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Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

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1answer
24 views

Consistent estimator problem

The Problem Check if estimator $Y_n=max(X_1,X_2,...,X_n)$ where $X_1,X_2,...,X_n$ ~ $U[0,X]$ with parameter $X$ is consistent or unbiased. We also assume that $X_1,X_2,...,X_n$ are independent. What ...
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70 views

Conditional probability with MLE of Poisson variable

I'm having some trouble with this study question and would appreciate any help. This may be a duplicate but I have not been able to find any others. Question: Leaves of plants are examined for bugs. ...
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16 views

Regressions with two independent variables where one is part of another.

I am doing simple regression analysis where I use one-way fixed effect model to estimate the effects of two variables on dependent variable. The question I am asking is how to interpret these two ...
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6 views

Estimation of eigenvalues for online convergence estimation

Given some matrix $A$ of the form: $$ A \equiv \left(H^\mathrm{H}\Sigma H + \Lambda\right) $$ with $\Sigma$ and $\Lambda$ full-rank diagonal real matrices, and $H$ a large rank deficient not-...
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1answer
37 views

How many rain drops in a storm?

According to Rain Wiki : Heavy rain — when the precipitation rate is > 7.6 mm (0.30 in) per hour,[106] or between 10 mm (0.39 in) and 50 mm (2.0 in) per hour According to USGS Water Science School ...
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17 views

Finding the bias of an estimator

Consider the following model: $$ y_i = a + b x_i + c z_i + w_i, $$ where $a,b,c$ are unobserved fixed parameters, $x_i$ and $z_i$ are fixed in repeated samples. Assume also $\mathrm{E}[w_i] = 0$ for ...
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1answer
41 views

Proof of Estimation Lemma (ML inequality) in Complex Analysis

I'm having trouble seeing why the estimation lemma used in complex analysis is true. I didn't get much out the Wiki article or anywhere else, so I'm asking for an explanation or proof here. ...
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6 views

Consistency of bootstrapping asymptotic variance

Let $(X_n)_{n \in \mathbb{N}}$ be an i.i.d. sequence with distribution $F$ and let: \begin{align*} U_n = T(X_1,\dots,X_n) \end{align*} be an estimator for some quantity $\theta$. Suppose that: \begin{...
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17 views

Non-parametric estimation for population distribution

I have a question: how can I estimate the population of the distribution when I only have the sample observations? After drawing the histogram, the sample distribution does not look normal, and the ...
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2answers
29 views

Showing $\hat{\theta} = \frac{x_1 + 2x_2 + x_3}{4}$ is a not a sufficient estimator for the mean of a Bernoulli-distributed population

Suppose $x_1$, $x_2$, $x_3$ are independant observations from a Bernoulli-distributed population with parameter $\theta$. I want to show that $$\hat{\theta} = \frac{x_1 + 2x_2 + x_3}{4}$$ is not a ...
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Find posterior distribution given beta prior

I have the following question: The output of a certain integrated-circuit production line is checked daily by inspecting a sample of 100 units. Over a long period of time, the process has maintained ...
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16 views

Estimate efficacy of shake to remove hair

A hairy man wants to get rid of all of his hair. He recently came up with a new technique to shake his head (three narrow circles followed by a rapid left-rgiht shaking), and he wants to test the long-...
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1answer
22 views

second moment of exp distributed rv and find a sequence of real numbers such that the estimator is unbiased

Let $ X_n $ be a sequence of iid $ \exp(\lambda) $ distributed rv´s with $ \lambda>0 $. a) We know that $ E[X_i] = \frac{1}{\lambda} $. Show that: $ E[X_i^2] = \frac{2}{\lambda^2} $. b) Compute $...
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Prove that the estimator $\delta(X) = X$ is not an admissible estimator when estimating the mean of a Gaussian

Let us observe a data point $X$ sampled from $N(\theta, 1)$ where $\theta \in \mathbb{R}^+$. We will consider squared/quadratic loss, and let our estimator be $\delta(X) = X$. How can I show that this ...
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1answer
30 views

Consistency of Maximum Likelihood Estimator for Gaussian R.V with Equal Mean and Variance

tl;dr: What is wrong with this MLE estimator? Does it not satisfy the conditions for consistency or did I make a mistake in the calculation. I am trying to compute the MLE estimator for parameter ...
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M-estimators: which function is more resistant to outliers? Huber or Bisquare?

I'm studying about M-estimators. The question is: Which function is more resistant to outliers, Huber or Bisquare? In the article of John Fox & Sanford Weisberg, October 8 2013, I read that "The ...
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1answer
36 views

Is Square of unbiased estimator is unbiased again?

