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

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28 views

How to predict/find a value based on historical data using statistics? [on hold]

I am not aware about statistics but want to implement statistical concepts. Following is scenario. I have set of 10 different values like - ...
1
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0answers
24 views

CV with zero mean?

Many of the papers I've been reading on glucose/insulin dynamics say: "measurement error was assumed to be independent, gaussian with zero mean and known standard deviation, (CV = 2%)." My question ...
-4
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0answers
68 views

model development or equation making [on hold]

I have three parameters $a,\ b,\ c$ and resultant $y$ Independently y depends on $a$ as follows. $y = a ^3 + 3 a^2 + 4 a + K $ $a$,$b$ and $c$ varies in coded form here $a$ varies from $-1$ to $+1$ ...
3
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1answer
26 views

Denominator term in Chi-Square-Test for association in a contingency table

The general formula for the Chi Square Test for association in a contingency table with $I$ rows and $J$ columns, and cell counts $n_{ij}$ is $$ \chi^2 = \sum_{i=1}^I \sum_{j=1}^J \frac{(n_{ij} - ...
0
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1answer
31 views

Independent and Identically Distributed(i.i.d.) Random Variables

The assumption that observations be i.i.d. tends to simplify the underlying mathematics of many statistical methods . However in practical applications of statistical modeling the assumption may or ...
2
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0answers
8 views

How to prove that the permutation of the points are the minimal sufficient statistics for Cauchy distribution?

I see this everywhere that the permutation of the samples $X_{(1)}, ..., X_{(n)}$ is the minimal sufficient statistic for the Cauchy distribution [1]. It is clear that it is a sufficient statistic,but ...
3
votes
1answer
30 views

identification of simultaneous equation model

Consider the following SEM can be identified: $$ y_i = x_i \alpha + z_i \beta + u_i\\ z_i = x_i \delta + v_i\\ $$ where we have $$ E[u_i] = E[v_i] = E[u_i x_i] = E[v_i x_i] = 0\\ cov(x_i, u_i v_i) = ...
0
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1answer
54 views

Does a degree in statistics make one a more rational thinker? [on hold]

All my life I have been plagued by mental illness, delusion, superstition. I have come to see the value of science and rationality and skepticism but don't really know how to think in a more ...
1
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0answers
14 views

Test to compare against a known value

I had a set of values of 40 observations, and I want to test their significance different with a known actual value. What test should I carry out in finding out the significance difference?
1
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1answer
24 views

Relate $Var(y)$ with $Var(y)_{(i)}$

How can I relate $Var(y)$ with $Var(y)_{(i)}$ where $Var(y)_{(i)}$ is de variance of the data with the ith item removed. It is necesary first relate $\bar{y}$ with $\bar{y}_{(i)}$ and it complicates ...
3
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0answers
19 views

Assumption about Systematic Errors

In Maronna, Martin and Yohai's Robust Statistics (2006, p.17), they describe a location model as follows. $$x_i = \mu + u_i,$$ where $x_i$ is the $i$th observation; $\mu$ is the hypothetical mean ...
0
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0answers
19 views

How can I calculate $$\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$$ [migrated]

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
1
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1answer
25 views

How to Get this Confidence Interval

One example in Maronna, Martin and Yohai's Robust Statistics (2006, p.2) is as follows. Given 24 measurements of certain quantity (see below) and their sample mean 4.28 and sample standard variation ...
1
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0answers
19 views

propagation of the error in the summation

I have a question regarding the propagation of the error during the summation. Please see the equation below. In this equation only quantity R has an error. How it will propagate to the final value ...
1
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1answer
20 views

GLS versus FGLS

Can anyone simplify how GLS estimation is different from FGLS estimation? I understand that the covariance matrix is estimated in FGLS using OLS. What are some other differences?
0
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0answers
14 views

Dominance analysis [closed]

How to run Dominance analysis using SPSS. I have ran logistic regression, to determine the relative importance of the predictors I am looking for Dominance analysis. I am using SPSS software. Please ...
0
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0answers
11 views

The asymptotic slope of the BLP

So I am given that $X$ is a binary r.v. and the following assignments, $E[X] = A, E[Y|X=1]=B, E[Y|X=0]=C, E[Y^{2}|X=1] = D, E[Y^{2}|X=0]=E$. I must express my answers in terms of these expressions ...
1
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1answer
21 views

Confidence interval for the variance when we know the mean of the population

We have $n$ random samples from a population which has normal distribution, and the mean of the population is known. How do we change the procedure of finding a confidence interval for the population ...
1
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1answer
50 views

Getting a meaningful metric for variation in this type of cyclical, panel data? WSS won't exactly cut it!

