Any statistical process which seeks to approximate an unknown value, such as a distribution, a point estimate (e.g. mean), or confidence interval.

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4 views

Help in state and parameter estimation when a time series model excited by pseudo binary input

An IIR system is excited by a pseudorandom binary signal $z_n$. The output of the system is corrupted by zero mean additive white Gaussian noise and this is observed, i.e., $y_n = \mathbf{h^Ty_{n-1}} ...
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0answers
15 views

Help in solving the expression in EM

An FIR system is excited by a nonlinear deterministic signal $z_n$. The output of the system is corrupted by zero mean additive white Gaussian noise and this is observed, i.e., $y_n = \mathbf{h^Tz_n} ...
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8 views

How to apply AIC to a situation where the mean of a multivariate normal is a 0-1 d-dimensional vector with exactly k 1's

I am trying to apply AIC to estimate mean in the following case: Let us consider that I have $n$ random variables $X_1, \ldots, X_n$, drawn i.i.d. from a normal distribution of mean $\mu\in\{0,1\}^d$ ...
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1answer
35 views

Bootstrap and MonteCarlo Method

I am trying to make sense of the bootstrap method. I am studying on Rice, "mathematical statistics and data analysis" Here it is its explanation of the bootstrap method: Imagine for the moment ...
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4 views

Dependence of PDF of LLR of symbols

I have a system model with $y=hs+\sum_i^n gx+n$ where h is rayleigh fading desired channel, g is interfering channel x is interfering symbols. $\hat{s}=w*y$ where w is MMSE filter. On what factor pdf ...
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31 views

Matlab: Problem in implementaion using toolbox while learning Kalman FIlter

I have AR(1) model with data samples $N=500$ that is driven by a random input sequence $x$. THe observation $y$ is corrupted with measurement noise $v$ of zero mean. The model is $y(t) = ay(t-1) + ...
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52 views
+50

Regression, Loss Function, Estimation Dilemma

I am trying to have an in depth understanding of predicting the value of a random variable given additional information. Suppose that $Y$ in $R$ and and $X$ in $R^n$. We would like to have a best ...
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1answer
25 views

OLS parameter estimation of an expression?

During my research for a class, I came across a paper that said they estimated an equation using OLS. But the parameter they were estimating appeared to be an expression that looked like this (not the ...
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0answers
18 views

Methods of Proving that a UMVUE does not exist?

Are there efficient methods of showing when a UMVUE does not exist? I can think of the trivial case when no unbiased estimators exist at all. But that's not really interesting. I feel like this ...
2
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2answers
42 views

How might Google go about estimating and updating traffic speeds?

This is, I guess, a specific example of a wider class of problem, one to which there must be a well-established solution, but which I, as a relative layman when it comes to statistics have thus far ...
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1answer
19 views

Deriving the common LIML estimator from first principles

David Hendry (1976) comments that deriving the LIML estimator is hard. I tend to agree. Guido Imbens has a nice expression here which reads \begin{eqnarray} \hat{\beta}_{LIML} = (X'(I - \lambda M_Z) ...
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0answers
15 views

Loss function for rank deficient covariance matrices?

I'm trying to compare the efficiency of different estimators of the covariance matrix of a particular type of multivariate normally distributed data. This comparison, as well as the estimation process ...
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1answer
28 views

Difference between restricted and unrestricted parameter space in MLE

I searched on the internet but I could not find any clues about my question. Can anyone just simply tell what is the difference between restricted and unrestricted parameter space in MLE? I have used ...
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1answer
21 views

Confidence interval and confidence region

Could you please tell me what is the difference between confidence interval and confidence region in the following sense? For example, we have s multiple linear regression model. For individual ...
2
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1answer
28 views

mle estimate for standard deviation of t-distribution

For a Student-t distribution, $t_{\nu}\left(\mu,s^2\right)$, let $\hat{s}$ be mle of scale and $\hat{\nu}$ be the mle of degrees of freedom. Functional invariance of mle implies that any linear or ...
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0answers
29 views

Generalised Tobit for clustered data (type 2?)