I am kind of new in here. I have a question that bothering me that the square of an unbiased estimator is an unbiased estimator. I know that it is not the case but how I can prove that? Using $V(o) =...
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2answers
47 views

Maximum Likelihood Estimate with different parameters

Suppose that X and Y are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator of $\theta$ $$ \tilde{\theta} = k_1 X + k_2 Y $$ where $...
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Show that T achieves the Cramer Rao lower bound

Problem Statement: Consider $T$ to be an estimator of $\theta$. Show that $T$ achieves the Cramer Rao lower bound if and only if $Z$ is a linear function of $T$ $Z=a(\theta)T+b(\theta)$ ...
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50 views

Maximum Likelihood Estimator for a exp(1/$\theta$) distributed rv

Let $n \in \mathbb{N}, \, X_i: (0,\infty)^n \rightarrow (0,\infty)$ so that $ X_i(x_1,...,n_n) = x_i \, \forall i \in \{ 1,...,n \}.$ Let $ ((0,\infty)^n, B((0,\infty)^n), (P_\theta)_{\theta \in (0,\...
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1answer
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Which of the following are consistent estimators.

Let $X_1,X_2,...X_n$ be random sample from $U(\theta,0)$ where $\theta<0$. Let $T_n=\min(X_1,X_2...X_n)$ the which of the following estimators are consistent estimator of $\theta$ $A=T_n$ ...
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2answers
50 views

Integral estimation - I am being an…

I would like to show $$\int_{x}^{\infty} \exp^{-\frac{1}{2}y^2} dy \leq x^{-1}\exp^{-\frac{1}{2}x^2}$$ I have tried integrating by parts and dropping the negative part but I didn't make it work. ...
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1answer
55 views

Finding the condition on $k_1$ and $k_2$ of an unbiased estimator

I'm taking a statistics course and am asked the following : Suppose that $X$ and $Y$ are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the ...
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110 views

How many prime numbers contain strictly increasing digits?

This was posed as an estimation problem - I'd be interested in both more accurate approximate methods (than my underestimate of 74 in the answer below) and a check of my exact answer (100, already ...
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1answer
10 views

norm of difference of similar matrice

Let $a,b \in C^n$; $A, B\in C^{n\times n}$. If $A$ and $B$ are similar matrices, i.e. there exists nonsingular $S\in C^{n\times n}$ such that $B=S^{-1}AS$, is it possible to proof an inequality in the ...
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14 views

How to write a function in R that will obtain this product?

Kaplan-Meier estimator: I've just started using $R$, so I don't understand how to build this function $($it is modified Kaplan-Meier estimator$)$. It should be something like: ...
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1answer
72 views

Show that $x \left(\sum\limits_{n\leq x} \frac{\ln(n)}{n^2}\right) +\theta(x)=O(x)$

I know that $$x\left(\sum\limits_{n\leq x} \frac{\ln(n)}{n^2}\right)+\theta(x) \leq x\left(\sum\limits_{n\leq x} \frac{\ln(n)}{n}\right)+\theta(x)=x\frac{1}{2}\left(\ln{x}\right)^2+Cx+O(\ln x)+O(x)$$ ...
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Calculate the confidence interval of parameter of exponential distribution with summarySE in R?

I am trying to calculate the confidence interval for a set of data with the assumption they follow Exp dist. To achieve this, I am merging this with this in R, but does not work as I am not very ...
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How many samples do I need to estimate the mean of a power law distribution?

Take the following power law distribution $$ p(x) = \frac{\alpha-1}{x_{\rm{min}}} \left(\frac{x}{x_{\rm{min}}}\right)^{-\alpha} $$ with $\alpha > 3$ so that the mean and the variance of the ...
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2answers
350 views

Can I estimate this integral like that?

I have the following integral $ \int_{2}^{\infty} \frac{1}{\sqrt[3]{x^{3}-1}} d x $ and I should solve it without calculate it directly. So if I find two function which is smaller and bigger than $ \...
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Kalman Filter with same A, B and C, D matrices

I am trying to work on a toy problem of EKF which consists of a Kinematic Model of a bicycle. The model is as follows:Kinematic Model I have linearized the model with inputs as ...
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2answers
39 views

Showing weighted average is consistent estimate

Here's the problem statement: Let $X_1$, . . . , $X_n$ be independent random variables with common mean $\mu$ and variances $σ_i^2$ . To estimate $μ$, we use the weighted average $T_n$ = $\sum_{i=1}^n ...
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1answer
54 views

Why does the sample mean of a Poisson distribution have minimum variance over all unbiased estimators?

Suppose $\bar{X}$ is the sample mean of i.i.d Poisson random variables with mean parameter $\lambda$, and $T$ is any other unbiased estimator. How do we prove without any theorem in Statistics that $E\...
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13 views

Maximum error for x correct decimals

What is the relationship between the maximum error and x correct decimals? For example: I want to estimate an answer with 6 correct decimals, how big can my maximum error be?
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21 views

Determine the value of $α$ for which the $MSE(T)$ is minimal.

