Spring 2014 Statistics 210 b ( Theoretical Statistics ) - Lecture One Aditya

Spring 2014 Statistics 210 b ( Theoretical Statistics ) - Lecture Two Aditya

The main idea of upper bounding (3) is to use Rademachers. A Rademacher variable σ simply takes the two values +1 and −1 each with probability 1/2. Let σ1, . . . , σn be n independent Rademachers that are also independent of X1, . . . , Xn and X ′ 1, . . . , X ′ n. For each i, note that the distribution of f(Xi)−f(X ′ i) is the same as the distribution of f(X ′ i)− f(Xi). Therefore, the distrib...

Spring 2013 Statistics 153 ( Time Series ) : Lecture Twenty One Aditya Guntuboyina

Spring 2013 Statistics 153 ( Time Series ) : Lecture Thirteen Aditya

Spring 2013 Statistics 153 ( Time Series ) : Lecture Twenty Three Aditya Guntuboyina

When we were fitting ARMA models to the data, we first looked at the sample autocovariance or autocorrelation function and we then tried to find the ARMA model whose theoretical acf matched with the sample acf. Now the sample autocovariance function is a nonparametric estimate of the theoretical autocovariance function of the process. In other words, we first estimated γ(h) nonparametrically by...

Spring 2013 Statistics 153 ( Time Series ) : Lecture Twenty Two Aditya Guntuboyina

Let {Xt} be a stationary sequence of random variables and let γX(h) = cov(Xt, Xt+h) denote the autocovariance function. A theorem due to Herglotz (sometimes attributed to Bochner) states that every autocovariance function γX can be written as: γX(h) = ∫ 1/2 −1/2 edF (λ), where F (·) is a non-negative, right-continuous, non-decreasing function on [−1/2, 1/2] with F (−1/2) = 0 and F (1/2) = γX(0)...