I would like to make a Tobit estimation where my dependent variable is stadium attendance, but there are observations from 18 different stadiums (different capacities). My thoughts are it may be type ...
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0answers
33 views

Huber's M estimator for contaminated Gaussian noise

Huber discussed in this seminal paper "Robust Estimation of a Location Parameter" link that if we have some observations $x_i$ as follows: $$y_i = \theta + \nu_i, ~~i=1,\cdots,N, \tag{1}$$ where ...
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0answers
16 views

Fisher information matrix of an arbitrary statistic

I have a proof (that is a bit long) that shows that for a single parameter $\Theta$, $J_\mathbf{X} (\Theta) \geq J_\mathbf{T} (\Theta)$ where $J_\mathbf{X} (\theta)$ is the fisher information of the ...
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0answers
13 views

Fitting multiple power laws, Zipf's law in the real-world

As a preface, the following questions are related: How to calculate Zipf's law coefficient from a set of top frequencies? How to estimate parameters for Zipf truncated distribution from a data ...
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1answer
24 views

Time Series Unobserved Components Model

I have real price data for 55 years and want to study its trends. for this i am trying to estimate the Unobserved Components (UC) Model. Which software will be better eviews or stata? Also what are ...
2
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0answers
13 views

Contaminated mark-recapture: estimating set size from sampled subsets

Someone poured marked balls in my urn! Simplistically, I think this is a capture-recapture problem where, after drawing and marking balls from the urn, somebody added an unknown number (approx 25% of ...
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1answer
27 views

Parameter Estimation vs Inference Error

I am having trouble reconciling (or maybe even understanding properly) the following issues: We have a dataset. We hypothesize a functional form for probability density. Then we estimate the ...
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0answers
19 views

Comparing Two Markov Processe with Known Transition Rate Matrices

I have a Markov process (Continuous time) with the transition matrix "A". I also have an estimation of the transition matrix named "B". How can I understand whether B is a good estimation for A. I ...
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1answer
38 views

RMSEP vs RMSECV vs RMSEC vs RMSEE

I am getting real confused now, What is the difference between, RMSEP (Root Mean Square Error of Prediction), RMSECV(Root Mean Square Error of Cross Validation), RMSEC (Root Mean Square of ...
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1answer
31 views

Given a sample of size n from a normal distribution, estimate the probability of picking a value larger than X from this distribution

Say I picked 10 random samples from a normal distribution with unknown parameters: 1.7, 2.6, 3.0, 4.4, 1.6, 2.1, 2.4, 2.7, 5.2, 3.3 What is the probability that I will pick a value larger than ...
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34 views

Is a coin fair? [duplicate]

Is a coin fair ? In other words, does a coin come up $50$% heads ? If a coin is unfair, then it would not come up $50$% heads. My thoughts : let's first identify the population and the parameter ...
2
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2answers
24 views

Estimation of probability mass function using finite samples [closed]

Suppose $X_1, X_2, \dots, X_N$ are $N$ random samples of a discrete probability distribution such that $X_i \in \{1, 2, \dots, K\}$. The probability distribution $p$ used for sampling is ...
2
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0answers
26 views

Estimating variances in orthogonal regression

In orthogonal regression it is assumed that both variables have noise. I'm interested in the simplest possible case. That is, I have a very large number of data points $(X_1,Y_1), ..., (X_n,Y_n)$. ...
3
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1answer
33 views

Estimating Size of a Set based on two Overlapping Subsets

I've searched everywhere for a similar question and many things come close but are not the same. I'm looking for a way to estimate the size of a set if two partially overlapping subsets are known ...
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0answers
14 views

Forecasting with a VAR estimated by GLS versus OLS

Suppose I have a VAR model with different regressors in different equations (this could be due to restricting some coefficients of a full VAR($p$) model to zero or having some different exogenous ...
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0answers
17 views

Estimating VAR by GLS versus OLS: efficiency

Suppose I have a VAR model with different regressors in different equations (this could be due to restricting some coefficients of a full VAR($p$) model to zero or having some different exogenous ...
1
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1answer
35 views

Estimating the true distributions from a sample of distributions

I am having a hard time formulating the following problem. Consider a company that runs a survey across several cities in the US to estimate the percentage of right-handed people and left-handed ...
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16 views

Estimating a complete income distribution from decile values

I haven't be able to find an answer to this so any suggestions would be really useful. I've been asked to post stratify a survey and I'm currently trying to do this for gross personal income. The ...
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0answers
28 views

Multivariate Cramér-Rao inequality: intuition for positive semidefiniteness

Here's what Wikipedia says about the Multivariate Cramér-Rao inequality: If $\boldsymbol{T}(X)$ is an unbiased estimator of $\boldsymbol{\theta}$, then the Cramér–Rao bound reduces to ...
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0answers
7 views

Estimating the covariance matrix in LDA ESL

I am reading Elements of Statistical Learning on LDA. On page 109, it talks about how we need to estimate parameters for Gaussian distribution. But why do we use this estimate for the covariance ...
3
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1answer
43 views

When would I be interested in a function of the parameter $\theta$ (i.e. $g(\theta)$) vs only $\theta$?

