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Questions tagged [estimation]

This tag is too general; please provide a more specific tag. For questions about the properties of specific estimators, use [estimators] tag instead.

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18 views

Parametrs of the Uniform Likelihood

Consider linear scalar model $y_k = x_k + w_k$ where $w_k \sim U(a,b)$. What we can say about the probability distribution $p(y_k|x_k)$? We can show that $p(y_k|x_k)$ have uniform density, but I ...
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9 views

Estimation and detection in communication problem?

I just want to get clear conceptually about the difference between detection and estimation in terms of a communication problem. Suppose I have a source and a destination. The source transmits binary ...
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0answers
8 views

Is this the right way to use 'two-parts model with recycled prediction?'

I'm researcher in health care study. 'Two-parts model' was used in severals healthcare studies. For example, when the events(e.g. readmission) occured in specific subgroup in population, only '...
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0answers
18 views

What is the relationship between minimizing prediciton error versus parameter estimation error?

With the advent of statistical learning techniques, people are talking a lot about prediction error, while in classical statistics, one is focusing on parameter estimation error. What is the ...
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8 views

Consistency of Jackknife and Bootstrap Estimates for Simple Statistics

For both, Jackknife and the simple Bootstrap I'm supposed to proof the consistency of the variance and bias estimators. (Only for the simple case, that our statistic is a function of the sample mean.) ...
2
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1answer
33 views

Uniform(0,$\theta$) ratio UMVUE?

Let $X_i$ be i.i.d $uniform(0,\theta)$ and $Y_i$ be i.i.d $uniform(0,\lambda)$. The problem is to find the UMVUE of $\frac{\theta}{\lambda}$. My attempt has been to use the fact that $(X_{(n)},Y_{(...
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0answers
18 views

How can I concentrate out parameters entering linearly in a partially non-linear regression?

I have model $$y_i = x_i^\top\beta + \delta \exp(w_i\eta) + \epsilon_i$$ in setting up a non-linear regression problem $$\min_{\beta,\delta,\eta} \frac{1}{2N} \sum_i^N (y_i - x_i^\top\beta - \delta \...
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0answers
17 views

Recent progresses in Maximum Likelihood for ARIMA?

I am looking for recent developments in optimization for MLE and conditional least squares especially w.r.t ARIMA. I am using ARIMA to make some forecasts, in order to do that I have to find the best ...
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1answer
50 views

Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?

It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
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23 views

Find $k \in R$ such that $P\left(\max\left\{\frac{{S_x}^2}{{S_y}^2}, \frac{{S_y}^2}{{S_x}^2}\right\} > k\right)= 0.05$

Let $\overline{X}$ and $\overline{Y}$ sample means and ${S_x}^2, {S_y}^2$ unbiased estimators for the variance of 2 independent random samples of size 7 with normal distribution with mean unknown and ...
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19 views

Finding a confidence interval for shifted exponential distribution

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\sigma ,\theta)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}, x\gt \theta$$ where $\sigma \gt 0 $ and $\theta \in R$ . a) if ...
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1answer
60 views

If $X_i \sim U(\theta-\frac{1}{2};\theta+\frac{1}{2})$, show that $[X_{(1)},X_{(n)}]$ is a confidence interval [on hold]

Let $X_1,...X_n$ random sample from $f(x;\theta)=I_{[\theta-\frac{1}{2};\theta+\frac{1}{2}]}(x)$. a) Show that $[X_{(1)},X_{(n)}]$ is a confidence interval for $\theta$. b) Compute the ...
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0answers
25 views

OLS basic doubt

In a multivariate OLS model : $ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$ My estimator for $\beta_1$ is given by which expression: $\hat \beta_1 = [X_1'X_1]^{-1} X_1'Y$ OR $\hat \...
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1answer
54 views

Multiple linear regression: am I interpreting the methodology right?

This is a follow-up question to 1 and 2. So we have the normal linear model \begin{align*} \textbf{Y} = \textbf{X}\beta + \epsilon \end{align*} where $\epsilon\sim\mathcal{N}(\textbf{0},\sigma^{2}\...
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2answers
9 views

Correct way of getting generalizaton performance of a model using the whole dataset

Standard practice is to split data into a train/test set, then use the train set for hyperparameter tuning / model selection, using for example cross-validation over the whole training set. Finally, ...
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1answer
26 views

Estimate mean of a population with multiple samples when the individual sample mean is biased

I am working with datasets of grades going ~15 years back for different classes. I am trying to determine if there is a difference in the average grade for odd years compared to even years. There is a ...
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0answers
15 views

