Mathematical theory of statistics, concerned with formal defintions and general results.
1
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1answer
71 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, ...
0
votes
0answers
23 views
How to score the options in a questionnaire and develop a scoring range in a risk profiler questionnaire?
I have developed a risk profiler questionnaire but don't know how to score the options and make a scoring range. Once the range is developed, I could categorize the client as conservative, moderate ...
4
votes
0answers
60 views
Are there applications for differential equations in statistics?
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 ...
0
votes
0answers
27 views
Derivation of the pdf of the Beta distribution [duplicate]
$$
f(x)= \frac{x^{(\alpha−1)} * (1−x)^{(\beta−1)}}{\mathcal B(\alpha,\beta)}
$$
Questions:
How can we derive this famous pdf?
What is the intuitive meaning of the Beta distribution? (Please ...
2
votes
1answer
51 views
Finding conditions on unspecified CDF that permit a solution to an equation
[A duplicate thread can also be found at http://mathoverflow.net/questions/131142/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ]
Let $F(\alpha) := ...
0
votes
0answers
12 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 ...
4
votes
1answer
57 views
Relation between the phi, Matthews and Pearson correlation coefficients?
Are the phi and Matthews correlation coefficients the same concept?
How are they related or equivalent to Pearson correlation coefficient for two binary variables? I assume the binary values are $0$ ...
2
votes
2answers
111 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 ...
0
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0answers
24 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 ...
0
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0answers
36 views
How to statistically compare two fibonacci algorithms?
I have two Fibonacci recursive algorithms and I want to check the efficiency comparing them with each other.
Does anyone know how to do it statistically?
Here's my data.
Recursive_1:
Recursive_2 ...
2
votes
1answer
34 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 ...
3
votes
2answers
132 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 ...
1
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1answer
79 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 ...
0
votes
0answers
12 views
Adjusting mixes of categorical variables by duplicating records
I have a large table with 4 columns, MONTH, COLOR, REGION and RESULT with the results of various contests that are repeated many times each month for the different colors. I can get the result to see ...
0
votes
1answer
52 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.
...
1
vote
1answer
40 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?
0
votes
0answers
13 views
What do a submodel and a supermodel of an exponential family mean?
I don't quite understand the meaning a submodel and a supermodel of an exponential family, even after reading Bickel and Doksum's Mathematical Statistics Vol 1 2007. I was wondering if someone could ...
1
vote
2answers
100 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 ...
1
vote
0answers
10 views
Conditions for the existence of UMP in one-sided two sample hypothesis test
Assume that there are two samples from, possibly, two distinct distributions.
I would like to find a UMP test $H_0: p_1 = p_2 = x$ versus $H_1: x < p_1 < p_2$. If that is not possible, I would ...
3
votes
1answer
81 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 ...
0
votes
0answers
20 views
completeness of a statistic and identifiability of a parameteric model of sample
From Wikipedia
(completeness of a statistic) is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the ...
1
vote
1answer
40 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 $T$ ...
2
votes
0answers
18 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 ...
1
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0answers
29 views
Neyman-Pearson Theorem Question
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
votes
1answer
34 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 ...
0
votes
2answers
42 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
votes
2answers
47 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 ...
2
votes
1answer
70 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 ...
0
votes
1answer
54 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) ...
0
votes
0answers
18 views
What do I take as gamma's argument?
I have collected wind speed data for over a year. I want to calculate the mean wind speed using the formula in which the mean speed is the function of the two weibull parameters and the gamma ...
1
vote
1answer
36 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 ...
3
votes
1answer
66 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.
1
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0answers
37 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 ...
0
votes
1answer
38 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 ...
2
votes
1answer
34 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: ...
1
vote
1answer
37 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
votes
2answers
57 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?
-1
votes
1answer
63 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 ...
2
votes
1answer
69 views
What does it imply when an estimate is not inside its 95% confidence interval?
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 ...
0
votes
0answers
34 views
Is a multivariate normal restricted to an affine set normal? [duplicate]
This seems like a basic question, but I've been confused about it anyway.
Let $X$ be a multivariate normal random variable in $\mathbb R^n$. Let $A$ be the affine set $\{x\in\mathbb R^n : ...
1
vote
1answer
41 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 ...
1
vote
1answer
28 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
votes
1answer
76 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.
...
1
vote
0answers
31 views
$X \sim $ Geometric$(p)$, $Y \sim$ NegativeBinomial$(2,p)$: find the MVUE for $p(1-p)$
Let $X$ and $Y$ be independent random variables with pdf
\begin{align}
f(x)&=p(1-p)^x,x=0,1,2,\ldots \\
f(y)&=(1+y)p^2(1-p)^y, y=0,1,2,\ldots
\end{align}
Find the MVUE for
$$
p(1-p).
$$
I ...
2
votes
1answer
45 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 ...
0
votes
1answer
77 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:
...
0
votes
0answers
47 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
vote
0answers
61 views
Entropy calculations and Process ranking
Still working on my financial datasets, I am now trying to observe the effects of the pre-processing on the results of my study.
To sum it up: I have thirty time series of prices (the main indices, ...
1
vote
0answers
31 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 ...
0
votes
0answers
31 views
monotone likelihood ratio
Let X is a random variable with density $f(x,\theta)$ given by
$f(x,\theta)=2x\theta +(1-\theta)$ if $0 \le x \le 1$ and $-1 \le \theta \le 1$ otherwise.
Show that this family of densities has a ...
