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0
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1answer
8 views

Variance estimator of quadratic term of Cox PH model

I have a failure time data set. I want to fit Cox PH model with quadratic term like $h(t;x)=h_0(t)*exp(x\beta_0+x^2\beta_1)$ What will be the mathematical formula for variance of quadratic term? ...
3
votes
1answer
48 views

Relationship between Gram and covariance matrices

For a $n\times p$ matrix $X$, where $p \gg n$, what is the relationship between $X^{T}X$ (scatter matrix, on which covariance matrix is based) and $XX^{T}$ (outer product sometimes called Gram ...
3
votes
0answers
61 views

Does a “pruned” i.i.d. multivariate sample behave as the i.i.d. sample?

Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
1
vote
1answer
46 views

Prove that the distribution of $Q$ is chi-squared with $p_2$ degrees of freedom

Suppose $X$ is a $p$-dimensional vector following $N_p(\mu,\Sigma)$ distribution, where $\mu$ is $p$-dimensional and $\Sigma$ is $p\times p$. Let $X=\left(\begin{array}{ccc}X_1\\X_2\end{array} ...
1
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0answers
29 views

Interpreting scaled betas for quadratic terms in a negative binomial regression

I created a negative binomial model where the final model included 5 quadratic predictors (each with a corresponding linear term). I am considering two ways to interpret the beta coefficients for each ...
0
votes
1answer
33 views

estimators with singular covariance matrix

Suppose I have 2 vectors of random variables $\boldsymbol\theta_1 \in \mathbb{R^n}$ and $\boldsymbol\theta_2 \in \mathbb{R^m}$ with asymptotic covariance $\Sigma_1$ and $\Sigma_2$ respectively. I want ...
6
votes
2answers
131 views

A strange step on a proof about the distribution of quadratic forms

The following theorem comes from the 7th edition of "Introduction to Mathematical Statistics" by Hogg, Craig and Mckean and it concerns the necessary and sufficient condition for the independence of ...
1
vote
0answers
54 views

Method to minimize a quadratic form

How should I minimize a quadratic form $g(x)^TA(x)^{-1}g(x)$ with respect to $x$, where $g$ is a vector which depends on the vector $x$? This quadratic form is obtained from a quasi likelihood ...
1
vote
0answers
62 views

Correlation between two quadratic forms of Gaussian random vectors

I want to approximately calculate the correlation between two quadratic forms of two Gaussian random vecotrs (of course these are in fact non-Gaussian densities). Does anyone know the derivation of ...
0
votes
1answer
50 views

For a quadratic form to minimize with a L2 regularization term, is the gradient of the solution collinear to the solution?

Say you minimize a quadratic form f with a L2 regularization term (g = f + L2_term). The solution of minimizing g is x*. Is the gradient of f applied to x* collinear to x* as the figure below ...
0
votes
1answer
47 views

Expectation of a fractional form of chi squared

I have been trying to calculate or find a result for the expectation $$\mathbb{E} \left[ \frac{w^\top D^2 w}{1 + w^\top D w} \right] $$ where $$w \sim \mathcal{N}(0,I_N),$$ and $D$ is a diagonal ...
4
votes
1answer
169 views

Parallel regression assumption

In ordinal logit models, do I violate the parallel regression assumption if I include a quadratic term for a nonlinear relationship? Because then it would mean that the coefficients are not in the ...
0
votes
0answers
59 views

Quadratic categorical variable

My model includes a categorical independent variable. The graphical analysis suggests that the relation between this variable and the dependent one could be quadratic. As regards including a squared ...
2
votes
0answers
30 views

Proving the given quadratic form is chi-squared $k$

Suppose $\underline{X}$ is an $m$-dimensional vector following multivariate Normal distribution i.e. $\underline{X}$~$N_m(\underline{\mu},\Sigma)$ where $\Sigma$ is positive definite. Let $B$ be a ...
1
vote
1answer
747 views

Plotting a polynomial regression with its confidence interval of 95% in R

I have been trying for a while plotting a polynomial regression using R. I have read several libraries, as ggplot2, qplot, etc, with no succeed. The next are my data: I normally use the R GUI called ...
1
vote
1answer
71 views

