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

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3
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18 views

Distribution of a binary matrix times a Bernoulli vector

Suppose we have the vector $\mathbf{Y} = (Y_{1},\ldots,Y_{n})$ where $Y_{i} \sim \textrm{Bernoulli}(p_{i})$ independently. For the applications I have in mind, $n$ will typically be several thousand, ...
0
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0answers
18 views

(binary) Matrix completion with less known data

Recently, I meet such problems, I call it matrix completion problem. For example, the row denotes the users and the column denotes items. And If one user like the ...
0
votes
0answers
50 views

Probabilistic model Vs Weight based model

Can someone please explain what is the difference between probabilistic model vs weight based model with respect to ML context. When to use one over the other?
0
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0answers
13 views

Best way to graph probabilities of feature vectors

I have a lot of feature vectors in the form of: v1=[x0, x1, x2, x3, x4] where x0, x1, and x2 can take binary values. either 0 or 1 x3 and x4 can take values from 0 up to 9 I have a lot of vectors ...
1
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0answers
17 views

how to determine outliers in sample affected by ascertainment bias

I don't know if this is a really silly question as I'm in no way a statistician and I don't know if this is something that's actually quite rudimentary... Thanks for reading in advance too it got kind ...
0
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0answers
24 views

Statistical nomenclature: what family does “head” and “tail” - in the sense of data runs belong

If I look at mean, variance, skew and kurtosis they are nth moments. If I look at percentiles, quartiles, and medians they are order statistics. Both work together to make a textbook summary. The ...
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0answers
17 views

Auto-correlation calculation [on hold]

Is auto correlation of lag k the same as auto correlation of order k? If not, how do their formulas differ? For lag k is the denominator: $\sum_1^n(x_t-\bar{x})^2$. For order k is the denominator: ...
5
votes
1answer
44 views

How should I evaluate the expectation of the ratio of two random variables?

Let $A$ and $B$ be random variables and $f(A,B)=\frac{A}{B}$. How should I approximate $E(f(A,B))$? I think a Taylor expansion may be in order, but I am not sure how to fire it off in this function. ...
0
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0answers
28 views

Combined standard deviation of geometric series

I have two series of trading profit results. I use the geometric mean to calculate the average in percent (CAGR). I would like to divide it by the standard deviation by combining the two series, but ...
0
votes
1answer
11 views

Predictive analysis based on history

Let me first say that I am a CS person and my knowledge about statistics is quite basic. I am trying to see what predictive analysis to use for a problem I am trying to solve. I will try to make my ...
0
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0answers
20 views

How to CORRECT un-reliable and un-stability in the prediction results

Currently, I meet such questions when building Random Forest model using my data set. My full data set: X_lab: 839 * 469 and y_lab: 839 * 1 which is for all labelled data and X_unl: 20346 * 469 which ...
1
vote
1answer
87 views

Mutual Exclusivity

Suppose that a study is being done on all families with one, two, or three children. Let the outcomes of the study be the genders of the children in descending order of their age. A. List sample ...
0
votes
1answer
17 views

Dirichlet distribution: Normalization of alpha values

I'm a programmer and currently trying to apply the Latent Dirichlet Allocation algorithm by Blei et al. on a text mining problem. I am using a library called gensim for this, which takes, among ...
0
votes
0answers
12 views

regarding the statistics to test the impact of some subset values on a given variable

For a given variable, there is a set of values. We know that these values can belong to different clusters. For instance, a lot of values center around a small value while other values center around a ...
0
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0answers
18 views

The way to evaluate the importance of an independent variable in a regression model

In a regression model, like y~( x1, x2, x3). Is there a test or a way to evaluate which independent variable, x1, x2, or x3, is ...
1
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0answers
22 views

Asymptotic consistency with non-zero asymptotic variance -what does it represent?

The issue has come up before, but I want to ask a specific question that will attempt to elicit an answer that will clarify (and classify) it: In Poor Man's Asymptotics, one keeps a clear distinction ...
1
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0answers
23 views

Clarification on proposition in paper

I'm referring to proposition 1, page 309 of [1] The proposition itself reads: Let $\pmb x_1,\ldots,\pmb x_n,\ldots$ be iid with $\pmb x_i=\pmb B\pmb u_i+\pmb t_0$ where $\pmb u_i$ has ...
0
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0answers
17 views

How to measure defense and attack of a soccer team using goals

i've been trying to come up with a way to measure attack and defense of a team in terms of goals scored and received. The only thing that has crossed my mind is to divide the number of goals ...
1
vote
1answer
31 views

How to quantify the importance of a win in soccer?

