# derivatives

1. ### Deriving Chi-Square for contingency table 2x2

Hi Given a 2x2 contingency table, with a b c d as the observed values and n as the total, I have to show from the fact that Chi-Square = Sum(((O-E)^2)/E) Where E is the expected value of the observed, that: Chi-Square= (n(ad-bc)^2)/(a+b)(a+c)(b+d)(c+d). I just cant seem to get there. Any links...
2. ### Restricted Maximum Likelihood (REML) Estimate of Variance Component.

Let, \mathbf y_i = \mathbf X_i\mathbf\beta + \mathbf Z_i\mathbf b_i+ \mathbf\epsilon_i, where \mathbf y_i\sim N(\mathbf X_i\mathbf\beta, \Sigma_i=\sigma^2\mathbf I_{n_i}+\mathbf Z_i \mathbf G\mathbf Z_i'), \mathbf b_i\sim N(\mathbf 0, \mathbf G), \mathbf\epsilon_i\sim N(\mathbf 0...
3. ### Differentiation Involving Determinant.

I have to compute the following differentiation : \frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times n}')^{-1}\mathbf X_{n\times p}], where \sigma^2 is a scalar, \det denotes determinant, \mathbf...
4. ### Probability Density Function of the derivative of a stochastic process

Dear all; What is the expression of the PDF of the derivative of a stochastic process? My problem is completely general and theoretical, so, let's assume that all statistics of the stochastic process are given (e.g. pdf, higher order statistics, and so on). In other words, if we know...
5. ### Optimimality condition for quantile estimators

*OPTIMALITY, sorry. I'm trying to understand a passage in Koenker's Quantile regression book (p.33). It says: (note that y,x, are vectors and w is the direction vector) With the first part of the outcome no problem: I apply the product rule for derivatives and the derivative of the...
6. ### 1st, 2nd and 3rd derivatives of seven parameter double sigmoid function

I want the 1st, 2nd and 3rd derivatives of the following seven parameter double sigmoid function Y = a + b /(1 + exp(-c(X-d))) – e/(1 + exp(-f(X-g))) a,b,c,d,e,f,g are the equation parameters. I can go as far as the 1st derivative: If z = (1 + exp(-c(X-d))) and h = (1 + exp(-f(X-g)))...