How to fit a single quadratic term to a regression

I have a high dimensional multivariate model and am fitting linear weights to each of the $N$ free variables using a classic stable SVD matrix solver. This works. I want to improve the fit by using a ...
0
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0answers
114 views

Single group with repeated measures using R? - Trend analysis (linear, quadratic effect)

I wonder how to get the repeated measures analysis of variance using R. My data is like this. ...
0
votes
0answers
55 views

quadratic endogenous variables in R

As part of ongoing research I'm to test a certain model on some data. One of the questions asked (c.f. one of the hypotheses) involves estimating the quadratic term of an independent variable (in R). ...
2
votes
1answer
99 views

Linear regression with an inequality constraint

I am looking for an efficient way of finding a linear fit $Mx = y$ subject to an inequality constraint: $\frac{|x_2|}{\sqrt{x_3^2 + x_4^2}} \geq a$, with $a \geq 1$. The rectangular matrix $M$ is ...
3
votes
1answer
80 views

Weird pdf of a quadratic function of a N(0,1) variable: miscoding or big rounding error?

I would like to calculate the pdf of a random variable y defined by : y=c+b*x+a*x^2 The pdf is a non-central chi-squared distribution. For a>0, it should be equal ...
0
votes
0answers
30 views

Can I determine whether two quadratic slopes differ significantly?

I know that there are ways to compare two linear functions to determine whether the slopes of the functions are significantly different. However, I'm wondering whether there is any way to compare ...
2
votes
2answers
36 views

Testing equality of two X values in quadratic regression

So let's say we have a quadratic relationship between two variables, y and x. Graphically, it is U-shaped. However, there is also a linear component to it, such that the left curve is lower than the ...
1
vote
1answer
105 views

Determinant of the covariance matrix in a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
5
votes
2answers
525 views

Quadratic Programming and Lasso

I'm trying to perform a lasso regression, which has following form: Minimize $w$ in $(Y - Xw)'(Y - Xw) + \lambda \;\text{norm}(w,1)$ Given a $\lambda$, I was advised to find the optimal $w$ with ...
1
vote
1answer
55 views

How should I interpret a factor that is significant both (1) in linear form in interaction with another factor and (2) in its quadratic form?

Our study aims to identify and understand how several ecological factors relate to parasite abundance in a colonial animal. However, we are uncertain on how to interpret a factor (density) that is ...
5
votes
3answers
177 views

What is the PDF of $[(X-a)^2 + (Y-b)^2]^{1/2}$ where $X$ and $Y$ are two non-standard normal random variables?

I have to conduct an experiment getting data from a system. These data are the estimated values, provided by the system, of a true value that we know beforehand. I then compare the estimated values ...
0
votes
0answers
108 views

Interpretation when only the squared term is significant

I am using a Poisson model. My theory suggests a positive $X_1$ and a negative $X_1^2$. However, the results show an insignificant, positive estimate for $X_1$ and a significant, positive estimate for ...
4
votes
2answers
415 views

Independence of a linear and a quadratic form

How can I prove the following lemma? Let $\mathbf{X}^ \prime$ = $ \left[ X_1 , X_2 , \ldots, X_n \right]$ where $ X_1, X_2, \ldots X_n $ are observations of a random sample from a distribution which ...
0
votes
0answers
46 views

Second order terms in regressions

I have a linear regression, with 3 covariates, one of which is categorical with 4 categories. I want to determine if second order terms have a significant effect in the model. I would like to ask if ...
1
vote
1answer
93 views

Linear Kernel in Baysian Linear Regression

I came up with http://mlg.eng.cam.ac.uk/duvenaud/cookbook/index.html and it is actually very useful. At some point it says If you use just a linear kernel in a GP, you're simply doing Bayesian ...
1
vote
0answers
94 views

Comparison between normal glm and glm.nb regression with quadratic term?