I've been thinking lately that the "importance" of a win in a soccer match depends on how good the opponent is. Let's take the spanish league for example. You have some elite teams like Barcelona FC, ...
1
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0answers
33 views

Reverse engineer a predictive model from a time series graph

I have found some real estate plots in a scientific article. These graphs mainly describe, the believes of the author of the development of the real estate market in the future for certain countries. ...
1
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0answers
33 views

Are there local maxima in the standard 2-pl model or is there proof that there aren't?

I'm looking to use iterative logistic regression to find the parameter values of the 2 pl model: $ P = \sigma(\alpha_{i} (\theta_{s} - \beta_{i})) $ where $ \sigma(x) = \frac{1}{1+e^{-x}} $ I know ...
1
vote
1answer
20 views

Finding Bias$(s^2)$ in incorrect linear model

I am unable to find the bias of the sample variance estimator in this problem. Unfortunately I keep coming up with Question: Suppose that the true linear model is $y = X_1\beta_1 + X_2\beta_2 + ...
2
votes
0answers
16 views

Finding the minimax estimator

Suppose $x$ comes from a Bernoulli distribution $f(x, \theta) = \theta^{x}(1-\theta)^{1-x}$ where $0 \leq \theta \leq 1$. Now suppose $y$ comes from the same distribution. An estimate for the ...
0
votes
1answer
40 views

Standardized regression coefficients

I have a question regarding standardized regression coefficients and the correlation matrix. I have a two part problem that I am working on, and I need to show that the correlation matrix R is equal ...
0
votes
1answer
16 views

Jags: How to create vector with probabilities that are based on one of its outcomes?

I am very new to Jags. I just started using it for my stats class. I have done other types of programming, but I am having trouble understanding how to use Jags properly. Right now, I have a vector ...
3
votes
2answers
62 views

Affine equivariance and consistency

Denote $$\pmb x_i\sim\mbox{Ell}(\pmb 0,\pmb\varSigma)|i=1,\ldots,n$$ with $\pmb\varSigma$ symmetric positive definite and $\mbox{Ell}(\pmb 0,\pmb\varSigma)$ denotes a $p$ variate elliptical ...
5
votes
1answer
129 views

What is the meaning of $\oplus$ and $\otimes$?

I am struggling to fully understand some notation in a book where they use a "crosshair" symbol - first like $\bigoplus\limits_{i=1}^n{} Z_j $ where the $Z_j$ are matrices and second like $I_n \otimes ...
4
votes
2answers
120 views

Characteristic function problem

Suppose $X_1$ and $X_2$ are independent random variables and suppose also that $X_1$ and $X_1-X_2$ are independent. Show that $$\mathbb{P}_{X_1}[X_1=c]=1$$ for some constant $c$. What I get so ...
2
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0answers
77 views

A Integral question

I am having problem with integral: $$\int_{-\infty}^\infty\frac{1}{2\pi} e^{-itx} \left( 0.5 \left( e^{-4t^2} + 1 +\cos(t) + \cos(2t) \right)\right) \,dt$$ @DilipSarwate I have proved lim sup(cf)=1 ...
0
votes
0answers
17 views

What is a convex support? (Bickel&Doksum, Mathematical Statistics, Basic ideas…Vol1)

Bickel&Doksum, Mathematical Statistics, Basic ideas...Vol1 page 122, just above Cor2.3.1, it says: Define the convex support of a probability P to be the smallest convex set C such that ...
0
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0answers
12 views

Application of Lévy–Khinchine formula

How can we express the characteristic functions of Wiener and Poisson processes by using the Lévy–Khinchine formula? I don't know how to find the characteristic functions of particular Levy ...
2
votes
2answers
52 views

Question on the correlation between two dependent variables

I'm working on this question and it's stumping me. Let $S_n = X_1 + \ldots + X_n$ (with $n>=1$) be a random walk with $X_1, \ldots, X_n$ be iid RV's. $$ E(X_k)=\mu,\,{\rm Var}(X_k)=\sigma^2. ...
2
votes
1answer
51 views

Does vector version of the Cauchy-Schwarz inequality ensure that the correlation coefficient is bounded by 1?