Model or equation for deciding who gets to play

I am a team manager for an amateur sports team. We play 2 games a week and use a game roster of 40 people. However, our team roster includes around 55-60 people and there is competition for spots for ...
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0answers
39 views

Density Estimation Efficieny

My Question Let's say a set training samples like D from a discrete distribution like p(x) over a discrete variable vector like x is available. We don't have any prior knowledge about the form of p(x)...
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0answers
29 views

Consistent estimator and distribution function

a general question: If the distribution function $F_n$ of some estimator $T_n$ suffices \lim_{n \rightarrow \infty} F_n(x) = 1 \text{ or } 0 \forall x}. Does that imply that $T_n$ is consistent? I ...
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0answers
17 views

unbiased estimator using Walsh Averages?

In class, the professor said the median of the Walsh averages, $\mu _w$, can be used as an estimator. Further $$\dfrac{\textrm{variance of}\;\mu _w}{\textrm{variance of} \;\bar {Y}} =\dfrac{1}{ARE} ...
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0answers
18 views

Gaussian process where the output is constrained to be 0 or greater

I am trying my hand at simple GP regression and in my case, the output variable can be greater than or equal to zero. How can one constrain GPs, so that the predictions always stay in the specified ...
0
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1answer
21 views

OLS - regression: how to interpret it? [duplicate]

I'm running an OLS and was wondering if the 'Estimate' in my SPSS output is the same as the beta coefficient in a linear regression? Are there specific assumptions required to run an OLS? I have age, ...
1
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1answer
28 views

Calculating bias of ML estimate of AR(1) coefficient

I am trying to develop adjustment factors for maximum-likelihood estimates of the auto-regression coefficient in an AR(1) process. By simulation I have discovered that the estimates are positively ...
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0answers
17 views

Estimate frequency based on given sample

I have a list of users (~25 million) and a dataset containing events created by the users in a specific time window (eg all tweets of all Twitter users that happened in a month - on avg I have 100K ...
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0answers
12 views

How to estimate the standard error?

I have to estimate this expression: $a=1- \mathbb{E}[D | Z=1]$. $D$ and $Z$ are binaries variables. I know that we can rewrite a as $$\begin{align*} a &= 1 - \frac{\mathbb{E}[DZ]}{\mathbb{P}[Z=1]...
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2answers
30 views

Score function of poisson distribution

I have a stupid question I haven't figured out. So when counting the score for poisson distribution I get the log likelihood $$S(\mu ) = \frac{\partial \ell(\lambda )}{\partial \lambda } = \sum_1^n \...
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0answers
22 views

Interpreting the likelihood estimate of transformed data

There is data $X(t)=[325200,500000,240000,130000,1200,10,10]$ where $t$ is time, $t=[0, 1 ,2 ,3,4,5,6]$ and I am fitting to these data assuming $X(t)$~Poisson$(X_0e^{\theta t})$ to estimate the ...
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0answers
14 views

Estimating weights in the assignment problem

How would you learn a function with the emphasis on feature interactions? I have the standard assignment problem: $$ \max_{x_{ij}} \sum_{(i, j)} w_{ij} x_{ij}, $$ where $w_{ij}$ is the weight of ...
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1answer
32 views

ARMA/GARCH estimation with standard errors

I want to estimate the parameters and standard errors of the following ARMA/GARCH model: $$y_t = a + by_{t-6} + cy_{t-8} + d\epsilon_{t-1} + \epsilon_t $$ $$\sigma^2_t = \omega + \alpha \epsilon_{t-1}^...
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0answers
27 views

How does the frequentist approach to probability estimation work when the number of outcomes is greater than 2?

I've read Checking_whether_a_coin_is_fair and I'm trying to find a resource which generalizes the frequentist approach to the case where there are $k$ distinct outcomes and $n$ trials. Can someone ...
1
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1answer
50 views

Estimate value with binomial distribution [closed]

We have some compound A diluted in a solution. In 200 trials, we find that when we mix $1$ $\mathrm{mm}^3$ our solution of A with some amount of some compound B, we get a reaction 185 times. How can I ...
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0answers
34 views

Maximum Likelihood with 2 Samples

I would like to know if my workings are correct. I have for $\theta\in(0,1)$, $$ f_\theta(x) = \begin{cases}\theta^2&,\text{ if }-1 \leq x \le 0 \\ 1- \theta^2 &, \text{ if } \;\,0 \leq x \...
1
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1answer
60 views

MLE 2 coins, 6 flips

I am a bit confused on the way to go for this exercise could you please help me : You have five coins in your pocket. It is known a priori that one coin has heads with probability 0.4 and the other ...
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0answers
20 views

Does intended model use affect Bayesian parameter estimation?

