Estimation and Inference of Discontinuity in Density

More documents

Recommendations

Info

in (3) are written as0 = 1 n0 = 1 nnX(1; X i;h ) (1 I i ) K (X i;h )i=1nX(1; X i;h ) I i K (X i;h )i=1ZxcZx
the binned data, the computational cost of `G () is relatively cheaper than that of ` () because thedimensions of the auxiliary parameters and (a l ; b l ; b r ) in ` () are higher than those of G and a l in`G ().One useful feature of our empirical likelihood approach is that it can easily incorporate additionalinformation. Suppose we have prior information about X speci…ed in the form of E [m (X)] = 0, suchas the mean, variance, or quantiles. Using the weights fp i g n i=1, this information can be incorporatedinto the likelihood maximization problem (10) by adding the restriction P ni=1 p im (X i ) = 0. In thiscase, the dual form (11) is re-de…ned by adding m (X i ) to the moment function g i (a l ; a r ; b l ; b r ). Theresulting empirical likelihood inference is more e¢ cient if the additional information is valid.Although this paper focuses exclusively on discontinuities of density functions, our empirical likelihoodapproach can also deal with discontinuities in derivatives of density functions. For example,suppose we are interested in conducting inference on the …rst-order derivatives of the density usingthe local likelihood-based empirical likelihood. Then the contrast 0 = fr0 fl 0 can be estimated by^ = ^b r exp (^a r ) ^bl exp (^a l ) using the solutions of (3), and the empirical likelihood function for 0 canbe de…ned as` () =min` (a l; a r ; b l ; b r ) :f(a l ;a r;b l ;b r)2A l A rB l B r:=b r exp(a r) b l exp(a l )gIn this case, one may add quadratic terms by X 2 i;h to the moment function g i (a l ; a r ; b l ; b r ) to reducethe boundary bias for the derivative estimates.2.3 Large Sample PropertiesWe now investigate the asymptotic properties of the proposed empirical likelihood statistics. We imposethe following assumptions.Assumption.1. fX i g n i=1is i.i.d.2. There exists a neighborhood N around c such that f is continuously second-order di¤erentiableon N n fcg. For the matrices V and G de…ned in (13), V is positive de…nite and G has fullcolumn rank.3. K is a symmetric and bounded density function with support [ k; k] for some k > 0.4. As n ! 1, h ! 0, nh ! 1, and nh 5 ! 0. Additionally, b=h ! 0 for `G () and nh 3 ! 1 for` ().5. A l and A r are compact, f l 2 int (A l ), and f r 2 int (A r ). Additionally, for ` (), B l and B r arecompact, fl 0=f l 2 int (B l ), and fr=f 0 r 2 int (B r ).9
Page 3 and 4: the empirical likelihood con…denc
Page 7: whereKlj G = K Xj;hG ( JX1 IkGk=1K
Page 11: type of derivation. Second, the emp
Page 15 and 16: AMathematical AppendixHereafter “
Page 17 and 18: w.p.a.1. On the other hand, an expa
Page 19 and 20: Let g i;1 ( 0 ) be the …rst eleme
Page 21 and 22: References[1] Angrist, J. D. and V.
Page 23 and 24: Table 1: Finite-sample biases, stan
Page 25 and 26: Table 3: Finite-sample sizes, quant
Page 27 and 28: Frequencies0.0160.0140.0120.010.008
Page 29: Densities0.015Binning EstimatorLoca

Estimation and Inference of Discontinuity in Density

Create successful ePaper yourself

Delete template?

Save as template?