Questions tagged [mathematical-statistics]

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

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1answer
796 views

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 (...
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1answer
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 ...
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2answers
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 ...
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1answer
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, ...
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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 ...
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3answers
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, ...
14
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1answer
1k views

Product of two independent random variables

I have a sample of about 1000 values​​. These data are obtained from the product of two independent random variables $\xi \ast \psi $. The first random variable has a uniform distribution $\xi \sim U(...
2
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1answer
151 views

Finding conditions on unspecified CDF that permit a solution to an equation

[A duplicate thread can also be found at https://mathoverflow.net/questions/131142/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ] Let $F(\alpha) := \mathbb{P}(\tilde{\...
3
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0answers
108 views

Implications of lower-bounded total variation distance on hypothesis testing

Let $\{X_i\}_n$ be a sequence of $n$ random variables independently and identically drawn from either $P$ or $Q$. Thus the sequence $\{X_i\}_n$ has a product distribution, which is either $P^n$ or $Q^...
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3answers
1k views

Having a conjugate prior: Deep property or mathematical accident?

Some distributions have conjugate priors and some do not. Is this distinction just an accident? That is, you do the math, and it works out one way or the other, but it does not really tell you ...
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3answers
48k views

Stats: Relationship between Alpha and Beta

My question has to do with the relationship between alpha and beta and their definitions in statistics. alpha = type I error rate = significance level under consideration that the NULL hypothesis is ...
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0answers
62 views

How is a statistic represented as a mapping with varying sample size?

Suppose there is a sample of varying size $n\in \mathbb N$, each sample point taking values in $\mathbb R$, and a statistic $T$. If I am correct, a statistic can accept arbitrary sample size. What are ...
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1answer
87 views

Question about proof for luce choice axiom w.r.t. conditional probability

In Luce (1959) the choice axiom is definied, that for a finite subset $T$ of $U$ such that, for every $S\subset T$, $P_S$ is defined. If $P(x,y)\ne 0,1$ for all $x,y\in T$, then for $R\subset S\...
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0answers
162 views

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 ...
3
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1answer
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 ...
4
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2answers
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 ...
2
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1answer
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 ...
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1answer
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: ...
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1answer
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?
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2answers
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 ...
4
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1answer
1k views

Sufficient statistic and hypothesis testing

Suppose I have a family of (continuous) distributions $\mathcal{P}=\{P_\theta(x),\theta\in\mathbb{R}^+\}$. I also have a statistic $T(x)$ that is sufficient for $\theta$. The value of the parameter $...
2
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1answer
412 views

Does Fisher's factorization theorem provide the pdf of the sufficient statistic?

From Wikipedia Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is $ƒ_θ(x)$, then $T$ ...
3
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1answer
69 views

Ancillary statistic not containing information about sample distribution?

From a note by Jun Shao If $V(X)$ is a nontrivial ancillary statistic, then $σ(V(X)) ⊂ σ(X)$ is a nontrivial σ-field that does not contain any information about $P$. I was wondering in what sense "...
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1answer
2k views

Neyman-Pearson Lemma for the exponential distribution

I have the following question for my homework: Suppose X~exp($\theta$). We want to test $H_0: \theta=1 vs. H_a:\theta=2$, based on a sample of size 2 - ${X_1,X_2}.$ a. Obtain the most powerful test ...
2
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1answer
126 views

Meaning of “a statistic $U$ is ancillary to another statistic $T$”?

From Wikipedia Given a statistic $T$ that is not sufficient, an ancillary complement is a statistic $U$ that is ancillary to $T$ and such that $(T, U)$ is sufficient. Intuitively, an ancillary ...
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2answers
377 views

How to detect a quasi separation problem for a data set?

Suppose that we have a two-column data set. One column consists of a hundred x=0 and a hundred x=1, whereas the other one consists of y's (1 or 0 response). Besides, suppose that the P(Y=1|X=0) = 0....
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2answers
124 views

Find the expected value of $(\bar X_n-p)^3$

Let $X_1,X_2,\dots, X_n$ be a random sample from a Bernoulli distribution with parameter $p$. Let $\bar X_n$ be the sample average given by $\bar X_n=\frac{1}{n} (X_1+X_2+\dots+ X_n)$). Find the ...
3
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1answer
348 views

Conditional expectation for the reciprocal of a normal

We know the expected value of $1 \over X$, where $X$ is a normal random variable, does not exist. But suppose we condition on an interval not containing zero. For example, if $ \mu_X = 10$ and $\...
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1answer
578 views

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) ...
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1answer
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 ...
19
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1answer
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.
2
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1answer
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 ...
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1answer
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 ...
3
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1answer
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: $\lambda$...
2
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1answer
2k views

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?
2
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2answers
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?
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1answer
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 ...
3
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1answer
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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2answers
1k views

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 ...
2
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1answer
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 ...
2
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1answer
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. ...
2
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1answer
144 views

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 ...
2
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1answer
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: ...
2
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1answer
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 ...
1
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1answer
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 ...
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1answer
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 ...
2
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2answers
122 views

Condition for Law of Large Numbers, Monte Carlo

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 ...
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1answer
370 views

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 ...