I have been trying to understand the proof that the correlation between two random variables $X$ and $Y$ is between $-1$ and $1$. For simplicity, suppose $X$ and $Y$ have mean zero. Then ...
2
votes
0answers
53 views

Show alias coefficient of Plackett Burman design equals $r_{ij}=\frac{c_{1i}'c_{2j}}{N}$

Consider a Plackett Burman design with N rows and let $\beta_{1}$ be the of the regression coefficients corresponding to the main effects and let $\beta_{2}$ be the vector of the regression coefficients ...
0
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0answers
36 views

Probability distribution of time series

I am unable to understand concepts related to the probability distribution of binary time series. This is from the book Binary time series by Benjamin Kedem, vol 52 Let $X_t$, t =0,1,... be a binary ...
0
votes
0answers
20 views

prior predictive distribution negative binomial in R

I have a prior distribution Gamma(1.71,1.05) from a Poisson(2.2), and I know that the prior predictive distribution will be a Negative-Binomial of Gamma parameters i.e. Neg-bin(1.71,1.05). I would ...
1
vote
1answer
59 views

If $\frac{(2n-1)s^2}{\sigma^2} \overset{}{\sim} \chi^2_{2n-1}$, can I take the variance of both sides to get an equality relation?

I currently have a standard result regarding variances and it looks like: $\frac{(2n-1)s^2}{\sigma^2} \overset{}{\sim} \chi^2_{2n-1}$ that is, it is approximately a chi-squared distribution with ...
2
votes
2answers
61 views

Jointly sufficient statistic?

A random sample $X_{1},...,X_{n}$ is pulled from a gamma distribution. Are there jointly sufficient statistics based on these observations for the two unknown parameters? The definition of a gamma ...
3
votes
1answer
50 views

Regession diagnostics

Why is it necessary to have the number of independent variables less than the number of data points in a regression setup? If suppose, it is the other way round, i.e. the number of independent ...
1
vote
1answer
27 views

how to calculate Suffcient Sample amount for one distribution

I am trying to answer the following question. I want to know if the distribution of my samples follows some well-known distributions. How I can compute the minimum required samples in order to be ...
-1
votes
1answer
30 views

to find the values of gamma distribution knowing the parameter value of a poisson

I have a variable X, and information available to me is that the parameter $\theta$ is around 2.2 $X\sim \mathbf{Poisson}(\theta)$ How can I determinate the value of the parameters $\alpha$ and ...
3
votes
2answers
35 views

Poisson as a limiting case of negative binomial

I was reading "Maximum Likelihood Estimation for the Negative Binomial Dispersion Parameter" by Walter W. Pieogorsch, and in the intro it says the Poisson distribution is a limiting case of negative ...
-1
votes
0answers
41 views

machine learning versus statistical technique [duplicate]

Is there any difference between machine learning and statistical techniques.If there is any difference please provide any research paper which justified it..
0
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0answers
86 views

How to derive OLS through MLE? [duplicate]

I am just curious on finding about this derivation
3
votes
1answer
29 views

Classical Normal Linear Regression model identity

The context of the following identity is in the Classical Normal Linear Regression Model, ie, $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+ \boldsymbol{u}$ where $\boldsymbol{u}$ is a $n \times ...
2
votes
1answer
93 views

PhD in statistics, but unsure of how much pure math is needed as requirements

I am applying to a PhD program in biostatistics, and my undergrad was in B.S. applied math and limited pure math background; only topology and some analysis. My question, though, is how much pure ...
2
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0answers
33 views

Regression Model Proof

In the standard regression model where $$ Y=Xβ+u\ ,$$ with normal errors, under the hypothesis $$\text{H}_0: Rβ=r$$ where $β$ is $k\times 1$, $R$ a $q\times k$ matrix with rank $q$: the OLS ...
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0answers
12 views

Three stage cluster sampling

I was just curious if anyone knew where to find information on three-stage cluster sampling. There is a significant amount of information on two-stage cluster sampling, but I am unable to find ...
2
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0answers
25 views

Change of variables with a continuous but not differentiable mapping

Suppose that Y is a continuous random variable with a density function $f_{Y}(y)$. We transform $Y$ by the following mapping \begin{equation} Y^{*} = \left \{ \begin{array}{ll} \alpha Y + \beta ...
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0answers
33 views

Significance of Eigenvectors of a Covariance matrix [duplicate]

In PCA and in many other problems of machine learning we use Eigenvectors of covariance matrix of the data. How do we visualize Eigenvectors of Covariance matrix? The Principal Eigenvector ...