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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10 views

How to estimate ratio of variances?

Suppose we have two time series generated by $$ x_t = \sigma_t\,e_t \\ y_t = c\,\sigma_t\, f_t $$ where $e_t$ and $f_t$ are IIDs with mean zero and unit variance. We don't know much about $\sigma_t$ ...
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15 views

Estimating the distribution of frequencies from a sampling of unique values

Let $L$ be a list of elements with repetitions, and $S$ be the set of elements appearing in $L$. I have two tools at my disposal: I have a sample of $k$ elements taken uniformly in $S$, and for each ...
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2answers
21 views

Generating parameters from lognormal or Gamma likelihood

I need to draw the gamma or lognormal parameters only from their likelihood functions. I'm using uniform prior, so the posterior distribution coincides with the likelihood functions. Is there a ...
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29 views

How do I calculate the Cramér–Rao bound of a function of a parameter?

I have, say, $Y_i \stackrel{iid}{\sim} \mathcal{B}(1, p)$ and I want to calculate the lower bound of variance of an estimator of $\theta = p(1 - p)$ (which happens to be $Var[Y_i]$). If $\theta = p$, ...
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1answer
38 views

Estimating accurately the mean of an autocorrelated bounded integer time series

I have a bounded integer time series $X_{1:\infty}$ ($1\leq X_k\leq M$), and I want to estimate the mean $$ s = \lim_{L\to\infty} \frac{1}{L}\sum_{k=1}^L X_k. $$ I'm assuming it exists and that $X_k$ ...
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1answer
30 views

Relation Between Bayesian Estimation and Maximum a posteriori estimation

Is maximum a posteriori estimation some kind of Bayesian Estimation? If yes, can you point out other Bayesian estimators? Edit: So I've come to know the following (don't know if they are correct): ...
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6 views

How to calculate sample size for testing H0: Y is related to X by a sigmoidal function?

I'm assuming I would need to specify a given functional form and conditional standard deviation for Y at each X. That would give me some parameters to be estimated and I would need to set a threshold ...
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1answer
20 views

Method For Calculating Performance from Average Scores

In a simplification of my problem, let's assume we have players who can get an integer value score $s$ considering $\{ s \in \mathbb{N} : 0 \leq s \leq 3 \}$. Same sample data for scores: Player A: ...
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1answer
40 views

Why does the parameter variance change when control variables are added to a regression model?

If I add a control variable to my regression, this changes the variances of the parameter estimates. Why is this the case? Is this because SSE increases and therefore the SSR decreases? So basically ...
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0answers
16 views

Delta Method vs. Lognormal

I have a single parameter $\theta > 0$ of a probability model I estimate with MLE on i.i.d. data. To get rid of the positivity constraint I instead estimate $\log \theta$ for which MLE gives me an ...
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1answer
42 views

How to get the maximum likelihood estimator of $U(\theta,\theta +1)$?

I know how to find the MLE for $U(0,\theta)$ but I am in trouble with this one. let $X_1,\dots,X_n$ be a random sample from $U(\theta,\theta +1)$. Consider the following three estimators for ...
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8 views

Negative of log-likelihood on test data decreases but the parameters Mean Square Error(MSE) increases. How to justify the situation?

We develop an EM algorithm to model a problem. We generate some synthetic data of the model with parameters $\Theta$. We call the data $\text{D}$ which is decomposed it into two separate sets, ...
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15 views

Estimating the effect of wrong input parameters in a model estimation

I've got a physical system that detect counts in an array of detectors. In each detector $y_i$ I expect to measure $\bar y_i = f_i(\bar \lambda)+\bar b_i$ counts. $b$ represent the vector of of counts ...
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1answer
35 views

If given statistic is unbiased estimator?

I am having trouble finding out and verifying estimators properties ( like unbiased, consistency sufficiency, efficiency ) but in this particular problem given below I have to find a constant such ...
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0answers
6 views

Naive bayes - parameters count

How many parameters do we have to estimate using naive bayes when the input features are conditionaly Independent and when they are not? Is there a formula that can fit - boolean, discrete and ...
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30 views

Analytical properties of sample quantiles in statistical packages

When studying and proving properties of sample quantiles, such as its consistency or its asymptotic normality, every text I have seen uses the standard definition of this estimators. This is, given a ...
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22 views

Probit Model: Interpretation of marginal effects if explanatory variables are proportions

How do I interpret the marginal effect of an explanatory variable that is a proportion in a probit model? For example if I get a marginal effect of 0.8 does this mean that if the proportion increases ...
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16 views

Non-random parameter estimation, alternative terminology?

