Questions tagged [estimation]
For questions about estimation and how and when to estimate correctly
1,726
questions
2
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1
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35
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Ergodic theorem under condition $E[\sup Y]<\infty$ instead of $E|Y|<\infty$?
In my lecture, the ergodic theorem is given as: let $X_t,t\in \mathbb{Z}$ be stationary and ergodic, $Y_t=f(X_t,X_{t-1},...)$ measurable with $E|Y_0|<\infty$, then
$$\frac{1}{n}\sum_{k=0}^{n-1}Y_k\...
- 125
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0
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23
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$\|f\|_{H^k}^2 \leq C \sum\limits_{|\alpha|=k} \|D^\alpha f\|_{L^2}^2$ on $H_0^k(\Omega)$
Let $\Omega\subset \mathbb{R}^n$ ($n\geq 1$) be open and bounded and let $f\in H_0^k(\Omega)$ for some natural number $k$, where
$H_0^k(\Omega)$ is the closure of all testfunction $C_c^\infty(\Omega)$ ...
- 740
-5
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21
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Advanced econometrics [closed]
Let $x_1, x_2,\ldots , x_n$ be a random sample from the uniform distribution on $(0, \theta)$, i.e., $f(x)=1/\theta$, where $\theta$ is an unknown parameter ($\theta > 0$). Then,
(a) What is the ...
1
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0
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56
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+100
Monte carlo estimation, but function value can only be estimated indirectly
I have the following setup.
I want to Monte Carlo estimate a big sum $$F=\sum_x p(x) f(x)$$ by drawing $\\{ x_1,\dots, x_M \\}$ from the distribution $p(x)$ and averaging $f(x_i)$.
However, in my case ...
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-1
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0
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15
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How to Estimate the Joint and Conditional probability distributions for discrete variables
I read the proof the of the law of large numbers where its states that the the sample mean converges in probability to the population mean and it its proven by Chebyshev's Inequality Here
I am curious ...
- 19
1
vote
1
answer
36
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Finding Constant for Unbiased Estimation of Standard Deviation in Normal Distribution
Let $\{X_1, X_2, \ldots, X_n\}$ be a random sample from a $N(0, \theta^2)$ distribution. We want to estimate the standard deviation $\theta$. Find the constant $c$ so that $\hat{\theta} = c \sum_{i=1}^...
- 1,313
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8
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Does the Tyler's M-estimator lose the estimator of scale?
I was learning some robust estimation methods dealing with outliers and heavy-tail. I noticed that Tyler's M-estimator, whose key idea is to standardize the sample data by the distance to the mean, ...
1
vote
3
answers
59
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How to justify this estimation?
I was looking (again) at the series $$\sum_{n = 0}^{+\infty} \frac{3^n}{n!}$$ to give an estimation of its sum withouth the knowledge of the exponential series, and following the steps of a ...
- 2,758
0
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0
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30
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Total sample size vs unique visitors problem
Suppose from a total population on 10 million people...
In scenario 1:
During week 1, 1M people visited a website.
4 weeks later, 1M people visited the same website...
And 100K of them were the same ...
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0
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0
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23
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Periodogram formula for sampling
The formula I know for the periodogram is
$$I\left(f\right)=\frac{1}{NT}\sum_{n=0}^{N-1}\left|x\left[n\right]\exp\left(-2\pi fjnT\right)\right|^2$$
but for this one question I have which states that $...
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1
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0
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33
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Shifting Denominator in a Finite Alternating Series [closed]
Suppose I have the following finite alternating series:
$ \sum\limits_{k=1}^{n} (-1)^k \frac{a_k}{k} = 0$
Where $a_k > 0$ Can anything be said about the bounds of the series with a shifted ...
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7
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5
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258
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Converting $68^8$ microseconds to years without calculator
The following problem is easy to solve with the help of a calculator. However, when doing it manually, we are faced with a very large exponentiation ($68^8$) and which, at first glance, does not seem ...
