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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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Best estimation of a fitting parameter to measured data

My goal is to estimate a parameter $\alpha_1 = (\alpha_{11}, \alpha_{12})$ which provides the best fit of certain measured data (a readout of some currents in a set of positions for a set of loads) to ...
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1answer
13 views

How to prove this Corollary regarding ratios of densities being sufficient

The following Corollary is used in "Theory of Point Estimation" by Lehmann to prove a theorem. However I'm unsure how to prove this Corollary (it's left as a problem, so proof is omitted). The ...
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12 views

How to estimate Breslow type Baseline Hazard for mixture cure model?

I have done the following bit myself: ...
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0answers
11 views

What are the effects of autocorrelation on logistic regression?

I need a simple way to estimate the probability of winning an auction as a function of bid amount. I modeled the auction using the LogisticRegression from pandas, ...
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13 views

Estimating vector error correction model - 1 or 2 steps?

I have $n$ cointegrated price series (prices of the same stock traded on different markets) and I'd like to fit a VEC model to this data: $\Delta p_t=B_1\Delta p_{t-1}+B_2\Delta p_{t-2}+\dots +B_q\...
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1answer
14 views
+50

Bunching estimation and RDD - what is the difference?

Both are quasi-experimental designs used to measure the effect of some running variable near the cutoff value. Bunching is becoming popular in economics, regression discontinuity design (RDD) is more ...
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18 views

MLE of $\mu_k$ and $\Sigma$ for the singular matrix $\{\mu_k \}_1^k$

This problem is number 4.8 from the Elements of Statistical Learning by Hastie, Tibshirani, and Friedman. Consider the multivariate Gaussian model $(X|G = k)\sim N(\mu_k,\Sigma)$, where $k$ is one ...
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1answer
48 views

MVUE is unique - wrong proof?

Here is the proof of "MVUE is unique" that my lecturer gave: Now I understand the following: The first expansion is done using the formula for the sum of correlated random variables (https://en....
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2answers
68 views

Find an estimator for $\theta$ when PDF is:

Find an estimator for $\theta$ when PDF is: $$ f(x) =(1-\theta)\mathbb{I}_{[-1/2,0)}(x)+ (1+\theta)\mathbb{I}_{(0,1/2]}(x). $$ I know that one way is to write the likelihood function then do Log-...
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3answers
160 views

How is it that an ML estimator might not be unique or consistent?

Christian H Weiss says that: In general, it is not clear if the ML estimators (uniquely) exist and if they are consistent. Can someone explain what he means? Do we not generally know the shape of ...
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1answer
99 views
+50

How to fit an autoregressive (AR(1)) model with trend and/or seasonality to a time series?

I want to test a model I have on a time series. The model is that the time series adapts to a trend $f(t)$ with a speed $\alpha$. There is also noise in the model. So, the time series is a function ...
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0answers
26 views

When to use all 3 variables in a graph to estimate a conditional expectation of 2

The title might not be perfect. But here goes: Suppose there are 3 variables $(A,X,Y)$. And they have the following dependencies : $Pr(Y,A,X)=Pr(Y\mid A,X)Pr(X\mid A)Pr(A)$ $A \rightarrow (X,Y)$ $...
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1answer
114 views

Maximum Likelihood Estimate for a likelihood defined by parts

Suppose $X_1$, . . . , $X_n$ are i.i.d random variables having pdf $$ f(x\mid\theta)= \begin{cases} \frac{4}{\theta}-\frac{4x}{\theta^2} & \frac{\theta}{2} \lt x \lt \theta \\ \frac{4x}{...
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0answers
15 views

Online estimation of drifting discrete probability

I recently come across (in a practical setting) to the following problem. Suppose I receive items from a finite set ,one at a time . At each moment one item is drawn independently from an unknown ...
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8 views

Estimating sales by extrapolating from one year to the next using the change in population and price

Would the below be a reasonable approach to extrapolate aggregate sales for item X from 2017 to 2018? Sales of item X in 2017 (in euros): 5,364 Population increase (of target population that buys ...
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0answers
13 views

The variance of the function that is composed of MLE that can't estimate analytically

I want to know how to calculate the variance of the function that is composed of MLE that can't estimate analytically. Specifically, I want to know the variance of the estimate of mean of GPD that is ...
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0answers
15 views

What properties should a variance estimator have, when we propose 95% CI?