Bayesian parameter estimation results in a posterior distribution for model parameters. The user may or may not be interested equally much in all properties of the distribution. Perhaps the user ...
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0answers
11 views

Problem with Hurst exponent estimation for ARFIMA models

guys. I try to realize my ARFIMA model identification script in R. I try to find the best method for unbiased Hurst exponent estimation (fractional difference parameter could be found as Hurst - 0.5) ...
1
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1answer
36 views

Consistency of the maximum likelihood estimator for the variance of a normal random variable when the parameter is perturbed with white noise

Let $X_1, X_2, \dots , X_n$ be normally distributed independent observations with known variance $\sigma^2$ and mean respectively given by $\mu_i = \mu + \epsilon_i$ where $\epsilon_i$ is white noise, ...
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0answers
16 views

How to get confidence interval for a z-score/percentile of an individual observation?

Let's say I want to calculate a z-score or a percentile of one subject with respect to a target population. To calculate it is easy, however I have only a sample from this population, so although I ...
2
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1answer
39 views

Estimate parameters of a normal knowing the density function

I have a matrix with 3 columns. The first column contains values of a variable $x_1 \in [-1,1]$. The second column contains values of a variable $x_2 \in [-1,1]$. The third column contains a variable $...
2
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1answer
29 views

A reference request for the consistency of the parameters of an autoregressive process estimated through maximum likelihood

Let $y_t$ be modeled as an auto regressive process of order 1, that is $$ \begin{aligned} y_{t} &= \alpha + \beta y_{t-1} + \epsilon_{t}, \\ \epsilon_{t} &\stackrel{iid}{\sim} N(0,1). \end{...
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1answer
28 views

Bayesian approach: ignoring the denominator leads to the conditional density equaling the joint density? [duplicate]

I know there are a lot of questions here about ignoring the denominator in a Bayesian approach, but I don't think mine is a duplicate of any of them. I am reading the book "Pattern recognition and ...
3
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1answer
67 views

What does it mean to “non-parametrically” identify a causal effect within the super-population perspective in causal inference?

I am wondering, within the context of causal inference, what it means to "non-parametrically" identify a causal effect within the super-population perspective. For example, in Hernan/Robins Causal ...
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0answers
27 views

Is it better to estimate a population parameter directly or via a proportion?

Situation I'll describe the situation through a made-up example. Suppose there is a village of 1,000 households where people collectively bought $F$ kg of food over some period of time, and ended up ...
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1answer
79 views

Gamma distribution parameters estimation [closed]

I have a set of samples taken from a population distributed with a Gamma distribution, so \begin{equation} f_X(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} \end{equation} I should ...
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1answer
54 views

Find UMVUE of $p^3$

Let $X_1, X_2, ..., X_n$ be a random sample from $Binom(1, p)$. I'm trying to find the UMVUE of $p^3$. Some thoughts: Apparently, $\bar{X}^3$ is not the answer, although it's the MLE of $p^3$. For ...
3
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1answer
212 views

MLE of the unknown radius

Consider this question, Suppose that $(X_1, Y_1),(X_2, Y_2), . . . ,(X_n, Y_n)$ are the coordinates of $n$ points chosen independently and uniformly at random within a circle with center $(0, 0)...
1
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1answer
40 views

Showing estimator is biased without assuming $X^TX$ is invertible?

I would like to show that the ridge regression estimator: $$\beta^R = (X^TX+\lambda I)^{-1}X^T Y$$ is biased, where $Y \sim N(X\beta, \sigma^2 I)$. If we assume that $X^TX$ is invertible, this can ...
4
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1answer
33 views

Should MLE estimation always be using penalizers?

I am referring to the family of estimation techniques like MLEs, least-squares, etc., that an l2 penalizer/regularizer can be added to. I'm not interested in NHST, but just estimation (say, of some ...
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1answer
24 views

Good Estimates of the Square of Bernoulli Probability of Success?

I am trying to understand the metrics of a good estimator. For example, the Bernoulli probability of success takes the parameter p. But for X1...Xn iid Ber(p^2) how would you estimate the p^2. How ...
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0answers
23 views

Minimizing the expected loss or the mean risk?

Is there a reason why one should choose to pick his Bayesian decision minimizing the expected loss or the mean value of the risk function? The expected loss function \begin{gather} \int \mathscr{L}(\...
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0answers
24 views

Scaling and Transformation of histograms

Basically I have some histograms of images which I want to scale and transform so that if I apply a single threshold then I would get a constant alarm rate.How can I go about doing this.Below are 2 ...