0
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0
answers
14
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Variance of the individual-specific estimator for linear mixed effects model
Formualation: Consider a special linear mixed effects model expressed in two stages. In stage I, the individual model is
\begin{equation}
Y_i=C_i\beta_i + e_i,\, \mathbb{E}(e_i|x_i)=0,\, \mathbb{V}(...
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0
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1
answer
48
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Estimating the amount of primes for a very large number
I want to estimate the amount of prime numbers up to $\sqrt{6657\cdot 2^{10000}+1}$ by manual calculation.
My attempt is as follows:
Due to the prime number theorem we know that $\frac{x}{\log(x)}$ is ...
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1
vote
1
answer
48
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Method of Moments Estimator and its bias
Let $X_1,.....,X_n$ be a random sample from population with density
$$f(x) =\frac{\beta}{x^{\beta+1}} \qquad x \geq 1$$
Use the method of moments to find an estimator for the parameter beta. Upon ...
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0
answers
33
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$\frac{X_{1:n}+X_{n:n}}{2}$ is unbiased for mean
For $X_{i}\sim UNIF(\theta_{1},\theta_{2})$, we know that $X_{1:n}$ and $X_{n:n}$ are jointly sufficient for $\theta_{1}$ and $\theta_{2}$.
Suppose that it is desired to estimate the mean $\mu = \frac{...
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1
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0
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81
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Estimating Gaussian data
Let $x\in \mathbb{R}^{d}$, $d\geq 1$ and $p>\frac{2}{d}$. For any $R>0$ and any $0<s<\frac{dp}{2}-1$
\begin{align}
&\int_{\mathbb{R}}\int_{|x|>R}\left(\frac{1}{1+t^{2}}\right)^{\...
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1
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1
answer
68
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If we use $\bar{X}$ to estimate $\mu$, what is the probability that the absolute error is less than $1$?
Here is the question:
A random sample of size $12$ from a normal population $\mathcal{N}(\mu,\sigma^2)$ has mean $\bar{x}=27.8$ and standard deviation $s^2=3.24$. If we use $\bar{X}$ to estimate $\mu$,...
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1
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30
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Understanding Least Square Estimation (LSE) for vector parameter
I am trying to understand the following part from book Fundamentals of statistical signal processing by Steven Kay.
It is written that if parameter $\theta$ is vector of dimension $p \times 1$ then we ...
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1
answer
30
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Understanding the following equation
I have the following expression from Linear Least Square part of Steven M Kay book titled Fundamentals of statistical signal processing.
$\hat{\theta} = \frac{\sum_{n=0}^{N-1}x[n]h[n]}{\sum_{n=0}^{N-1}...
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1
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0
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19
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Consistent estimator of sum of squared sample covariances
This following paper made the following statement without giving proof.
https://www.jstage.jst.go.jp/article/jjss/35/2/35_2_251/_pdf/-char/ja
$$a^{20}=\frac{n}{p(n+2)}\sum_{i=1}^{p}s_{ii}^2$$
$$\alpha^...
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0
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45
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All about estimators biased and unbiased
I am unable to understand the difference between an estimator and a distribution, for example what is the difference between variance of a distribution and the variance of an estimator, and what is ...
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3
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0
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39
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Best estimator of a matrix signal with binary entries
Setup: Given that we have a noisy matrix signal $\breve{B}\in\mathbb{R}^{p\times L}$ of the true signal $B\in\mathbb{R}^{p\times L}$, where the empirical distribution of the rows of $B$ converge to $\...
- 730
1
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1
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130
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Regarding the proof of James-Stein estimator
I'm currently struggling to understand the james stein estimator.
For $N \ge 3$ and the James-Stein estimator $\hat \mu^{JS} = (1-\frac{N-2}{\sum z_i^2})z$, where $z \sim N_N (\mu , I)$,
$$E[\Vert \...
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0
answers
40
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Derivation of least squares solution using SVD for Lee-Carter mortality model
The Lee-carter model aims to model central mortality rates using the following
$$
log(m_{x,t}) = a_{x} + b_{x}k_{t} + \epsilon_{x,t}
$$
where $\epsilon_{x,t} \sim N(0,\sigma^{2})$
The following ...
