The estimation tag has no wiki summary.
2
votes
1answer
54 views
What is the probability of the number 1 and number 2 employees getting the bonus at a call center?
Two weeks ago, a friend working at a call center told me about their staff bonus policy. Here I paraphrase it.
Suppose employee A answers the maximum number ($N_1$) of calls among the staff, and ...
1
vote
1answer
53 views
$\mathbb E[\frac{\partial}{\partial\theta}\log f(X;\theta)]^2$ and $\mathbb E[\frac{\partial^2}{\partial\theta^2}\log f(X;\theta)]$
$f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$; $-\infty<x<\infty,\quad-\infty<\theta<\infty$
$\log f(x;\theta)=\log (\frac{1}{\pi[1+(x-\theta)^2]})$
$\Rightarrow \log ...
0
votes
0answers
16 views
Data estimation based on progression
Given a data-set $x$ and $y$.
x | y
------------------
153,000 | 0.058848
332,641 | 0.36352
506,629 | 0.53
If $x$ being the number of database records ...
0
votes
0answers
14 views
Smoothing technique for parameter estimation
I have a real-world web-graph and am trying to check the formula
$P = cd^{-\gamma}$, where $d$ is the degree.
I have a problem that there are too many verticies with unique d, so one cannot calculate ...
0
votes
1answer
28 views
MLE estimation for number of customers.
A clerk in a shop has noticed two customers arrived at the shop between 12:00 and 12:45. Another clerk noticed only one customer between 12:15 and 13:00. Assuming a Poisson distribution on the number ...
0
votes
0answers
23 views
MLE problem - measuring water pollution.
A lake is tested for pollution by taking some one-liter and some two-liter samples of water and checking for presence of bacteria. From the 1l samples, $n_1$ contained bacteria and $m_1$ didn't. From ...
2
votes
1answer
38 views
Maximum likelihood estimation of $a,b$ for a uniform distribution on $[a,b]$
I'm supposed to calculate the MLE's for $a$ and $b$ from a random sample of $(X_1,...,X_n)$ drawn from a uniform distribution on $[a,b]$. But the likelihood function, ...
2
votes
2answers
19 views
Estimating Poisson $\theta$ only from which percentage of intervals have events
Radioactive particles are emitted randomly over time from a source at an average rate
of per second. In $n$ time periods of varying lengths $t_1,t_2,\dots,t_n$ (seconds), the numbers of particles ...
3
votes
1answer
48 views
Maximum likelihood estimation - why is $\mathcal{L}$ not the joint pdf?
Here's an excerpt from my notes:
Define the likelihood function: $$\mathcal{L}(\vec{x};\theta)=\prod_{i=1}^{n} f(x_i;\theta)$$
Where $f$ is the pdf of the distribution we're sampling the $x$'s ...
4
votes
1answer
107 views
Estimate the scale of $e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m}$
I would like to estimate the scale of the following series,
$$S(m,t)=e^{-(m+1) t} \sum _{k=0}^{\infty } \frac{t^k}{k!}\left(\sum _{r=0}^k \frac{t^r}{r!}\right)^{m},$$
where $e$ is the base of ...
1
vote
1answer
27 views
Statistical inference, estimation, conceptual trouble
I've just begun learning about statistical inference and I'm having a bit of trouble understanding the concepts at hand. The exercises I've done and lectures I've read kind of gloss over the details ...
0
votes
0answers
25 views
Statistical estimation problem, basics
A lab measures the thickness of laminated surfaces. 100 measurements were made at random points of an unlaminated surface and then of a laminated surface, yielding mean thicknesses: $\overline{x}$ and ...
0
votes
1answer
389 views
Finding an unbiased estimator for the negative binomial distribution
Consider a negative binomial random variable Y
as the number of failures that occur before the r
th success in a sequence of independent and identical success/failure trials. The pmf of $Y$
is ...
0
votes
0answers
15 views
Kernel distribution estimation
Due to an assignment I need to implement a algorithm based on KDE to schedule an input data in different servers.
So far, I studied statistics in my bachelor but we did not go that far and they did ...
1
vote
0answers
37 views
Fast way to estimate cardinal number of subset
I have a large set $S$ of items, but the set is not exactly known. All I know are the cardinal numbers of categories i.e. a number of disjoint subsets,
$ \vert{S_1}\vert \dots \vert S_n\vert$ with ...
4
votes
1answer
121 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
29 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
45 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
43 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
18 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
68 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
25 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
43 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
36 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
22 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
96 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
40 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
38 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
23 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
46 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
61 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 $$a\leq ...
0
votes
2answers
34 views
Correlation bound
Let x and y be two random variables such that:
Corr(x,y) = b, where Corr(x,y) represents correlation between x and y, b is a scalar number in range of [-1, 1]. Let y' be an estimation of y. An ...
1
vote
1answer
34 views
Interpret the terms in Strang's second lemma
The second lemma of Strang states that for a certain choice of $V_h$, $a$, $u$ and $f$ there exists a $c>0$ such that
$$||u-u_h|| \leq c (\inf_{v\in V_h} ||u-v|| + \sup_{v\in V_h} ...
-1
votes
1answer
44 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
170 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
38 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
87 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
122 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
73 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
26 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
32 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
56 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
96 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 $$f(t)=\dfrac{9}{2 \pi ...
2
votes
1answer
66 views
How to asymptotically estimate a lower bound of this function?
The function is given as
$$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 ...