This is related to a previous question I have asked, but I am not after visualization but rather a meaningful summary statistic. Situation: I have many (150k) customers. Each generates his own ...
1
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0answers
17 views

Variance of the influence function of the parameters of an over-dispersed poisson model

Assume an over-dispersed Poisson model: \begin{align} E[C_ij] &= m_ij \quad\text{ and } \\ {\rm Var}[C_ij] &= ∅E[C_ij]= ∅m_ij \\ \log(m_{ij}) &= n_ij \\ n_{ij} &= c+ α_i + β_j ...
1
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0answers
26 views

Mean square convergence

I am working on an example in my book and cannot figure out an expectation. Let $$E(T_n)= \frac{n\theta}{n+1}$$ $$E(T_n^2)= \frac{n\theta^2}{n+2}$$ $$g(t)=\frac{nt^{n-1}}{\theta^{-n}}$$ Then ...
1
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2answers
15 views

Distribution Enquiries

I have a set of cycle time data for a processing counter. Since the cycle time is less than 1 minute, so the time taken are all less than 1 (ie 0.14m). I am trying to fit into distribution, but ...
3
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1answer
119 views

Convergence of likelihood implying convergence of marginal likelihood?

I will ask my question through a toy motivating example. It is well known that a Poisson process is the continuous time analog to a Bernoulli process (for example: ...
0
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1answer
19 views

Determine Number of observations required

How to determine whether the number of data required to be taken for a experiment? eg: I need to take a set of data for the noise mapping, how many should I take ?
0
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0answers
38 views

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ . find $ E(U|W,X)$

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ find $ E(U|W,X)$ and conditions under which $E(U|W=w,X=x)$ is an increasing ...
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41 views

How can I verify that variance(factor)=1 from Exploratory factor analysis results?

I am reading upon Exploratory factor analysis. One of the assumptions of the Orthogonal factor model is that $$ \sigma^2(factor)=1 $$. Reference via "Applied Multivariate Statistical Analysis-by ...
0
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0answers
3 views

How to show the given expression is geometric mean [migrated]

Let $a_1,a_2,\dots,a_n$ be any $n$ positive real numbers. Show that $$\lim_{t \to 0^+}\left[\frac1n \sum_{i=1}^{n}a_i^t\right]^{1/t}$$ is the geometric mean of $a_1,a_2,\dots,a_n$. I know ...
0
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1answer
18 views

minimizer weighted linear regression

In a regression problem, with $y=X\theta+\epsilon$ and $X$ is an $n$ by $p$ matrix the ‘weighted least squares estimate is the minimizer $\theta^{*}$ of ...
0
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6 views

Is a Bayes randomized rule always a Bayes nonrandomized rule?

Given a prior distribution on the parameter space, is a Bayes rule among all randomized rules (nonrandomized rules are special randomized rules) always a nonrandomized rule? I.e. Is a Bayes randomized ...
0
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20 views

Are these independent: the sample, randomized rule, and random variable having the prior distribution on the parameter space?

In section 1.3 of Bickel and Doksum's Mathematical Statistics 2006, the risk function of a nonrandomized rule $d$ is the expectation of loss of the rule wrt the random sample. $$ R(\theta, d) := ...
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1answer
83 views

What does “import” mean for the relation between a proposition and a theorem?

