The estimation tag has no wiki summary.
4
votes
1answer
93 views
Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$
By integral test, it is easy to see that
$$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$
converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$]
I am ...
0
votes
1answer
24 views
Generalisations of the Gronwall's lemma
Suppose we have the following differential inequality
$F''(w)\le \frac{p-1}{p}\frac{(F'(w))^2}{F(w)}$ on $w\in(w_0,w_0+\varepsilon)$, $w_0>0$, $\varepsilon>0$, $p>1$. In addition, ...
2
votes
1answer
44 views
Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?
We know that
$\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and
$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ ...
2
votes
3answers
41 views
Series evaluated to $m$ terms, approximating the error
Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms?
$$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$
...
1
vote
0answers
17 views
Is there a way to estimate the range of fitting coefficients from only the data?
Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain
$f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$
where ...
0
votes
2answers
57 views
Mental Math - Estimating Logarithms
How can we estimate logarithms with different bases? Take $\log_2 10$ ($1\over\log_{10}2$$\approx3.32192809$) for example. If we convert $10$ to binary, we get $1010_2$. So $\log_21010_2$ can clearly ...
1
vote
2answers
24 views
Expected value of total accumulated lifetime (understanding gap in proof)
Problem:
I understand the first line $E(T) = ...$
However, I don't get the next two steps. I feel like I almost get it.
It's like we are factoring out a $\sum_{j=1}^{20}$ but how did he ...
3
votes
1answer
42 views
Best estimate for random values
Due to work related issues I can't discuss the exact question I want to ask, but I thought of a silly little example that conveys the same idea.
Lets say the number of candy that comes in a package ...
0
votes
1answer
34 views
Likelihood of the mean of one random variable with unknown parameters greater than another
Assume we have two random variables $X$ and $Y$ that are gamma distributed (or normally distributed, if it makes the math easier) with unknown parameters. We have samples $x_1,x_2,...,x_m$ and ...
0
votes
1answer
19 views
Approximation question
Cars and buses arrive at a bridge according to the independent Poisson processes at a rate of $3$ cars/minute and $1$ bus/ minute. What is the chance that strictly more buses arrive than cars in a ...
1
vote
3answers
88 views
Let $ X_1,X_2,…,X_n$ be i.i.d. $N(\theta_1, \theta_2)$, please prove that $E[(X_1-\theta_1)^4] = 3\theta_2^2$
If $X_{1}$, $X_{2}$, ..., $X_{n}$ is sampled from $N(\theta_1, \theta_2)$, how can I prove that $E [(X_{1} - \theta_1)^{4}] = 3 \theta_2^{2}$?
I started off this question finding the completely ...
1
vote
0answers
30 views
Cramer-Rao bound for $\chi^2$ distribution parameter estimates.
I've stuck in unpleasant problem with noncentral $\chi^2$ distribution.
I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
0
votes
0answers
35 views
Numerical calculation of fisher information
I am trying to obtain numerically the fisher information. Given a likelihood function
$$ f(X,\theta),$$
with $X \in [0,1]$.
The fisher information is given by
$$ ...
0
votes
1answer
21 views
Estimate growing graphs
Lets make my scenario not generic just so that i could use particular terms
Say i have a graph of population per year of someplace over some decades
Lets say the graph is like this
How can i ...
0
votes
0answers
44 views
Big Oh Equality
I am stuck on a proof, having encountered this bit I can't figure out:
$$e^{O(1/\log x)}=1+O \left( \frac{1}{\log x} \right)$$.
Why is this?
Thank you very much in advance!
0
votes
1answer
59 views
Non-Linear regression
Imagine that I have a function $ f(x,y) $ to model a physical phenomenon.
I believe that functions is defined by $$ f(x,y) = A*x + B*y + C*x*y$$
I have many values for $ (x,y,f(x,y)) $, how can I ...
1
vote
1answer
50 views
How to establish the estimate?
I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
0
votes
2answers
34 views
Correlation bound
Let x and y be two random variables such that:
Corr(x,y) = b, where Corr(x,y) represents correlation between x and y, b is a scalar number in range of [-1, 1]. Let y' be an estimation of y. An ...
1
vote
1answer
26 views
Interpret the terms in Strang's second lemma
The second lemma of Strang states that for a certain choice of $V_h$, $a$, $u$ and $f$ there exists a $c>0$ such that
$$||u-u_h|| \leq c (\inf_{v\in V_h} ||u-v|| + \sup_{v\in V_h} ...