I have a book "Navigation Signal Processing for GNSS Software Receivers" by Thomas Pany (2010) that describes non-random parameter estimation as a fundamental method used by GPS receivers for ...
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1answer
21 views

Deconvolution of two Gaussians

Assuming $X$ and $Y$ are two Gaussians with parameters of $\mu_X,\Sigma_X$ and $\mu_Y,\Sigma_Y$ then for their convolution we know that (reference) : $Z=X*Y$ is also a Gaussian with parameters of ...
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0answers
9 views

Nested cross validation vs. split dataset into train, validation and test for parameter selection and performance evaluation

The goal is to get the unbiased performance estimation of the 'algorithm' (or model), e.g. precision and recall. And get a final model for practical usage. From what I read online, nested cross ...
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20 views

Practicality of sparse inverse covariance matrix assumptions

For a set of $p$ datapoints in $m$ dimensional space, if the features are packed in a $p\times m$ matrix $X$, then $C = XX^T$ is the covariance matrix and $K = C^{-1}$ is the inverse covariance ...
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34 views

Relationship between CLT for estimator of sample and number of combinations for subsample?

What is the relationship between the Central Limit Theorem as applied to the expected value of an estimator of a parent sample and the number of possible combinations of a subsample used to calculate ...
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1answer
17 views

Likelihood for dependent data above a threshold

Let $(Y_t)$ a real-valued stationary Markov chain and $u$ some positive threshold. We assume that for $y>u$, $$Y_{t+1}|\{Y_t=y\}\sim\mathcal{N}(\alpha y+\mu y^\beta,\sigma^2 y^{2\beta})$$ I want ...
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1answer
89 views

How can I use bayesian reasoning to explain a Scrum team if they initial estimates were right or wrong and if the project is currently delayed or not?

Imagine a software development team estimates they are going to be able to complete 80 user stories in 5 sprints using Scrum: Sprint 1: 16 stories Sprint 2: 16 stories Sprint 3: 16 stories Sprint ...
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1answer
38 views

justification of Monte Carlo integration

I came across the following justification of Monte Carlo integration, where $p(x)=\frac{1}{b-a}$ $E[F_N]=E\bigg[ \frac{b-a}{N} \sum \limits_{i=1}^{N}f(X_i) \bigg]$ $= \frac{b-a}{N} \sum ...
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1answer
47 views

Why do we resample in bootstrap estimation? [duplicate]

Why do we need to resample from an initial set of samples when using bootstrapping? Why don't we just take fresh sets of samples from the original distribution? What is the justification behind ...
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19 views

Sampling property for linear regression estimation

Least Squares Estimation for linear regression is that, $$\hat{\beta} = (X^{T}X)^{-1}X^{T}y$$ Also, it is easy to see that, $$\hat{\beta} \sim \mathcal{N}(\beta, (X^{T}X)^{-1}\sigma^2)$$ If we assume ...
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1answer
33 views

Different methods, different confidence intervals

A definition of a confidence interval could be: A confidence interval for the parameter θ, with confidence level or confidence coefficient γ, is an interval with random endpoints ($u(X)$, ...
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28 views

Is the intercept estimation affected by multicollinearity?

Suppose I am running a regression $$x_t = \alpha + b_1y_{1t} + \dots + b_m y_{mt} + \varepsilon_t$$ where the $y_{i}$ are potentially linearly correlated (Some have an IVF bigger than 4; generally ...
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2answers
68 views

How to estimate real series from smoothed moving average?

Suppose I have an observed time series $y_t$, which I suspect has been smoothed out. It appears to be significant autocorrelation at lag 1 and 2, therefore I suppose that the observed series $y_t$ is ...
3
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1answer
116 views

MLE uniform distribution

Let $X_1, X_2, \ldots, X_n$ be a random sample of discrete random variable with Uniform distribution on set of integers $\{-\theta, -\theta+1, ... ..- 1, 0, 1, \theta-1, \theta\}$ where $\theta$ is ...
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1answer
34 views

When would you want to reduce variance?

