# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### Coupling and Total variational distance

Suppose we have two distributions: $\mu$ and $\upsilon$ on $\{1,2,3\}$. $\mu(1) = 1/2, \mu(2) = 1/3, \mu(3) = 1/6,\upsilon(1) = 1/3, \upsilon(2) = 1/6, \upsilon(3) = 1/2$. Could anyone explain to me (...
699 views

### The probability density function of a rescaled / transformed chi-squared random variable

If $X\sim\chi^2_{6}$, what is the probability density function of $T = \frac{(X-6)}{\sqrt12}$? The problem I'm confronted with is that the chi-squared random variable, $X$, can assume only positive ...
172 views

### Linear regression intercept proofs

Consider that $b_{0} = Y_1 - b_1X_1$ where $Y_1$ and $X_1$ here are the mean values for each. How would I derive the variance and other properties for $b_0$? I'm not very good at doing proofs in ...
133 views

### Which statistical method should I use?

I wish to analyse the following : Independent Variable (IV): User of online financial reports Perceived usefulness and Perceived Quality (Sub to Relevance, Reliability, Understandability, ...
7k views

### Are there applications for differential equations in statistics? [closed]

So I know we statisticians don't use differential equations as heavily as e.g. engineers. Actually, I have never seen or needed them in my studies. I'm curious to learn about them now, and I'd be ...
2k views

### Can Neyman-Pearson lemma apply to the case when simple null and alternative don't belong to the same family of distributions?

Can the Neyman-Pearson lemma apply to the case when a simple null and a simple alternative don't belong to the same family of distributions? From its proof, I don't see why it can't. For example, ...
1k views

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### Unbiased estimate of the semi-partial correlation

Is the sample semi-partial correlation a biased estimate of the population semi-partial correlation? If it is biased, what is an unbiased estimator of the population semi-partial correlation? Are ...
497 views

### Proof of a PD covariance matrix for conditional Gaussian

I was looking at the formula for the conditional covariance of a partitioned matrix. I understand the intuition behind the equation for the conditional covariance, but I'm not sure how to show that ...
588 views

### Exponential family in testing and estimation

In the Annals of Statistics paper "Defining the curvature of a statistical problem(with applications to second order efficiency)" by Bradley Efron, he claims the following two statements in the first ...
748 views

### How to show that $E[(\hat\theta -\theta)^2]<Var(\bar X)=\dfrac{1}{n}$?

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \le \theta \le \theta_1$, where $\theta_0 \lt \theta_1$ are two specified numbers. Find the MLE of $\theta$ and show that it is better ...
81 views

### Show that $G(x)$ is a distribution function and find mean

Let $F$ be a distribution function on $\mathbb{R}$ with $F(0)=1$ and $\mu$ be its mean.Show that $$G(x)=\frac{1}{\mu}\int_{0}^{x}[1-F(t)]dt$$ is a distribution function. Also find its mean. Trial: ...
394 views

### What is the distribution of the ratio of two t-distributed random variables?

x is t-distributed; y is t-distributed. How is x/y distributed? Does it have a closed-form formula?
694 views

### In R, coefficients of MA function are wrong?

I'm currently sifting through my copy of Analysis of Financial Time Series 2nd Edition by Ruey Tsay, and one of the sections involves fitting a MA model to certain data (data set is here). Here's the ...
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### Does Neyman-Pearson Lemma consider the case when the likelihood ratio equals the critical value?

Here are three different versions of Neyman-Pearson lemma. They differ in that the first two (books) ignore the case when the likelihood ratio equals the critical value, while the last one (Wikipedia) ...
127 views

### how to calculate E[vech(x x')vech(x x')']?

Supposing a vector x follows normal distribution. I want to calculate the expectation of the "fourth moment" in a vector form, meaning $\text{E}[\text{vech}(x x')\text{vech}(x x')']$, given that we ...
28k views

### Expected value and variance of log(a)

I have a random variable $X(a) = \log(a)$ where a is normal distributed $\mathcal N(\mu,\sigma^2)$. What can I say about $E(X)$ and $Var(X)$? An approximation would be helpful too.
140 views

### Can pivot be used for testing

A pivotal quantity $Q(X, \theta)$ can be used to construct a confidence interval. I was wondering if it can be used to construct a test statistic and rejection region? In simpler cases involving a ...
2k views

### What is the purpose of introducing nuisance parameters?

From Wikipedia: a nuisance parameter is any parameter which is not of immediate interest but which must be accounted for in the analysis of those parameters which are of interest. Suppose ...
3k views

### Need help calculating a Bayes estimation for a Poisson

My study group and I are stuck on this Bayes' estimator problem. The question is: Let X~Pois($\lambda$) Find the Bayes estimator for $\lambda$ with respect to: (i) The prior distribution: $\lambda$...
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### Ripley's K Function and L Function for Point Patterns

The following is a spatial point pattern: and these are the corresponding Ripley's K function and L function for this data: How are these functions interpreted?
193 views

### What functions qualify to be called measures of dispersion by statistician?

What functions qualify to be called measures of dispersion by statisticians? Why there are so many of these?
642 views

### Why we need variance in this world? [duplicate]

Why we need variance in this world? What is the purpose to make such function and how does it work? I know that it is a measure of how the data spread, but why we don't just use the absolute ...
724 views

### What does it imply when an estimate is not inside its 95% confidence interval? [closed]

What does it actually imply when a 95% CI does not contain an estimate (coefficient or parameter). Is there some model assumption that has not been satisfied? Or it means something else? I know when ...
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### Tests for spatial stationarity (homogeneity)?

There are many models for spatial point patterns and spatial marked point patterns that assume spatial homogeneity or stationarity. i) Is there a statistical test for determining this, where the ...
535 views

### Relation between minimum contrast estimate and minimum distance estimate?

What relation are between minimum contrast estimate and minimum distance estimate? If I understand correctly, these two are different methods? or are they equivalent? Thanks and regards! Minimum ...
153 views

### How is the Df computed in a mixed model?

The following is the output for a mixed model example. The only difference between fm1 and fm2 is the random factor "URBAN", why the df for fm2 is 5 but not 4? Any help would be great. ...
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### Why is it reasonable to consider deviation from null mean as test statistic?

In a testing task, suppose we have already chosen a test statistic $T(X)$, and know its distribution under null hypothesis. Let $\mu$ be the mean of the null distribution of $T(X)$. Why is it ...
7k views

### Question about reading an output for using ANOVA to compare two linear models

I tried to compare the following two models using "anova.lm()" in R: ...
2k views

### How to calculate 95% CI for a random effect?

The R code "intervals()" gives confidence intervals for fixed effects only in a mixed model. *Is there a reason why only fixed effects' confidence intervals are provided? *Is there any way to get ...
69 views

### Divergence interpretation

I'am getting familiar with the statistical notion of Divergence. The word "divergence" is also used in physics (or vector analysis, see here http://en.wikipedia.org/wiki/Divergence). As I was more ...
1k views

### How to prove that Manova is a special case of mixed models?

I am writing my master's thesis in quantile multilevel regression. My professor all of a sudden decided to change the subject of my thesis into something that could be called "quantile multilevel ...
In some lecture notes I am reading, there is the following; Consider $X_{1},...,X_{n}$, each with pdf $g$ (the instrumental distribution). Our aim is to estimate $E_{f}[h(X)]$ where $h(X)$ is some ...
### Does $y'y= \hat y' y + e'e$ hold for the least square model?
I want to show that $y'y= \hat y' \hat y + e'e$ hold for the least square model. I found out that: $\hat y= X b$ with $b$ the least squares estimator of the coefficient vector and $e$ the residual ...