0
votes
1answer
43 views
Estimating component variance for a sum of random variables
Say I have two zero mean single variate independent random variables $X$ and $Y$, and a third variable $Z = X + Y$. I can draw samples $z_i$ with $i = 1..n$ from $Z$ and I know $Var(Y)$. How can I ...
1
vote
1answer
160 views
Sufficiency and UMVUE for Poisson distribution
I need to show that $\hat\lambda = \bar X$ is a sufficient estimator for a Poisson distribution iid $X_1...X_n$, show that $\hat\lambda$ is the UMVUE for $\lambda$ and that $\hat\lambda$ is a ...
0
votes
1answer
34 views
Parameter estimation with GMM
I have estimated the parameters of normal distribution with GMM and got the following results:
$mean = -0.01168 , p-value = 0.83519, Sd = 1.77 , p-value = 0.00000.$
I'm bit confused in ...
1
vote
1answer
82 views
Bayes Estimator
Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf:
$f_{X}(x|\alpha,\beta) = ...
1
vote
1answer
113 views
How does this calculation of $\int_0^\infty(\sin^2x)/x^2\ dx$ work?
I'm having trouble with two steps in a calculation of
$$\int_0^\infty\left(\frac{\sin x}{x}\right)^2\ dx$$
in a book.
They take the contours $C_R$ composed of upper half-circles ...
5
votes
2answers
68 views
Is there a lower-bound version of the triangle inequality for more than two terms?
The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$
Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
0
votes
1answer
25 views
estimation of limit / reducing of limit
I am trying to recalculate an exam and the solution for my problem is shown in the picture. How ever
Could someone please explain to me why the first limit can be reduced to that second one, and ...
0
votes
0answers
30 views
how to compute Wald Statistic for $\beta_2=0$ and $\beta_2-\beta_1=0$
$\sqrt{T}(\hat{b}-b)\sim N(0,E)$, where $E$ is the matrix $\begin{pmatrix}1&0.5\\0.5&3\end{pmatrix}$ and $b= (b1, b2)$.
Q: what is the distribution of AX?
Q2: what is the asymptotic ...
0
votes
1answer
23 views
How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define?
How to obtain estimate of covariance matrix that will be guarantee to be semi-positive define ? (Is CrossValidated better place for this question ?)
2
votes
1answer
26 views
Multi-dimensional MLE Guassian
I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
1
vote
1answer
48 views
How do I use student's-t distribution without the sample size?
Here is my question (homework obviously):
A sample from a normal population produced variance 4.0.
Find the size of the sample if the sample mean deviates from the population
mean by no more than 2.0 ...
2
votes
1answer
19 views
Estimation for large $k$.
I have a function $f(k)$ defined on the set of natural numbers and I managed to show that $f(k)>n-\binom n k(1-n^{-2/3})^{k(k-1)/2}$ for all integers $n\ge k$. I am hoping to get a further ...
1
vote
1answer
39 views
Some estimate concerning hyperbolic functions
I want to show that $|\sinh(az)|\leq|\sinh(z)|$ for all $z\in\mathbf{C}$ (or at least for all $z\in\mathbf{H}$, the upper half plane), provided that $0<a<1$ However, I am not even certain ...
2
votes
1answer
82 views
Inverse Laplace transform and Jordan's Lemma
I'm trying find the inverse Laplace transform $f(t)$ where I have $F(p)=\dfrac{9}{p(p+3)^2}$. I know $f(t)$ already to be $1-3te^{-3t}-e^{-3t}$. I have the integral $$f(t)=\dfrac{9}{2 \pi ...
2
votes
1answer
59 views
How to asymptotically estimate a lower bound of this function?
The function is given as
$$f(x)\geq \sum_{i=1}^{[x/2]}f(i)+1$$
The boundary condition is $f(0)=0$.
What I can get is this function grows faster than any polynomial function, and grows slower than ...
0
votes
0answers
21 views
Combining function estimates
I have two piecewise linear estimates for two different realisations of the same random variable.
What are some techniques that I could use to combine these function estimates into a single ...
3
votes
1answer
40 views
Estimate the given sum.
This is the question from "Data Structures and Algorithm Analysis in C" By Mark Weiss. It is the question 1.7. It goes as follows:-
Estimate the sum ...