I have a estimator $\hat{\mu}$ for $\mu$, and I know the estimator follows the normal distribution asymptotically, with the mean $\mu$. But the problem is, we don't know the true variance of $\hat{\mu}...
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1answer
9 views

A generic terminology for a given object to be estimated?

If one has to suggest, what is it one would call a given object that is to be estimated? We already have a generic term "estimator" on the one hand. When the context is clear, usually there is no ...
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0answers
19 views

Estimator for repeated sampling and fitting

Say I have a Normal distribution $\mathcal{N_1}(\mu_1,\sigma_1)$. Now I will sample $N$ samples $X_1$ from this distribution, and use estimators for $\hat\mu_2$ and $\hat\sigma_2$ to fit a new ...
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0answers
17 views

exponentially decaying information in time as independent variable [closed]

I want to set up a model incorporating the time-varying effect of information. It means, the new information effect is exponentially decayed over time and the accumulated effect of information is one ...
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0answers
8 views

Estimation of copula parameters for $C(u, v) = \min(u^a,v^b)\min(u^{1-a},v^{1-b})$

Given a bivariate copula, say $C(u, v) = \min(u^a,v^b)\min(u^{1-a},v^{1-b})$, $0 < a,b < 1$, how would we use the method of maximum likelihood to estimate the copula parameters, in this case $a$ ...
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Finding the minimum number of entries that will give me a confidence in the result

I have a big number of hotels, and each hotel has an number of price matches. There are two types of price matches : exact price match and price mismatch. The number of price matches per hotel could ...
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16 views

Would randomly permuting a quasirandom sequence successfully avoid otherwise intrinsic correlations in estimation?

Let $A$ be an $n \times n$ (Ginibre) matrix of complex-valued entries, the real and imaginary parts of which are, thus, standard normal variates, and $U$ be an independent such matrix, the rows and ...
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0answers
16 views

Appropriate Distribution for Diagonal Covariance Matrices

Let's say I have a model like: \begin{align} X\mid\mu,\Sigma_X &\sim \mathcal{N}(\mu,\Sigma_X)\\ \mu\mid m, \Sigma_\mu &\sim \mathcal{N}(m,\Sigma_\mu) \\ \Sigma_X\mid \Psi, c &\sim \...
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20 views

What is the logic behind the calculation of sample size in the attached problem?

how to arrive at this n=543..till 276.87 logic looks good. Reference Problem 7-62 Statistics for management Levin,Rubin,Rastogi,Siddiqui.
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47 views

What is this parameter estimation strategy called?

Let $X_1, X_2, \ldots, X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Consider the problem of estimating $P(X > 100)$. One way to accomplish this is ...
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2answers
98 views

Can an optimal weighted average ever have negative weights?

I've got a few measurements $\vec{x}$ for some real-world value $\hat{x}$. These measurements have some uncertainty, and are correlated. Given these estimates, and their covariances, I want to take ...
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0answers
43 views

How do I estimate $P[|v-\mu| > \epsilon]$ as a function of $\epsilon$?

Problem from: Learning from Data Say I flip $1000$ coins $10$ times each and choose three coins: $c1$ = the first coin, $c_{\text min}$ = the coin with the minimum frequency of heads, and $c_{\text{...
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0answers
11 views

Estimation of emission distribution parameters of HMM using Baum-Welch

It seems like the count-based Baum-Welch method described elsewhere is concerned only with the categorical emission distribution. In Hidden Markov Models, I have normal transition matrix between ...
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27 views

Computing standard errors after estimating mean and variance using MLE

I am trying to make estimates of mean and variance using MLE (Maximum Likelihood Estimate) for 1000 random samples following normal distribution. I started by generating random sample of size n = ...
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17 views

Uncertainty of earthquake location

Maximum of the so-called image function in the figure below interpreted as earthquake location (in XY plane in this case). The shape of this function will depend (upon other things) on frequency of ...
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1answer
28 views

Maximal observation r.v. from Binomial r.v

Supposing $W_1,W_2,...,W_m$ is an iid and has dist Binomial with parameters $(t,p)$. What is the $p(t^*\neq t)$ with estimator $t^*=\max_{i=1,...,m} W_i$? What I did: found it is equivalent in idea ...
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1answer
173 views

Deriving the Maximum Likelihood Estimation (MLE) of a parameter for an Inverse Gaussian Distribution

Given the following likelihood function $$f(y|x,\tau) = \prod_{i=0}^Nf_T(u_i-x_i-\tau) \tag{1}$$ where, $f_T(t)$ is the probability density function of an Inverse Gaussian distribution given by ...
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1answer
30 views

