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.
0
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0answers
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 ...
0
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0answers
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 ...
0
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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 '...
1
vote
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 ...
0
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0answers
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
votes
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_{(...
0
votes
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
$$\min_{\beta,\delta,\eta} \frac{1}{2N} \sum_i^N (y_i - x_i^\top\beta - \delta \...
1
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0answers
17 views
Recent progresses in Maximum Likelihood for ARIMA?
I am looking for recent developments in optimization for MLE and conditional least squares especially w.r.t ARIMA.
I am using ARIMA to make some forecasts, in order to do that I have to find the best ...
1
vote
1answer
50 views
Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?
It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
0
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0answers
23 views
Find $k \in R$ such that $P\left(\max\left\{\frac{{S_x}^2}{{S_y}^2}, \frac{{S_y}^2}{{S_x}^2}\right\} > k\right)= 0.05$
Let $\overline{X}$ and $\overline{Y}$ sample means and ${S_x}^2, {S_y}^2$ unbiased estimators for the variance of 2 independent random samples of size 7 with normal distribution with mean unknown and ...
0
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0answers
19 views
Finding a confidence interval for shifted exponential distribution
Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$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 ...
0
votes
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 ...
0
votes
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 \...
0
votes
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}\...
0
votes
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, ...
1
vote
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 ...
0
votes
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 ...
0
votes
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)...
1
vote
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 ...
0
votes
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} ...
0
votes
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
votes
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
vote
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 ...
0
votes
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 ...
0
votes
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
$$\begin{align*}
a &= 1 - \frac{\mathbb{E}[DZ]}{\mathbb{P}[Z=1]...
2
votes
2answers
30 views
Score function of poisson distribution
I have a stupid question I haven't figured out.
So when counting the score for poisson distribution I get the log likelihood
$$S(\mu ) = \frac{\partial \ell(\lambda )}{\partial \lambda } = \sum_1^n \...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
32 views
ARMA/GARCH estimation with standard errors
I want to estimate the parameters and standard errors of the following ARMA/GARCH model:
$$y_t = a + by_{t-6} + cy_{t-8} + d\epsilon_{t-1} + \epsilon_t $$
$$\sigma^2_t = \omega + \alpha \epsilon_{t-1}^...
1
vote
0answers
27 views
How does the frequentist approach to probability estimation work when the number of outcomes is greater than 2?
I've read Checking_whether_a_coin_is_fair and I'm trying to find a resource which generalizes the frequentist approach to the case where there are $k$ distinct outcomes and $n$ trials. Can someone ...
1
vote
1answer
50 views
Estimate value with binomial distribution [closed]
We have some compound A diluted in a solution. In 200 trials, we find that when we mix $1$ $\mathrm{mm}^3$ our solution of A with some amount of some compound B, we get a reaction 185 times. How can I ...
1
vote
0answers
34 views
Maximum Likelihood with 2 Samples
I would like to know if my workings are correct.
I have for $\theta\in(0,1)$,
$$ f_\theta(x) = \begin{cases}\theta^2&,\text{ if }-1 \leq x \le 0 \\ 1- \theta^2 &, \text{ if } \;\,0 \leq x \...
1
vote
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 ...
1
vote
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 ...
0
votes
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
vote
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, ...
0
votes
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
votes
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
votes
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{...
0
votes
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
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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
votes
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
vote
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
votes
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 ...
0
votes
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 ...
0
votes
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}(\...
0
votes
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 ...
