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1answer
18 views

lower order term has positive prediction in simple regression but negative prediction in quadratic regression

I am using SPSS to conduct a quadratic regression. The IV is positively and significantly related to the DV in a simple linear model. However, after the squared IV is put in the model, the lower order ...
4
votes
1answer
62 views

Distribution of a quadratic form, non-central chi-squared distribution

Definition. Suppose $\mathbf{y} \sim \mathcal{N}(\boldsymbol{\mu}, I_{n \times n})$. Then $$w = \mathbf{y}^{T}\mathbf{y} = \|\mathbf{y}\|^2 \sim \chi^{2}_{n}\left(\theta = ...
2
votes
1answer
42 views

Gaussian distribution sample norm versus mean norm

Consider a Gaussian distribution with mean $\boldsymbol\mu$ and covariance matrix $\mathbf{Q}$, and suppose I take a sample from this distribution and compare its norm with the mean norm. My goal is ...
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0answers
17 views

How to estimate a Simple Quadratic Equation

Hello i am quite new to economic application and have a really simple question about estimating a quadratic equation. I have a time series data set with 55 points and want to estimate a quadratic ...
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0answers
6 views

Independence of quadratic forms of random variables [duplicate]

If $X$ and $Y$ are two vectors of random variables, such that $X$ and $Y$ are independent, is it true that $X^TX$ and $Y^TY$ are independent? If yes, how do we prove it?
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0answers
22 views

covariance of a normal distribution and a quadratic form [duplicate]

I'm working on this problem and am a little lost, can anyone lend their assistance? $$X\sim N_k(\mu, \Sigma)$$ show $$cov(x, x'Ax)=2\Sigma A\mu$$ I tried ...
2
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1answer
29 views

First two moments of a quadratic form in which the vector and matrix are random (though independent)

$\DeclareMathOperator\tr{\mathrm{tr}}$Let $x$ be a standard normal $p$-variate random variable, which is independent of a symmetric positive definite random matrix $Y$. I would like to compute the ...
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0answers
44 views

Quadratic equation system solving analytically or numerically

I have such a nonlinear equation system $x_i=\frac{\sum_{j\neq i}a_{ij}\times\sum_{k\neq i}x_k}{\sum_{k\neq i}x_k-\sum_{j\neq i}a_{ij}}$ where $a_{ij}$s are known coefficients in $[0,1]$. And $x_i$s ...
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0answers
38 views

Mean of a truncated quadratic function of a normal variable

Suppose you want to compute the mean of the following random variable: $$ V=\left(\frac{z-y}{z}\right)I\left(y<z\right)-\left(\frac{z-b_0}{z}\right)I\left(b_0<z\right) $$ where: $$ ...
3
votes
1answer
63 views

Expectation of Euclidean Norm and Quadratic Forms

If $\boldsymbol{\beta} \sim \mathcal{N}_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, can someone please help me understand why $\mathbb{E}[||\boldsymbol{\beta}||_2^2] = ||\boldsymbol{\mu}||_2^2 + ...
3
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0answers
74 views

Modeling quadratic function parameters as a function of variables

I'm modeling a response variable that varies seasonally. The pattern is relatively well described by a quadratic function. As the data have been collected over many (~50) years, I'm interested to ...
0
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1answer
15 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? ...
4
votes
1answer
123 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
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0answers
68 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
51 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} ...
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0answers
61 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
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1answer
38 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 ...
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2answers
137 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
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0answers
58 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 ...
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0answers
157 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
80 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
56 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 ...
5
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1answer
328 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
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0answers
92 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
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0answers
40 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
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1answer
1k 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
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1answer
89 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
144 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
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0answers
77 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
131 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
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1answer
87 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 ...
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0answers
41 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
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2answers
43 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
129 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 ...
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2answers
1k 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
65 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
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3answers
204 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 ...
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0answers
153 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 ...
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2answers
522 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
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0answers
58 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
131 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 ...
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0answers
105 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
208 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
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1answer
2k 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
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1answer
204 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 ...
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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 ...
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2answers
746 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
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1answer
122 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
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0answers
184 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} ...
2
votes
1answer
9k views

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

What causes the different results below? ...