Estimate number of objects in an environment given agent's observations

I'm solving a reinforcement learning-like problem, where I have an agent trying to survive in a 2D room. These room contains a finite and constant number of moving objects that interact with an agent. ...
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1answer
47 views

Why, *intuitively*, in regular parametric problems, does uncertainty go down at a $\sqrt{ n }$ rate on the SE/posterior SD scale?

consider the simplest regular statistical inference problem: $( y_1, \dots, y_n | F ) \sim$ $\text{IID}$ from a cumulative distribution function $F$ on $\mathbb{ R }$ with mean $\mu$ and finite ...
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1answer
34 views

Regression with multidimensional output variable Y

Say we have an $N \times q$ matrix $Y$ with $N>q$. Also, we have an $N \times p$ data matrix $X$. We are interested in a model of $Y = X \times W + \epsilon$, where $W$ is a $p \times q$ matrix ...
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15 views

Bivariate probit with a fixed rho

In a series of papers by Altonji, Elder and Taber -- for example 1 -- they check the robustness of bivariate binary choice models by seeing what values of $\rho$ (the correlation coefficient between ...
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7 views

Hoping to find out how each factors are related to final results. What method should I use?

My current project is to find out how much accumulated current have been flown into the main motor in manufacturing some particular product. In doing so, I’ll have to map [Data 1], which consists of ...
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1answer
36 views

Jointly sufficient statistics of a multi-parameter exponential family

Let $f_X$ be a joint density function that comes from an $s$-parameter exponential family with sufficient statistics $(T_1, T_2, \dots, T_s)$ so that the density $f_X$ can be expressed as $$f_{X|\...
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19 views

Estimation of the mean and the the variance in Shapiro-Wilk test

I have started to learn about the non-parametric tests recently. And in my textbook, the introduction to the Shapiro-Wilk test ( a test of normality) is as below: " ... Let $X_1,...,X_n$ be i.i.d. We ...
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28 views

Is an unbiased estimator based off multiple complete sufficient statistics also UMVUE?

If $T(X)$ is a complete sufficient statistic such that $ET(X) = \sigma^2$, then $T(X)$ is the UMVUE estimator of $\sigma^2$. My question is, suppose $\tau (T(X),W(X))$ is an unbiased estimator of $\...
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33 views

What's the best estimator for expectation if we can draw samples iid *and* we know the likelihood of each sample we receive

Suppose that $X_1, X_2, \ldots X_n$ is a sequence of random variables on a set $S$, drawn independently according to the pdf $p : S \to [0,1]$. Part I: Given some $f : S \to \mathbb{R}$, I want to ...
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25 views

On the meaning of mutual information and on how to test the convergence of an estimation

As the result of a Molecular Dynamics simulation, I have the time series of two variables, $X$ and $Y,$ and I am interested in computing the mutual information of these two random variables. I've ...
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48 views

Variance of 2 Protocols: Sampling Coloured Balls with Dots

Suppose, we have an urn where each ball has one of $M$ colours and some balls have a dot. We would like to estimate the proportion $p$ of balls that have a dot. We have two experimental protocols: We ...
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84 views

Binomial Distribution - assumptions, parameter estimation, and probability calculations

I am working with transaction data at a adtech DMP. I believe that the number of transactions that are attributed to my company (as a proportion of the total number of transactions a client observes ...
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2answers
24 views

Linear Least square estimate of $x^3$ given $x$ and the moments

I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments E$(x^n)=\mu_n$. I need to find the ...
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1answer
30 views

regression parameter estimation for correlated variables

Usually we don't want to include correlated variables in regression model as problems with estimation and variable significance arise. I have always thought that a major problem is that the estimates ...
4
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1answer
49 views

Predictor and error are independent

I have been trying to understand the proof of the bias/variance decomposition formula, and I came across a gap that I haven't been able to fill. I will use the notation of The Elements of Statistical ...
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12 views

Bias of a function of the avarage

It's well known that $\overline{x}$ is an unbiased estimator of the exact average. Now, we let's imagine that we want to estimate some function of the average $f(\langle x \rangle)\equiv f(X)$. My ...
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1answer
41 views

Sufficient statistic for p in a binom(1,p)

this might be a stupid question but I don't really understand why the statistic $T = \sum_{i=1}^{n} X_{i}$ is a sufficient statistic for p , for $X_{1}, ... X_{n}\sim^{iid} Binom(1,p)$. Shouldn't it ...