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3
votes
1
answer
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Feynman's estimation of a cube root
In Lucky Numbers, Feynman described how he beat a man with the abacus to find the cube root of the number $1729.03$.
The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic ...
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3
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0
answers
74
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Expected value of the parameter $P$ of a Bernoulli having observed $s$ successes over $t$ trials, when $P$ is uniform on $[a,b]$ with $0\le a<b\le 1$
Using the relation between Beta and Gamma functions
$$B(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$
where
$$B(m,n) = \int_{[0,1]} p^{m-1}(1-p)^{n-1}\operatorname{d}p$$
and the fact that for any $n ...
- 5,603
3
votes
1
answer
134
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Estimate for $\sin(x)/x$ and all its derivatives
Consider the function $f(x) := \frac{\sin x}{x}, f(0) := 1$. It is easy to see that $$\lvert f(x) \rvert \leq \frac{2}{1+\lvert x \rvert}. $$
I am trying to prove that more generally, for each $k\in \...
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1
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Is $\exp (-\alpha\log (n)^\beta(1+o(1)))=o(1)$?
Referring to Butucea, $\hat s_n$ is an estimator of $s_k$.
In (5) on p.902 there is the statement that
$$\mathrm{P}\left(\hat{s}_n \neq s_k\right) \leq \exp \left(-\frac{A^2}{4} 2^{2 \beta^{\prime} / \...
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1
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0
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An estimate for a logarithm occuring in analytic number theory
Let $s$ range over the complex numbers and write such numbers as $s=\sigma+iT$ with $\sigma,T$ real. In textbooks on analytic number theory, I have found the following estimate:
$$\frac{\log|s|}{|s|} \...
- 2,405
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0
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10
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Size of sample to extend regression model
I have the results of a constructed logistic regression model in which the objective function is $Y = Y(X_1, ..., X_k)$. By result I mean here the values in the interval $[0,1]$ obtained by the ...
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0
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53
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how to estimate this expression (comes from probability) asymptotically?
Consider
$$(n-1)\exp\{(n-1)(\frac{1}{2}-p(n))\}(2p(n))^{\frac{n-1}{2}}$$
where $n$ is positive integer, $p(n)$ is a function of $n$ that is undefined. This expression in my context represents a ...
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1
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0
answers
46
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Non-sequential German Tank Problem
I have a bag containing $n$ coins. Each coin has a unique integer on it in the range $[1, m]$. As a corollary, $n \le m$.
We may sample $k$ coins without replacement where $k < n$. Given that we ...
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3
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2
answers
138
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Estimating how many of the first $10,000$ Fibonacci numbers start with the digit $9$
Consider the problem of estimating how many of the first $10,000$ Fibonacci numbers begin with the digit $9$.
The only ideas I have so far:
Obviously, if we assume that the every first digit is ...
2
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0
answers
39
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Under-damped cosine signal estimation
I have N measurements of an under-damped cosine signal, represented by the equation $$ x(t) = Ae^{-\alpha t} \cos(2 \pi ft) $$ where $ A $ is a known constant. My objective is to estimate the ...
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15
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Modeling Regional Quality Based on SUM of GCL / SUM of TAP - Comparing Two Approaches
I'm learning Data Science and I'm currently working on analyzing loan data across different regions. My goal is to build a model that can assess the quality of a given region based on the ratio $$\...
0
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0
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28
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Behavior of Partial Sums in a One-Dimensional Dynamical System
Consider the following sequence:
$$f_{n+1}=(\lambda f_n + \frac{1}{2})\mod 1 - \frac{1}{2}.$$
For the initial $f_0$, you can select any non-zero number from the open interval $(-\tfrac{1}{2}, \tfrac{1}...
3
votes
3
answers
276
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Drawing marbles from a box
Say we're drawing marbles from a box. The marbles can be labeled X, Y, or Z and can be either black, brown, or white. The probability of drawing a marble with each letter label is unknown but fixed ...