1
vote
1answer
77 views
arguing away - complex analysis
Probably a trivial question but I can't understand how to argue away the value of integrals in complex analysis.
I am trying to find the inverse Laplace transform of $F(s)=\frac{1}{s(s+1)}$. The ...
2
votes
1answer
81 views
How many citations to read before convergence?
So I have the following question assuming I start with N academic papers, though I was thinking to make this simple I start with one academic paper. And say it has C citations, and each one of these C ...
0
votes
0answers
36 views
Strichartz estimates and operator from $L^{2}_{x}$ to $L^{6}_{x,t}$
I want to prove that the operator $T=| \nabla|^{1/6} e^{-t\partial ^{3}_{x}}\tilde{P}_{N}$ takes functions from $L^{2}_{x}$ to $L^{6}_{x,t}$. The hint is to first prove for Schwartz functions, and ...
0
votes
0answers
42 views
Some kind of trace inequality
What is the trick, to prove
$\| u\|_{L^2(\Gamma)} \leq k \frac{1}{r}\| u\|_{L^2(\Omega)} + r \| \nabla u\|_{L^2(\Omega)} $ ?
$\Gamma$ is one side of $\Omega:= [0,r] \times [0,r] $.
I tried partial ...
1
vote
1answer
30 views
machine learning for a rule
I have data in the following form , where x is an integer and r is 0 or 1.
I know that
if $x < C$, then $r = 1$,
if $x\geq C$, then $r= 0$.
How can I automatically estimate the value of $C$? ...
1
vote
1answer
53 views
Inverse estimate of gradient of Sobolev function
I need an estimate for $\| \nabla w\|_{L^2{(\Omega \subset \mathbb{R}^n)}}$, such that it is $< c\| w\|,\ w \in H_0^1(\Omega)\ $. Is this possible?
2
votes
1answer
100 views
finding maximum likelihood estimate from dependent binomial rvs
let $X_{1}$, $X_{2}$, $X_3$, $X_{4}$ be iid bernoulli rvs with $\mathbb{P}(0)=0.5$, $\mathbb{P}(1)=0.5$.
$Y_{1} = X_{1}+X_{2}+X_{3}$ and $Y_{2}=X_{1}+X_{2}+X_{4}$
$Y_{1}$, $Y_{2}$ are dependent ...
1
vote
1answer
45 views
Estimating a sample mean using sub-sample means
we collect a lot of data on a daily basis via an API, and part of this data includes fields that represent sample means. Specifically, we get provided with a sample mean and the sample size, lets call ...
0
votes
1answer
72 views
How can I make estimates on large powers and logarithms such as $e^{10}$?
Just wondering, are there any useful tricks to make estimates of large powers or logarithms just by hand such as for $e^{10}$? Any such ways to get an error less than 1?
1
vote
1answer
48 views
How can I quantify the amount of space required to store all possible 128kilobit mp3s?
Somone has suggested that
Within, say, a collection of every possible 30 second long MP3 file encoded at 128kbps, I'd probably be infringing on a few thousand copyrighted works.
128kilobits per ...
1
vote
1answer
69 views
Real Positive Zeros of Equation
During my research on physical problem, I faced the following simple equation:
$r^{2k+1}+ab\,r-a=0$
With:
$-1\leq k\leq1\:,\:0<r\:,\: a,b\in\mathbb{R}$
I need to put bounders on $a,b,k$ such ...
0
votes
0answers
32 views
Any relationship between Weighted Least Squares and Weighted Minimax?
I got a question that I would like expert opinions on. Here it goes
I like to use Weighted Least Squares because it has a close-form expression when I want to find the optimum solution. i.e.
...
-1
votes
1answer
27 views
Estimate: $|f^{(3)}(i/3)|$.
Suppose $f:D(0,1)\longrightarrow \mathbb{C}$ is holomorphic, where
$D(0,1)=\{z\in\mathbb{C}∣|z|<1\}$,
and assume the maximum $|f(z)|\leq 2$.
Estimate: $|f^{(3)}(i/3)|$.
I just don't understand how ...
1
vote
2answers
65 views
Independent tests bound. (Chernoff/Azuma?)
I have a series of N Bernoulli tests (p, 1-p).
I need to calculate a probability of passing more than N/2 tests, depending on N and p.
The obvious solution is Chernoff bound: $\varepsilon \leq ...