Let $X_1$ be an estimator for the probability $θ$ of unauthorized access. Let $X_2$ be another estimator for $θ$. Assume that $X_1$ and $X_2$ are independent, unbiased estimators for $θ$. ...
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Does pointwise convergence of estimator imply consistency

Let $n \in \mathbb N$ and $\Omega=\mathbb N^{n}, \mathcal{F}=2^{\Omega},\mathcal{P}:=\{P_{\vartheta}:=\operatorname{Geom}(\vartheta)^{\otimes n}:0<\vartheta<1\}$ Find the Estimator $\hat{\...
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1answer
72 views

$X_i$ follows Bernoulli distribution find UMVUE of $\theta(1-\theta)$

Let $X_1,X_2,X_3 ...X_n$ be a random sample from Bernoulli distribution with parameter $\theta$.Find UMVUE of $\theta(1-\theta)$. I know that $T=\sum_{i=1}^{n}X_i$ is complete sufficient statistic ...
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1answer
14 views

Concluding consistency of estimators

Say we have a set of $n$ iid rvs with variance $\sigma^{2}$ and an estimator T of some parameter $\theta$. If we know that $Var(T) = {\sigma^{2}\over n}$, is that enough for us to conclude that our ...
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29 views

On a nonlinear regression problem

Consider the function $f\colon \mathbb{R}^2\to \mathbb{R}$, $f(x_1,x_2)=x_1^2 +x_2$. Assume that I don't know the form of $f$ and I only have a set of $N$ independent "input-output" data $\{(x_1^{(i)},...
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Is absolute value of the empirical mean unbiased/biased?

Given a random variable $x$ defined on $\mathcal{X}$ and corresponding i.i.d. observations $X_i$, $i=1,...,n$, could we estimate a following probability? $$P\left(\left| \left|\mathbb{E}(x)\right|-\...
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59 views

Solve for Y in classic bond formula

I'm trying to set up an excel spreadsheet that solves for $Y$ in the classic bond formula: $P = \frac{C}{(1+Y)}+\frac{C}{(1+Y)^2}+\frac{C}{(1+Y)^3}+_{......}+\frac{(C + Q)}{(1+Y)^n}$ Where "C" is a ...
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38 views

Deriving the least squares estimate of $\beta_{k-1}$

Let $y_i=\Sigma^k_{j=0} x_{ij} \beta_j+\epsilon_i$ $\epsilon_i$ is $NID(0,\sigma^2)$ and $x_{ij}, i=1,...,n, j=0,...,k$ is the $(i,j)^{th}$ elelement of the $n \times (k+1)$ matrix $X$, which is of ...
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1answer
15 views

How to show that MSE of ML estimator is greater than that of Bayesian posterior mean?

This question is based on problem 9 from chapter 4 of Gelman et al.'s Bayesian Data Analysis. Suppose we observe $y\sim N(\theta,\sigma^2)$ and wish to estimate $\theta$, with $\sigma^2$ known. We ...
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0answers
33 views

Mean estimator of a Gaussian variable with positive mean for quadratic loss

Suppose $\phi, \Phi$ are PDF and CDF for a $1$-dimensional normal Gaussian, and $X\sim\mathcal{N}(\theta,1)$, in which $\theta>0$ is positive but othrewise unknown. We want to estimate $\theta$ ...
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1answer
33 views

centroid algorithm robust to missing poins

I need to find a center point of a person given the coordinates of all the joints. The joints of a person can be represented as a nodes of a graph with a fixed structure. The catch is some of the ...
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51 views

Identifying all the 1s in a sea of 0s

I have an $N$-length binary string, the total number of 1s in the string is D. Assume N is much larger than D. Also, D is fairly larger than 1. (For example, N=1000, D=40). Now, my aim is to find ...
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22 views

Modelling conditional distribution based on multiple variables of various types?

I have a looking basic statistics problem: basing on a large sample of multivariate data, model conditional probability distribution (continuous) of one variable based on the remaining ones: a few ...
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1answer
108 views

Finding UMVUE for uniform distribution $U(\alpha, \beta)$

Let $X = (X_1, X_2, \ldots, X_n)$ be a sample from uniform distribution $U(\alpha, \beta): \alpha, \beta \in \mathbb{R}, \alpha < \beta$. I am to find UMVUE for the parameters $\alpha, \beta$. ...
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1answer
55 views

How do you estimate the remaining degree distribution?

Let $q_{j,k}$ be defined as the joint probability distribution of the remaining degrees of the two nodes at either end of a randomly chosen edge. Let $G=(V,E)$ be an undirected graph with nodes $V=(...
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20 views

Is there any relationship between efficiency and correlation coefficient?

Let $t_1$ be the most efficient estimator and $t_2$ be the less efficient estimator with efficiency $e$ and let $r$ be correlation coefficient between the two estimator $t_1$ and $t_2$.Define ...