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1
answer
83
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Formula to increase concentration in one solution using 2 substances simultaneously
Using the following formula:
V2 = ((D-C1)/(C2-D))*V1
It is possible to estimate the amount of higly concentrated V2 with value C2 that has to be added to the base ...
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1
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2
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59
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Showing the big Oh bound for the logarithmic integral $\operatorname{Li}(x)=x/\log x + x/\log^2 (x)+O(x/\log^3 (x))$
Consider the logarithmic integral $\operatorname{Li}(x):=\int_2^x \frac{dt}{\log t}.$
Then I found a result stating that we have $\operatorname{Li}(x)=x/\log x+O(x/\log^2(x))$ and another integration ...
- 5,233
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0
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22
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Innovation error covariance decreasing but state error covariance inceasing
I am trying to implement Kalman Filter to estimate some random variables. I see that for the system I am using, the innovation error is zero for all times and the innovation error covariance matrix is ...
- 901
2
votes
2
answers
59
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Is there another simple way to find the limit using Stirlings formula?
$n\in\mathbb{N}$, find the limit
\begin{equation}
\lim_{n\to\infty} e^{\frac{n}{4}}n^{-\frac{n+1}{2}}(1^1\cdot2^2\cdot\cdots\cdot n^n)^{\frac{1}{n}}
\end{equation}
I calculate the limit in the ...
- 619
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votes
0
answers
78
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Computing the inf-norm of the inverse of a matrix
Consider a regular matrix $A \in \mathbb{R}^{n \times n}$. The normal $\infty$-norm of $A$ is given by
$$
\|A\|_{\infty} = \max_{\|x\|_{\infty} = 1} \|Ax\|_{\infty}
= \max_{i \in [n]} \sum_{j \in [n]}...
- 171
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votes
0
answers
24
views
Find Confidence Interval for a given sample mean without sample standard deviation
The question is:
Assume that a single-digit random number is generated for 100 trials. Let X denote the number generated per trial. Suppose that x(sample mean) assumed the value 3.85 for these 100 ...
- 25
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votes
0
answers
10
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Can conclude from the values from maximum likelihood estimation.
Determine the maximum likelihood estimates of a and λ when X1, ..., Xn is a sample from the Pareto density function
f(x) = λαλχ-(2+1), if x ≥ a
0 ,if x <a
I tried to solve it
The Pareto ...
- 1
0
votes
1
answer
36
views
All cliques of a graph
Given an arbitrary undirected graph $G$ with $n$ vertices, I am interested in counting the number of cliques.
(For example, the number of cliques of a complete graph $K_n$ is trivially $2^n-1$;the ...
- 113
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votes
0
answers
57
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Why does $Y'=\Theta+(W/3)$ have the same estimation and MSE as $Y=3\Theta+W$?
This is from MIT's 6.431x.
For the model $X=\Theta+W$, and under the usual independence and normality assumptions for $\Theta$ and $W$, the mean squared error (MSE) of the LMS estimator is
$$\frac{1}{...
- 1,448
1
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1
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75
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Understanding the implication of a result
In the following, I am referring to this paper, p. 14, line (3.14):
It is said that from
$$
\frac{\Vert f^{n+1}\Vert_{2,\gamma}^2-\Vert f^n\Vert_{2,\gamma}^2}{2\Delta t}+\frac{1}{\varepsilon^2}\Vert f^...
- 505
0
votes
0
answers
32
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Bound for MSE of the Kernel Density Estimator
On page 911 of the Paper Adaptivity in convolution models with partially known noise distribution they say: By using classical
results on this estimator, we have
$$\mathbb{E}_{f, s_k}\left[\left|f_n(x)...
- 121
0
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0
answers
33
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Decay estimate of Fourier transform with a log function
Assume $f(x)$ ($x \in \mathbb{R}$) is a smooth function with a compact support. Assume $0 < \epsilon < 1$ and $i = \sqrt{-1}$ is the imaginary unit. Can we show that there exist a constant $C$ ...
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