a) at'a given ar The ponparametrie estimate of the leftside is the proportion of the that fall in the i 1.11) To write this more formally, consider the indicator sta a)-o otherwise, Then a somparanmetric estimator of a) is Since the sample items are identically distributed, as h 0. Hence (r) is approximately an unbiased estimator of the density n density estimation terminology, the indicator function I, is called a rectangular kernel with bandwidth 2h. See Chapter 6 of Lehmann (1999) for a discussion of density estimation. Let be the realized values of the random sample. Our histogram estimate of f(x) is obtained as follows. For the discrete case, there are natural classes for the histogram, namely, the domain values. For the continuous c though, classes must be selected. One way of doing this is to select a positive integer m, an h>0, and a value a such that a < min ai, so that the m intervals (a-h,a+h, ath,a+3h, (a+3h, a+5h... .a+(2m-3h,a+(2m-1)h] (1.13) cover the range of the sample min , maxz]. These intervals form our classes. For the histogram, over the interval (a+(2i-3)h, a +(2i-1)h,i 1,..., m, let the height of the bar be the density estimate given in expression (1.12) at the midpoint of the interval, i.e., fla +2(i 1)h). The height of the bar is thus proportional to the number of rs that fall in the interval (a (2i -3)h, a (2i -1)h). Over the interval (a + (21-3)h, a + (2i-1) , our histigram estimate of the density is ha +2(i 1)h. To complete the histogram estimate of f(r), take it to be t0 for z a and for > a + (2m-1)h. Denote the intervals of the partition by I; = (a + (2i-3)h, a + (2i-1)h], i 1, , m. Then we can summarize our histogram estimate of the pdf by elsewehere ce the estimator is nonnegative and, as Exercise 1.9 shows, it integrates to 1 (-00,00). So it satisfies the properties of a pdf. over 214

As follow above the (1.12), what is the variance of the equation? I hope you solve it in details.

Answer 1

Ym all Xi'n ane independat and it wan given xi'D one identicaldy distauibutad ん Ans 2.