On p.171 in Bickel and Doksum's Mathematical Statistics 2006, does "The main practical import of minimax theorems is, in fact, contained in a converse and its extension that we now give͝" mean that ...
0
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0answers
6 views

Creating a normalized histogram of log intensities with GEO datasets [migrated]

I'm trying to create a density/log intensity histogram of a normalized GEO microarray dataset. I have tried doing this several different ways, but unfortunately I have had little to no luck. What is ...
5
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1answer
114 views

Cox discrete time regression model question

Cox's 1972 publication Regression Models and Life Tables links logistic regression to an extension of the discrete time proportional hazard model. I do not understand how Equation (21) in the ...
0
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0answers
20 views

PDF of the largest observation in a sample [duplicate]

Intro Let $P(X=x)$ be a Probability Density Function (PDF). Assume we were to perform $n$ observations from a population that is distributed according to $P(X=x)$ (sample size = $n$). I would expect ...
0
votes
1answer
43 views

Post Hoc test after lme [closed]

I am trying to run a Post Hoc test (glht) after a linear mixed model (lme) in R. I was ...
5
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2answers
132 views

Transformation Chi-squared to Normal distribution

The relationship between the standard normal and the chi-squared distributions is well known. I was wondering though, is there a transformation that can lead from a $\chi^2 (1)$ back to a standard ...
1
vote
2answers
43 views

'Punishment Function' in Number of Knots in Splines?

I was considering using natural cubic splines for my prediction problem when I had a thought: In Ridge Regression, you set out to minimize the equation; \begin{equation} F(X)=\lambda\sum_i ( b^2)+ ...
0
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0answers
21 views

Deriving the maximum likelihood for a generative classification model for K classes

In Christopher Bishop's book "Pattern Recognition and Machine learning", there is the following question: Consider a generative classification model for $K$ classes defined by the prior class ...
0
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0answers
14 views

EM algorithm - data imputation

Suppose we have iid sample of random vector $(X,Y,Z)$, where $x_i, y_i, z_i$ denotes corresponding realizations for the components. Assume also, that $(X,Y,Z)$ has a three-dimensional Gaussian ...
0
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0answers
37 views

Unsupervised learning algorithems to detect anomaly in waves

I have a sample of graphs (more then 10000...). that look like in the image below: I am searching an Unsupervised learning algorithems thet can help me to detect Anomaly observations. Here what i ...
0
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1answer
23 views

Fisher information always 0? What's wrong with this argument?

We exchange derivative and integral while proving that the expected value of the score is 0. Here is the proof which does so: \begin{align*} \int \left( \frac{\partial}{\partial \theta} \log ...
0
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0answers
32 views

Is the mvrnorm function in Julia unstable?

I am trying to sample multivariate samples using mvrnorm in julia. Arguments are set as: mean5 = zeros(5) cov5 = the cor(z1, z2, z3, z4, z5), where z_i is calculated as x1,x2 ~Normal(0, 1) iiid ...
6
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1answer
93 views

What are some good references on how probability theory got mathematically rigorous?

I am working on a term paper for an analysis course and I thought it would be interesting to talk about the connection between analysis and probability theory. Honestly, it would also benefit me a lot ...
8
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1answer
130 views

Why is the definition of a consistent estimator the way it is? What about alternative definitions of consistency?

Quote from wikipedia: In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter $θ^*$—having the property that ...
0
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7 views

Maximum entropy distribution >0 with vanishing probability at zero?

I know that the maximum entropy distribution if x > 0 and the mean is known is the exponential distribution. However, a large percentage of the probability for this distribution is close to zero ...
2
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1answer
112 views

Deriving the maximum likelihood for the parameters in linear regression

Notation: $\textbf{w}$ is an M-dimensional vector of parameters (including the bias parameter), $\textbf{x}_n$ is an M-dimensional vector of the features of each training example, $\textbf{t}$ is an ...
1
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14 views

Kullback Leibler divergence “efficient” upper bound

For a distribution of N values, how can I efficiently upper-bound the largest divergence between all non-negative distributions over the same random field? For example, for all distributions of a ...
1
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1answer
43 views

What statistical analysis should I use for my study?

In my psych class we were testing to see if groups of men or groups of women would be more helpful in an emergency situation (we are testing for the bystander effect), and we are testing to see if ...
0
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0answers
8 views

Weighted Standard Deviation [duplicate]

I got a problem calculating a weighted standard deviation for a municipality and its precipitation on various landcover types (LC). I got the mean values of precipitation for several days for each ...