In a sampling-estimation context, low variance of the estimate is a goal. Several things I've read suggest (though I can't quite connect the dots) that lowering variance in the data will improve the ...
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0answers
21 views

Asymptotic normality for nonsmooth objective functions

Assume that $f ({\bf x}; \theta): \mathbb{R}^p \times \Theta \to \mathbb{R}$, where ${\bf x}$ is the vector of inputs (with some distribution) and $\theta$ is the vector of parameters. Also, assume ...
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1answer
29 views

Estimate mean and variance from multiple realizations of Gaussian process

I have a certain number of realizations of the same Gaussian process. I want to get the mean and the variance of this process. How can I do that? To better explain the question lets suppose I have my ...
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30 views

Bayesian Linear Regression for Noise-Free Data (using PyMC)

I was surprised to find out that fitting a straight line into "perfect", i.e. noise-free, data, results in rather large uncertainties for the estimated parameters. Example and Estimation Results ...
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8 views

interpreting R plot of 100 t-confidence intervals on same graph. [duplicate]

I have drawn 100 sample of size 10 from standard normal distribution using r . after that for each repetition the 95% t-confidence intervals have been plotted on the same graph paper and those ...
3
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2answers
54 views

The concept of efficiency

I have some problems in understanding the concept of efficiency as related to an estimator. My sources (Mukhopadhyay, 2000 and Casella, Berger, 2002) do not treat this argument as I expected since ...
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0answers
10 views

Estimating the search space for a hand of Spades

I am wondering if it is possible to estimate how many possible permutations there are for a given hand of Spades (or maybe an average hand in general). Spades is played with 4 players and each player ...
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33 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\infty}_{\infty} f(x)p(x) dx$ (with p(x) ...
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0answers
35 views

Proof of why sample mean is best estimator for population mean

I am learning about the idea of estimators in statistics and understand that the role of an estimator is to estimate a parameter of the population distribution. In my book it says that the estimation ...
3
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0answers
28 views

Relationship beween SEs of two point estimates

Good evening everybody! There's a very odd question from an old exam in Introductory Statistics that has been preoccupying me for a couple ofhours: What is the relationship between the Standard ...
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0answers
18 views

Sample Properties for Estimation

Newbie here. I want to estimate the nationality of a person from the nationalities of his family. I am looking for ideas/pointers for the following: How can I know if his family is a good sample ...
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1answer
49 views

How should I estimate the variance using a sample (not knowing the mean)?

The unadjusted sample variance is biased, yet it has a smaller mean squared error. Also, if we assume the sample comes from a normal distribution, the maximum likelihood estimator for the variance is ...
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1answer
18 views

Efficient estimators and CRLB

An estimator is efficient if it reaches the Cramér-Rao Lower Bound and since it is efficient, it is also the UMVU estimator of the parametric function $\tau(\theta)$. But Cramér-Rao inequality and the ...
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0answers
53 views

Logistic Regression - Are estimators biased when there's low variance in X?

I am studying logistic regressions and I wonder why are estimators biased when the independent variables have low variance (maybe low variance compared to its mean, but anyway). I simulate the ...
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2answers
50 views

Uses and estimation of Student-t distribution

My question relates to a confusion between how the Student-t distribution is often documented versus how it is used. In the documentation the Student-t is used (from Wikipedia): when estimating ...
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29 views

Estimating Confidence Intervals

I have built a model that estimates the number of users who bought an item. I have data for two time periods, August 2015 and October 2015. I want to compare the number of users who purchased the item ...
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44 views

Estimate GARCH parameters using maximum likelihood pseudocode

I have to estimate the GARCH parameters using maximum likelihood in Scilab. I have tried many ways and so far nothing works properly. I have $$ x_t = \sigma_t y_t, \ \ \ \ \ y_t \sim N(\mu, \sigma) ...
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4 views

Queuing observations where only average number of active serves are observed

Related to a problem in computer science a look at a problem that can be modeled as a queuing model where infinitely many servers are available, i.e. in the general case a G/G/infinity model. I want ...
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1answer
20 views

Computing the expected value for a sample

Assume that we have a dataset $D$ having $N$ instances: $X_1,\cdots,X_N$. We select $n$ items $(Y_1,\cdots,Y_n)$ using sampling without replacement. We use $\bar{Y}$ as an estimate for $\bar{X}$, and ...