S. Portnoy, Local asymptotics for quantile smoothing splines, Annals of Statistics, 25, 414--434 (1997).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 ) while g ;L1 is a quadratic spline (piecewise quadratic function, termed L1 ) They recommend a Schwarztype information criterion for choosing the smoothing parameter. Efficient computation of both via linear program is described in Ng [21] while consistency results can be found in 10 Portnoy [22]. We estimate both an L 1 and an L1 version of the employment density quantile splines using the S plus implementation provided in He and Ng [11] The quantile smoothing spline has two advantages over the cubic smoothing spline. As discussed above, the quantile spline can be used to estimate any ....

S. Portnoy, Local asymptotics for quantile smoothing splines, Annals of Statistics, 25, 414--434 (1997).

....of the original data during the selection of the smoothing parameter . The sub sample size is chosen to be n = 670 log(n) 3 rounded to the nearest integer. Assuming that the true function is twice differentiable, we know that the optimal smoothing parameter is in the order of n 1=5 , see Portnoy (1997). The SIC based choice of from the subset can be adjusted by multiplying a factor of (n=n ) 1=5 , and then the whole data set is used in the final model fitting stage. As noted in the previous section, the storage space needed in the regression B spline (with lambda= 0) is of the order much ....

Portnoy, S. (1997), `Local asymptotics for quantile smoothing splines', Annals of Statistics 25, 414--434.

....distribution. There are also cases where some lower and upper quantile curves are of direct interest. Details on the regression quantiles can be found in Koenker and Bassett (1978) The use of B spline approximations to (unconstrained) quantile functions was discussed in He and Shi (1994) Portnoy (1997), and He (1997) In some applications, the fitted curves satisfy certain boundary conditions. For example, it helps to impose bounds 0 g(0) g(1) 1 in estimating, say, a probability curve. When we estimate the power curve of a level ff test for testing the hypothesis of = 0 versus 0, a ....

....Theorem 3.1 are the same as the optimal rates established by Stone (1982) However, if we choose k n n 1= 2m 1) as in He and Shi (1994) the uniform convergence rate will be sub optimal by a power of log n. On the other hand, this latter choice leads to the optimal global rates of convergence. Portnoy (1997, Theorem 2.3) gave the rate of uniform convergence in the special case of linear splines. He further obtained the asymptotic distributions for the function estimates at certain points. Due to the arguments made earlier, we have the following result for the monotone B spline estimates. Theorem ....

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Portnoy, S. (1997), Local asymptotics for quantile smoothing splines, Ann. Statist., 25, 414-434.

....symmetry properties, see Koenker Bassett (1978) and Efron (1991) both in the parametric (cf Bassett Koenker (1978) and in the nonparametric cases (cf Chaudhuri (1991) it has reasonable asymptotic properties. This approach has also been applied locally and we refer to Welsh (1994) and Portnoy (1997), respectively, for the most recent kernel and spline estimators based on conditional empirical quantiles. Let us note that the Koenker Basset estimators and the regression quantiles estimators in Welsh (1994) and Portnoy (1997) when applied to the parametric translation models, result in ....

....has also been applied locally and we refer to Welsh (1994) and Portnoy (1997) respectively, for the most recent kernel and spline estimators based on conditional empirical quantiles. Let us note that the Koenker Basset estimators and the regression quantiles estimators in Welsh (1994) and Portnoy (1997), when applied to the parametric translation models, result in empirical quantiles. Hence they have larger variance than our M estimators of a quantiles. In section 5 we show that in the local model similar relations remain valid in the case of kernel estimators of conditional a quantiles. Some ....

Portnoy, S. (1997), `Local asymptotics for quantile smoothing splines.', Ann. Statist.

....Conn (1980) Consistency results of mean smoothing splines require the existence of g (x) E [ZjX = x] but no restriction on the distribution of (X; Z) is necessary for quantile smoothing splines. Recently, Shen (1997) obtained the global convergence rates of the quantile smoothing splines and Portnoy (1997) presented some local asymptotic properties. When the response Z depends on two variables X and Y , we first assume, for the sake of convenience, that z ij is observed at each (x i ; y j ) i = 1; Delta Delta Delta ; m; j = 1; Delta Delta Delta ; n) over the 3 partition, x 1 x 2 ....

Portnoy, S. (1997). Local Asymptotics for Quantile Smoothing Splines. Annals of Statistics, 25, 414-434.

....) u 11 Gamma g (x 1 ; y 1 ) 0, dn 0 as n 1 and v 11 Gamma g (x 1 ; y 1 ) 0, e m 0 as m 1. Therefore, w 11 must converge to the same limit. To obtain uniform consistency of the univariate quantile smoothing splines, consider a univariate function, h, and use Theorem 2. 1 of Portnoy (1997) with some simple modifications. Let j be the index for sub intervals in which the fits are linear. By condition (C3) the number of observations in the j th sub interval is at least of the order n 2=3 . Let the j th linear segment be represented by h j (x) ff j fi j (x Gamma x j ) ....

....which the fits are linear. By condition (C3) the number of observations in the j th sub interval is at least of the order n 2=3 . Let the j th linear segment be represented by h j (x) ff j fi j (x Gamma x j ) where x j is a point in the j th sub interval. Then (2.7) and (2. 8) of Portnoy (1997) imply that ff j Gamma h ( x j ) o p (1) and fi j Gamma h 0 ( x j ) O p i M Gamma3=4 j log n j O i n=M 3=2 j j uniformly in j with M j being the number of points in the j th sub interval. It is easy to see that sup x fi fi fi h (x) Gamma h (x) fi fi fi = sup j jff j ....

[Article contains additional citation context not shown here]

Portnoy, S. (1997). Local asymptotics for quantile smoothing splines, Ann. Statist., 25, 414-434.

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Portnoy, Stephen, (1994), Local Asymptotics for quantile smoothing splines, preprint.

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