The classical

Let

In higher dimensions, the counterpart to procedure (

Procedure (

The rest of the paper is organized as follows. Section

It is well known that the sample mean in the

Now we consider a special univariate “projection depth-trimmed mean” (

Let

Let

We calculate

Finally, we simply delete first

Now we conduct simulation study to examine the performance of the new and classical

We will confine attention to the average length (or area) of the confidence interval (or region) from both procedures as well as their coverage frequency of true parameter

Table

Average coverage (length) of 95% CI's by

method | |||||
---|---|---|---|---|---|

.9390 | (.3819) | .9515 | (.5855) | ||

.9550 | (.3967) | .9605 | (.6607) | ||

.9470 | (.3842) | .9505 | (.5902) | ||

.9520 | (.3956) | .9520 | (.6618) | ||

.9410 | (.3864) | .9470 | (.5916) | ||

.9490 | (.3959) | .9540 | (.6582) | ||

.9435 | (.3876) | .9550 | (.5943) | ||

.9480 | (.3960) | .9545 | (.6589) |

Inspecting the table immediately reveals that the bootstrap number

Figure

In our simulation studies, we also compare our new procedure with the existing

Our experiments with

In addition to the distributions we considered in Table

We first display the typical single run results of

Here in Figure

On the right-hand side are

Of course, the single run results may not represent the overall performance of the two procedures. So we conduct a simulation over

Average coverage (length) of 95% CIs by

method | |||||
---|---|---|---|---|---|

.9455 | (.3922) | .9585 | (1.135) | ||

.9595 | (.4070) | .9760 | (57.76) | ||

.9525 | (.3953) | .9765 | (1.115) | ||

.9590 | (.4081) | .9825 | (20.10) | ||

.9530 | (.3972) | .9715 | (1.164) | ||

.9600 | (.4073) | .9775 | (31.87) | ||

.9485 | (.3994) | .9725 | (1.168) | ||

.9525 | (.4077) | .9830 | (25.61) |

Inspecting the table immediately reveals that the classical

In higher dimensions, with the multivariate version of

We first display single run results of two procedures at bivariate standard normal distribution

Of course, single run result may not represent the overall performance of the two procedures. To see if the single run results are repeatable now we list the average of coverage and the area of the confidence regions based on two procedures in

Average coverage (length) of 95% confidence regions by

method | |||||
---|---|---|---|---|---|

.9501 | (.1492) | .9515 | (.3045) | ||

.9524 | (.1935) | .9495 | (.5137) | ||

.9395 | (.1582) | .9577 | (.3247) | ||

.9515 | (.1947) | .9507 | (.5204) | ||

.9436 | (.1672) | .9475 | (.3438) | ||

.9547 | (.1949) | .9353 | (.5111) | ||

.9403 | (.1736) | .9586 | (.3554) | ||

.9470 | (.1949) | .9488 | (.5202) |

Inspecting the Table reveals that the two procedures are indeed (roughly)

From the last section we see that the new procedure has some advantages over the classical (seemingly optimal) procedures. But we know that we cannot get all the advantages of the new procedure for free. What kind of price we have to pay here? For all the advantage of the new procedures possess over the classical ones, the price it has to pay is the intensive computing in the implement of the procedure. In our simulation study, there are 4 million basic operations (the case

A natural question is Why the new procedure has advantage over the classical one? The procedure clearly depends on bootstrap and data depth. Is it due to bootstrap or data depth? Who is the main contributor? If one just uses bootstrap, can one have some advantages? The answer for the latter is positive, Indeed, in our simulation we compare the classical one with the bootstrap percentile procedure, it reveals that the bootstrap percentile one does have some mild advantage over the classical one but still is inferior to our new procedure. So both bootstrap and data depth make contributions to the advantages of the new procedure. But remember, it is data depth that allow the bootstrap percentile procedure (which originally was defined only in one dimension) implementable in high-dimensions: to order sample bootstrap mean vectors. Without the data depth, it is impossible to implement the procedure in high-dimensions. So overall, it is data depth that makes the major contribution towards the advantages of the new procedure.

We also like to point out at this point that there is different new procedure introduced and studied in Zuo [

Our empirical evidence for the new procedure in one and higher dimensions is very promising, but we still need some theoretical developments and justifications, which is beyond scope of this paper and will be pursued elsewhere. A heuristic argument is because the bootstrap percentile confidence interval has advantage over the classic confidence interval procedure in term of at the same nominal level it can produce an asymptotically shorter interval (see Hall [

One question left about our new procedure in practices is how does one choose the

There are a number of depth functions and related depth estimators (see Tukey [

Finally, we comment that findings in this paper are consistent with the results obtained in Bai and Saranadasa (BS) [

This research was partially supported by NSF Grants DMS-0234078 and DMS-0501174.