Lecture Notes 3
1
Uniform Bounds
n
i=1
Recall that, if X1 , . . . , Xn Bernoulli(p) and pn = n1
inequality,
2
P(|pn p| > ) 2e2n .
Xi then, from Hoedings
Sometimes we want to say more than this.
Example 1 Suppose that X1 , . . . , Xn have cdf F . Let
1
Fn
Lecture Notes 4
1
Random Samples
Let X1 , . . . , Xn F . A statistic is any function T = g (X1 , . . . , Xn ). Recall that the sample
mean is
n
1
Xn =
Xi
n i=1
and sample variance is
2
Sn
1
=
n1
n
(Xi X n )2 .
i=1
2
Let = E(Xi ) and = Var(Xi ). Recall tha
Lecture Notes 5
1
Statistical Models
A statistical model P is a collection of probability distributions (or a collection of densities). An example of a nonparametric model is
P=
p:
(p (x)2 dx < .
A parametric model has the form
P=
p ( x; ) :
2 /2
where R
Lecture Notes 6
1
The Likelihood Function
Denition. Let X n = (X1 , , Xn ) have joint density p(xn ; ) = p(x1 , . . . , xn ; ) where
. The likelihood function L : [0, ) is dened by
L() L(; xn ) = p(xn ; )
where xn is xed and varies in .
1. The likelihood
G80.3042.002 Fall 2007
Statistical Analysis and Modeling of Neural Data
Homework 4
Due: Wednesday, 19 Dec 2007
Your results should be in the form of a MATLAB le (typically, the lename should have an
extension of .m). Email your solutions to [email protected]
G80.3042.002 Fall 2007
Statistical Analysis and Modeling of Neural Data
Homework 3
Due: Friday, 7 Dec 2007
Your results should be in the form of a MATLAB le (typically, the lename should have an
extension of .m). Email your solutions to [email protected] a
G80.3042.002 Fall 2007
Statistical Analysis and Modeling of Neural Data
Homework 2
Due: 31 Oct 2007
Your results should be in the form of a MATLAB le (typically, the lename should have an
extension of .m). Email your solutions to [email protected] and bija
G80.3042.002 Fall 2007
Statistical Analysis and Modeling of Neural Data
Homework 1
Due: 10 Oct 2007
Your results should be in the form of a MATLAB le (typically, the lename should have an
extension of .m). Email your solutions to [email protected] and bija
function [e,v]=dpsschk(tapers)
%DPSSCHK Helper function for multitaper routines.
%
E = DPSSCHK(TAPERS) returns the DPSS array based on the input
%
TAPERS. If TAPERS is a cell containing the DPSS array then E
%
just returns the array.
%
If TAPERS is in the
Lecture Notes 2
1
Probability Inequalities
Inequalities are useful for bounding quantities that might otherwise be hard to compute.
They will also be used in the theory of convergence.
Theorem 1 (The Gaussian Tail Inequality) Let X N (0, 1). Then
P(|X | >
function [spec, f, err] = dmtspec(X, tapers, sampling, fk, pad, pval, flag)
% DMTSPEC calculates the direct multitaper spectral estimate for time series.
%
% [SPEC, F, ERR] = DMTSPEC(X, TAPERS, SAMPLING, FK, PAD, PVAL, FLAG)
%
% Inputs: X
= Time series ar
Lecture Notes 1
Brief Review of Basic Probability
(Casella and Berger Chapters 1-4)
1
Probability Review
Chapters 1-4 are a review. I will assume you have read and understood Chapters
1-4. Let us recall some of the key ideas.
1.1
Random Variables
A random