# Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

1,207 questions
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### Least Square Estimation

In the following example question (from Bertsekas, edition 1), i have two questions: Why the value of fX|Y(x|y) is 1/2? Can anyone explain in more detail about the graph on the right (especially the ...
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### Proving independence between Beta estimated and Delta in OLS

I know that in ordinary least squares $b$(beta estimated) and $\delta^2$(variance estimated) are independent, but how do I prove that?
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### Proving a gradient estimate for a continuous function in an open ball

I need to prove the estimate $|\partial_{x_i}u(x_0)| \leq \frac{N}{R}u(x_0)$ where $u \in C(B(x_0,R))$ and $u$ is nonnegative and harmonic in $B(x_0,R)$, and $i=1, \ldots,N$ I found this ...
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### What adaptive controller can be used in embedded system with low RAM?

This is not a question for data science, hardware or programming languages. This is a more practical question about adaptive control for embedded systems, but still a math question. I have tried to ...
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### Find the time gap between two vehicle based on km/h of the vehicle

I am currently researching on VANET for identifying the road capacity based on vehicle moving speed. Below is my question, If a car moving on 8km/h and distance to the next car is 15 meters, what ...
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### How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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### Find the Maximum Likelihood Estimator of a binomial distribution

X ~ Binomial (5,p) , where X is taken with replacement from a simple random sample $X_1 =x_1 , ... X_n = x_n$ (I guess it is until $X_5$ since n=5) Here is the full question in case the context is ...
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### Text book recommendations for statistical estimation theory -specifically MLEs and confidence intervals

I am looking for a textbook on statistical estimation theory. In particular I am interested in a book that explains MLEs and confidence intervals. Preferably accompanied by exercises. The book should ...
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### Solution of $\int_x^1y^{a-1}\left(1-y\right)^{b-1}dy = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$

When is $f=g$ on $(0,1)$ for $f = \int_x^1y^{a-1}\left(1-y\right)^{b-1}dy$ $g = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$ Let me show their graphs. They are small, so I multiplied ...
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### estimating $\prod_{i=1}^k a_i\leq \max_i a_i^{p_i}$, where $\sum_i 1/p_i=1$

Let $1<p_1,\dots,p_k<\infty$ such that $\sum_{i=1}^k\frac{1}{p_i}=1$. Moreover, let $a_1,\dots,a_k\geq 0$. I have to show that $$\prod_{i=1}^k a_i\leq\max_i a_i^{p_i}$$ I want to use induction ...
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### Where and Why do we use measurable functions in modern probability theory?

I am a newbie to the measure theoretic framework for probability. While studying MMSE Estimation I came across the term "borel function" which upon further googling made me understand that it's a "...
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### How can I write $\rho$ in function of $\Sigma$?

Be $\Sigma = (1-\rho) I_p + \rho 1_p 1_p^\top$. I need to write $\rho$ in function of $\Sigma$ to calculate the maximum likelihood estimator of $\rho$ and I know that the maximum likelihood estimator ...
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### Sufficient statistic for Double Exponential

Let $X_1,X_2,...X_n$ be a random sample from $f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$.We know by Factorisation theorem that $\frac{\sum |X_i|}{n}$ is sufficient for $\theta$. But can ...
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### Compute the bias of $\frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}}$

I found the maximum likelihood estimator of $x$. Now, how to compute the bias of: $$\hat{x}_{ML} = \frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}}$$ Where $\hat{x}_{ML}$ ...
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### error covariance of MMSE estimator relation to other error covariance estimators

I'm trying to prove the following: let $\Lambda_{e}$ be the error covariance of an estimator $\,\hat{\theta}(y)$ of $\,\theta$ based on $\,y$. I want to show that the error covariance of MMSE ...
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### Cardinality estimation using ordered statistics

In cardinality problem (count-distinct problem) the goal is to estimate the number of unique elements in a set. HyperLogLog is one such algorithm [ref] Another approach is using order statistics, such ...
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### Estimating an integral involving the n-th power of the cdf of the normal distribution.

I am trying to solve integrals that look like the following: $\int_{-\infty}^{\infty} \phi(cx+d)^n e^{-(x-a)^2}dx$ where $\phi$ is the cdf of the standard normal distribution. I have no idea how to ...
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### Let $X_{x,1}$ and $X_{x,2}$be two unbiased estimators for a real parameter $X$ [closed]

Let $X_{x,1}$ and $X_{x,2}$be two unbiased estimators for a real parameter $X$. Let us define $X_{x}=\alpha X_{x,1}+\beta X_{x,2}$ with $\alpha,\beta\in \mathbb R$. For which values of $\alpha$ and ...
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### Converse of ML inequality for contour integrals

If the ML inequality estimates a region of an integral in the complex plane to be zero then that's the actual value of the integral, and I've been using this for evaluating some integrals along the ...
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### 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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### 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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### 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 ...