I have solved the part (a) and part(b). But really stucked on part(c) even after
ID: 3176130 • Letter: I
Question
I have solved the part (a) and part(b). But really stucked on part(c) even after hours of attempt. Any help and/or correction of my solution is highly appreciated.
My solution for part (a) and part(b):
3. Let (zi, yi) be paired observations from model g(zi) ei (1) Vi where the are any zero mean error terms. The Nadaraya Watson kernel regression estimate (z) g(z) is given by Kh (z zi (2) wherever the expression is defined. Usually, Kh (z) oc p(z/h) where is a zero mean density, and h 0 is the bandwidth (a) Suppose p(z) IHIzl s 1/2), that is, the uniform density on interval 1/2, 1/2 Set Kh(z) p(z/h) in estimator (2). Let Nh (r) be the set of indices from n defined by zil s h/2), and denote cardinality nh (z) Nh(z)l. Express the gh (z) explicitly in terms of (zi,yi), i 1,...,n, making use of Na(z) and nh (z) (b) Suppose the error terms ei in (1) are an iid sample from N(0, 2). Give an explicit expression for the variance Va(z) and bias Bh(r) of gh (z) (c) We next consider a specific model. In (1) let on some interval z E I-M, M], M 0, for two constants Bo, B1 0. A Poisson process of rate A on any interval ICR is a random set of points XCI which possesses the following properties (in addition to others): (i) The number of points N' from in interval a, a h c I has a Poisson distribution with mean Ah (ii) Given that there are N' points from in interval a,a h], these points form an iid sample from a uniform distribution on a, a h (See Section 7.1 from CSC/DSC 262/462 lecture notes). Then assume the predictor values N) are generated by a Poisson process on M, M with rate A (this means that Nis a Poisson random variable with mean 2MA). Then, once a is generated, responses va are given by where the error terms Ei are an iid sample from N(0, o2) In general, the conditional expected value of X given event A, denoted ELXIA, is the expected value of X under the distribution POX EE A). Accordingly, the quantity E Bh(z)2 nh (z) nj is the expected value of Bh (z)2 calculated after assuming that nh (z) n, so that the property (ii) of a Poisson process given above may be applied. Then calculate EMSEh (z) EDVh (z) n] ELBh (z) (z) nh You can assume that M is large enough so that h/2,r+ h/2 c M, M1Explanation / Answer
Solutions for both the parts a & b are absolutely correct. the application has to be made of the parts a & b to solve part c.
It is related to Poisson process.