Let's say I have a function to simulate data for negative binomial regression: ...
1
vote
2answers
167 views

Adding quadratic term changes the sign of the variable

Number of books published in a year (noBook) is my dependent variable and I have independent variables including the age of the author (age). The coefficient of age is positive and it is significant. ...
1
vote
1answer
1k views

Transforming a relationship from quadratic to linear

Assume that I'm running a path analysis and I have discovered that certain relationships, empirically, are quadratic rather than linear. In order to model the data as such, I want to transform the ...
2
votes
1answer
192 views

Comparison of two linear regression models with squared regressors

I need to compare two linear regression models that include the regressors in levels and in squares: $Y=a_1x^2+b_1x+c_1$ and $Y=a_2x^2+b_2x+c_2 $ Specifically, I need to test whether these ...
7
votes
2answers
1k views

How to include a linear and quadratic term when also including interaction with those variables?

When adding a numeric predictor with categorical predictors and their interactions, it is usually considered necessary to center the variables at 0 beforehand. The reasoning is that the main effects ...
8
votes
2answers
559 views

Sum of two normal products is Laplace?

It is apparently the case that if $X_i \sim N(0,1)$, then $X_1 X_2 + X_3 X_4 \sim \mathrm{Laplace(0,1)}$ I've seen papers on arbitrary quadratic forms, which always results in horrible non-central ...
4
votes
1answer
119 views

Quadratic form of a bivariate normal

This is a homework problem. Let $(X,Y)\sim N(\mu_1,\mu_2,\sigma^2_1,\sigma^2_2,\rho)$. Show that if $\sigma_1,\sigma_2 >0,|\rho|<1$, then $$ ...
4
votes
0answers
157 views

How to do centering if I have a quadratic term?

I have been trying to run a multilevel model with both a linear and a quadratic term for income as my main variables of interest. It looks something like: \begin{eqnarray} ...
1
vote
1answer
6k views

Quadratic models with R. The use of poly(..) and I(..) functions (R-language)

What causes the different results below? ...
5
votes
2answers
2k views

Expected Value of Quadratic Form

On page 9 of Linear Regression Analysis 2nd Edition of Seber and Lee there is a proof for the expected value of a quadratic form that I don't understand. Let $X = (X_i)$ be an $n \times 1 $ random ...
5
votes
1answer
422 views

Adding a quadratic term: should I use the squared original (and not the squared standardized?)

In a multiple logistic regression I need to standardize one of the variables because I need to add a quadratic term. Whether I add the quadratic term as the squared original or the squared ...
3
votes
1answer
148 views

Covariance of $\mathbf{x}$ and $\mathbf{x}^\prime A \mathbf{x}$ when $\mathbf{x} \sim N_{m}(\mu, \Sigma)$

This is part of problem 5.23 in A First Course in Linear Model Theory, Dey and Ravishanker. It was on a previous midterm and I didn't know how to do it, but now I am studying for the final and would ...
5
votes
1answer
178 views

Optimal sampling to estimate a quadratic function

Suppose I have a quadratic function: $$ f(x) = a^T x + x^TBx $$ with $x \in \mathbb{R}^n$. Given a point $x$ I can measure $f(x)$ up to some noise, that is I can get a measurement: $$ \hat{f}(x) = ...
2
votes
2answers
627 views

Interpretation for simple slope analysis for curvilinear regression with interaction effects

When regressing income ($Y$) on age ($X$) moderated by gender ($Z$), I not only find significant effects for age ($X$), age squared ($X^2$), gender ($Z$), the interaction of age and gender ($XZ$), and ...
3
votes
1answer
398 views

Exponentiate a quadratic form, which MVN?

I read that if you have a quadratic $f(x) = ax^2+bx+c$ and providing the leading coefficient $a < 0$, then $e^{f(x)}$ is the pdf of a normal distribution with mean $\mu = -\frac{b}{2a}$ and ...
2
votes
1answer
97 views

How to specify and estimate the parameters of a model that is quadratic in several variables

I am trying to find how the quadratic model for a multiple featured dataset will be. Suppose that my training set is $X_1,...X_n$ with each $X$ of dimension $4$. Now suppose I want to fit a quadratic ...
2
votes
0answers
148 views

What is the distribution of the 'achieved' $R^2$?

I am interested in the distribution of/performing inference on the 'achieved' $R^2$ coefficient in multiple linear regression. Suppose that $y\sim x\beta + \epsilon$ with $\epsilon \sim ...