Questions tagged [estimation]

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5 views

estimating transformation of linear function non-parametrically

Suppose I have a regression model as follows $$Y_i = f(X_i + \varepsilon_i), $$ where $\varepsilon_i$ is standard normally distributed. I want to estimate function $f(\cdot)$ non parmetrically. I ...
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22 views

Is it harder to estimate the variance of a Gaussian compared to its mean?

In various ML talks I keep hearing that variance estimation is harder than mean estimation but I never really get why the above statement is correct. Is there a theoretical argument or a published ...
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Homework Problem Consistancy of Estimator

We have to show: Let $\theta_0$ be a k-dim vector. Show, that the following statements are equivalent: (1) $\hat{\theta}_n$ is consistent for $\theta_0$. (2) For each component $i= 1,...,k$: $\hat{\...
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130 views

Why do we multiply log likelihood times -2 when conducting MLE?

When we are performing maximum likelihood estimation (MLE) to estimate parameters, the fit function is often to -2 * LL, rather than just LL. I also see this "-2LL" term expressed as "...
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Should there be a transpose in calculating the parameters of SVM

I am trying to code SVM from scratch using a small toy problem that involves five support vector values. In the code below, there are 5 support vectors arbitrary chosen and denoted by the variables <...
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1answer
22 views

Link between Maximum Likelihood and Maximum Probability [duplicate]

How can I see that the maximum likelihood approach finds the parameter values of the probability distribution that maximize the probability of the observed sample? Maximum likelihood is not the ...
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12 views

Confusion about Maximum likelihood estimation [duplicate]

How can I see that the maximum likelihood approach finds the parameter values of the probability distribution that maximize the probability of the observed sample? Maximum likelihood is not the ...
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20 views

Point estimation of parameters

Let $\theta$ be a parameter with values in $\Theta$ that should be estimated by some given data $X$. The corresponding estimate is denoted as $\hat\theta = \hat\theta(X)$. Several times I read that ...
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22 views

What-If Scenario Regression Modelling

I'm pondering a scenario involving some insurance data but this could be relevant in many fields. The idea is that I have a total count of some event. Let's imagine this count is the # of attorney ...
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1answer
14 views

Non-adversarial robustness

One measure of an estimator's robustness is the breakdown point, which tells us how many adversarial observations are necessary to make the estimator useless. However, is there a notion of non-...
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4 views

Creating Proxy variable to minimise estimation error

As part of my econometrics project, I am investigating wage structures for female workers. The econometric model I am using follows the Mincer earnings function. The dataset available includes ...
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1answer
23 views

Non linear regression on 2D features space using SVR failed

I try to estimate the execution time of some application based on the mCPUs (fraction of physical CPU) used for the run and the overall size of input data that the application is processing. The data ...
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1answer
47 views

Did I correctly apply the factorisation theorem in this example?

Suppose that we have a density $f(x,\theta)=c(\theta)\psi(x)\unicode{x1D7D9}(x \in]\theta,\theta+1[)$ and the random variable $\mathbf{X}=(X_1,\ldots,X_n)$ are independently identically distributed ...
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107 views

variance estimation using order statistics

I have four largest samples drawn from a distribution of N i.i.d Gaussian R.V. with standard deviation (Sigma) where sigma is unknown. N is known to be between 50-200. Mean is given to be 0. How do ...
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1answer
26 views

Fitting ARMA-GARCH sequentially vs simultaneously

I am interested in fitting an ARMA-GARCH model to my data. After reading a few pages online I did so sequentially by first applying ARMA and then feeding the residuals into GARCH. I then took the ...
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Nonparameteric Empirical Estimator For Stochastic Process

Motivation: If $X$ is a random-variable defined on some probability space $(\Omega,\Sigma,\mathbb{P})$ then Glivenko-Cantelli lemma guarantees that the empirical distribution $\frac1{N}\sum_{n=1}^N \...
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What's the maximum likelihood estimation of $\theta$ in this density?

Suppose we have a n-sample $X=(X_1,..,X_n)$ with a distribution $f(x,\theta)=exp(\theta - x)\unicode{x1D7D9}_{x \geq \theta}(x)$. Find the maximum likelihood estimator $T$ of $\theta$ and prove that $...
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Finding the UMVUE of $P(X_1\ge k)$ [duplicate]

Let $X_1,X_1,...,X_n$ be a random sample of size $n$ from $N(\mu, \sigma^2)$ where both $\mu$ and $\sigma$ are unknown. It is required to find the UMVUE of $P(X_1\ge k)$. In this problem I am able to ...
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Distribution of hat sigma in simple linear regression model [duplicate]

How can I prove that the distribution of estimated σ^2 (the distribution of the estimated variance of errors εi, in case of homoschedasticity) is equal to a chi-squared with n-2 df * σ^2/(n-2) ?
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How to prove that the t distribution doesn't belong to the exponential family?