Let $\theta$ be an unknown parameter that describes the distribution of the data $X$. In the following sense: $$ X \sim P_{\theta} $$ I was learning about estimators and found the following quote on ...
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1answer
41 views

probability distribution estimation from correlated samples [closed]

I am looking to solve the following estimation problem. Consider a blackbox where (given below) given an scalar input $X \in \mathcal{X}$, its $N$ observations, $Y_1, \cdots, Y_n \in \mathcal X$ are ...
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0answers
27 views

When is the MLE equivalent to the MM estimator

I know that in natural exponential families the general method of moments estimator we get from working with the sample and the sufficient statistic gMM is equivalent to the maximum likelihood ...
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0answers
24 views

Estimate of linear problem

I know that $xL(y) = m_x+K_{xy}K_y^{-1}(y-m_y)$. I have that the $E[x/y]=y^2$ and also $y\sim N(0,s^2)$. How can I calculate the linear estimator?
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34 views

Doubt: Is the parameter a random variable?

My problem is that I am unable to determine if the measure $\theta$ is random variable or deterministic. That is why I have mentioned the following references of the definition of $\theta$. Please ...
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0answers
40 views

Maximum likelihood in the GJR-GARCH(1,1) model

In the standard GARCH(1,1) model with normal innovations $\sigma^2_t=\omega+\alpha\epsilon^2_{t-1}+\beta\sigma^2_{t-1} $ the likelihood of $m$ observations occurring in the order in which they are ...
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1answer
23 views

Consistency of density estimation under marginalization

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be samples from some unknown distribution $p(X,Y)$ and $\hat{p}(X,Y)$, $\hat{p}(Y)$ density estimates of the joint and marginal distributions (i.e., for the estimation ...
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1answer
22 views

Probability Distribution of Binary Operation Applied to Elements of Two Populations

Given two populations from which one can draw finite samples, how does one estimate the probability density of the result of some binary operation applied to elements of the two populations? The ...
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0answers
19 views

How to estimate density of multi dimensional data

Doubts are based on MLE of intrinsic dimension by E. Levina. Please correct me where wrong. The Authors propose ML estimator for the dimension $m$ using the nearest neighborhood distance information. ...
3
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23 views

Estimating parameters in a truncated binomial distr

I would like to find the estimates of the parameters in a truncated (at zero) negative binomial distribution.Suppose $Z$ has this distribution with parameters ($\alpha,\beta$). (The parametrization ...
2
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0answers
30 views

Antithetic Variables Approach

Given the function $\sqrt{x}/\sqrt{x+1}\exp(x^2)$, I am trying to approximate the integral over $[1,5]$ using the antithetic variables approach with five intervals. First, from my understanding of ...
2
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0answers
58 views

Finding the MLE of the Pareto distribution and distributions

The Pareto distribution $P(a,c)$, with positive parameters $a$ and $c$, has density function $$ p(x;a,c) = \frac{ac^a}{x^{a+1}} $$ for $x \geq c$. Then, if $X_1, \dots, X_n$ is a random sample from ...
3
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1answer
80 views

How would you fit this model?

Section 2.1 of "Exegeses on linear models" (p. 3) describes a local linear model based on a second-order Taylor expansion. How would I go about fitting the model? I see that the first two terms ...
4
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1answer
34 views

Determining actual number of observations in a dataset

I have two datasets one is a dataset with doctors in which I have the procedures they have performed at a given hospital where the actual number of procedures is not captured by this data since it is ...
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1answer
23 views

estimation of correlation coefficient of reduced data

Suppose (X,Y) $\sim$ Bivariate Normal Distribution. We know only the signs of X's and Y's. How can we estimate the correlation coefficient $\rho$ using this reduced data?