Or in other words, is there anyway prove that the t distribution doesn't belong to the exponential family without going through all that calculation? Since the density has the gamma function in it ...
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Finding maximum likelihood solution for mean when data is given which share the same mean but have different variance

I have some 'X sample points say (x1,x2,x3 ...) each of the samples form a Gaussian distribution with mean 'm' and variance v1,v2, ... All the distributions have the same mean but differ in variance. ...
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37 views

MLE and MOM estimates coincide in Normal distribution

Let $(X_1,Y_1),...,(X_n,Y_n)$ be a sample form $N(\mu_x,\mu_y,\sigma^2_x,\sigma^2_y,\rho)$ population. If $n\geq 5$ and $\mu_x$ and $\mu_y$ are unknown, I want to conclude that the estimates of all ...
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23 views

Can you use a given prediction interval to find the predicted value of Y on the estimated regression line?

Suppose you don't know the estimates of a linear regression model (b0, b1 are unknown) and you are given a 95% prediction interval at x = 6 to be [5,15]. Then, since prediction intervals are ...
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1answer
45 views

Geometric Waiting Time MLE

If the time is measured in discrete periods, a model that is often used for the time $X$ failure of an item is:$$P_{\theta}[X=k]=\theta^{k-1}(1-\theta), k=1,2,...$$ where $0<\theta<1$. Suppose ...
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1answer
73 views

MLE exists and is unique for iid series

Let $X_1,...,X_n\in R^p$ be i.i.d. with density, $$f_{\mathbf{\theta}}(\mathbf{x})=c(a)\exp(-|\mathbf{x-\theta}|^a), \mathbf{\theta}\in \mathbb R^p, a\geq 1$$ where $$c^{-1}(a)=\int_{R^p}\exp(-|\...
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unbiased estimation of the variance of $p$ (proportion) of a random sample without replacement

Given a random sample without replacement of size $n$ from population of size $N$ and $p$ is the estimator of the proportion $P$. How could one show that: \begin{equation*} \frac{N-n}{N(N-1)}pq \end{...
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37 views

Finding MLE of this confusing setup of distributions

I have been given two random variables $X$ and $Y$ following exponential with means $\lambda_1$ and $\lambda_2$. Let $Z_1 = min(X,Y)$ and $Z_2$ will be 0 if $Z_1 = X_1$ and it will be 1 if $Z_1 = X_2$...
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What is the covariance between function and its argument?

For example, we have log-likelihood function $\ell(\theta)$ and maximum likelihood estimator $\hat \theta_{ML}$ obtained as $\ell(\theta)' = 0$. What would be $$Cov(\ell(\theta), \hat \theta_{ML}) ?$$ ...
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14 views

Bayes estimate of upper limit of uniform distribution with exponential prior

Let $X_{1}, . . . , X_{n} > 0$ be a random sample from $U(0, \theta)$. Suppose $\theta$ has the prior $\pi(θ) = e^{-\theta} ; \theta > 0$. Find the Bayes estimate of $\frac{1}{\theta}$ with ...
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1answer
24 views

What Cramer-Rao bound should I use?

I have been researching about the Cramer-Rao bound and I have found two inequalities: $$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
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1answer
50 views

Confidence interval for $\mu_i$ (multivariate normal): some $\mu_j$ are known, unknown $\sigma^2$ and an estimation of the correlation structure?

Say we have a multivariate normal with m dimensions (let's say that m=1,000), the mean vector $\mu$ is known only for the first 100 elements, but unknown for the remaining 900. We have a single ...
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1answer
40 views

Find covariance of estimator and derivative of the log-likelihood function

Problem: Given and estimator $\hat k$. The estimation method is unknown (so, it can be max. likelihood, method of moments or another method), however, we know that $bias(\hat k) = 0$. Let $L$ be the ...
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1answer
164 views

What is the score function of two parameters?

According to this wikipedia article, score is the derivative of the log-likelihood function. However, I don't understand what if we have two parameters? For example, the logarithm of pdf has the ...
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12 views

What are good resources/packages for implementing copulas?

Say I have bivariate data $[X_i,Y_i]$ with $i=1...n$ and suppose I know the marginal distributions of $X$ and $Y$. I wish to estimate their joint distribution. I read that copulas can be used for this....
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1answer
20 views

How to check the consistency of OLS estimator in macroeconomic models

Problem: We have a model $$C_t = a + b Y_t + e_t$$ and $$ Y_t = C_t + I_t$$ It's known that $Cov(I, e)$ is zero. A student estimates the following model: $$C_t = a + b Y_t + e_t$$ Are the estimators $\...
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1answer
65 views

How to show that $d(X)$ is unique Bayes with respect to any prior whose mean is $\frac12$ and variance is $\frac18$?

Consider estimation of $\theta$ where $X\sim \text{Bernoulli}(\theta)$. Under squared error loss, I am asked to show that $d(X)=\frac{2X+1}{4}$ is unique Bayes with respect to any prior for which $\...
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7 views

Intra-run and inter-run variability formula

I have 1 machine that can perform 15 runs with 10 measures per run (so I have 150 results in total). I would like to estimate the variability (in terms of CV) of the results in order to evaluate the ...
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1answer
66 views

Confusion about prior used in Recursive Bayes Filter

I'm currently using this thesis to understand key concepts about probabilistic inference in computer vision which is being a great source. The frame of the question is the following: Let us assume we ...
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1answer
38 views

MLE with uneasy density function

I want to find the MLE of $\theta$ with the following density. $ (1- \lvert x - \theta \rvert) \mathbb{1}_{[\theta-1, \theta+1]} (x)$, to confirm : $\widehat{\theta_{n}} = \overline{X_{n}}$ found ...
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37 views

Derive the closed form of a probability expression

Consider the following probability $$ A\equiv Pr\Big (\delta_1+v+\lambda \epsilon_1\geq \delta_2+v+\lambda \epsilon_2 \text{, } \delta_1+v+\lambda \epsilon_1\geq \epsilon_0 \Big ) $$ where $\lambda\...
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1answer
39 views

Determine probability of an event using maximum likelihood estimation

Problem I have a bag of many red and green balls. To find out the ratio between the two, I randomly picked balls with replacement. Out of the 100 outcomes, 60 were red balls and 40 were green balls. ...
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Involving data length as a decision variable in a optimization for parameter estimation

Consider I have $N$ historical data as outputs from a unknown time-varying system. Assuming within any small time interval $T$, the time-varying system can be approximated by a linear time invariant (...
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15 views

How do I choose between regression methods for inference?

Suppose my goal is to understand the relationship between variable $y$ and covariates $X$. Let's say $y$ is a rate, the number of success in $n$ trials, therefore bounded between $0$ and $1$. Now I ...
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2answers
83 views

Why do OLS and logistic regression coefficients have opposite sign?

I have data $y$ which is the rate of success in $n$ trials. I also have covariates $X$ that I want to regress against $y$ to understand the relationship between them. I tried 2 different approaches. ...
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1answer
38 views

Where plus 1 came from in variance estimation [duplicate]

While $$ \mathrm{E}(\tilde{\mathrm{y}})=\alpha+\beta \tilde{\mathrm{x}} $$ Subject is Regression Analysis and this formula is from the "Features of Estimation ". and y is a neutral variable. ...
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49 views

How are the number of actual rape cases estimated?

I see in news articles of number of rapes reported and an estimate for actual number of rape cases. I am interested in knowing about statistical techniques used for making such estimations. Also, are ...
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1answer
50 views

Practical method to do MLE for natural parameters in exponential family

I encountered the following question in my research and I hope this is the correct place to post it. I'm following the notation in this lecture note by Michael I. Jordan. Assume random vector $X$ ...
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10 views

estimating the total number of balls (n) after 2 draws of k balls

The problem is predicting the total number of balls (n) after 2 draws of k balls. After the first draw of k balls, we have marked them so we can see if we had them before. The balls are returns and we ...
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20 views

Is the mundlak estimator equal to the within estimator?

Mundlak has proposed to estimate the following correlated random effects model: $$ y_{i t}=\boldsymbol{x}_{i t}^{\prime} \boldsymbol{\beta}+\overline{\boldsymbol{x}}_{i}^{\prime} \gamma+\omega_{i}+u_{...
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18 views

Giving a job lecture on “minimax statistical estimation”. What kind of questions to expect?

As part of a job application process, I am giving a short "mock lecture" for students on minimax estimation in statistics. I will introduce the basic concept and give an example. I am able ...

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