The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
0answers
11 views

Show that the found value is the MLE

Let $ X_1, ... X_n$ i.i.d with pdf $$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$ It is asked to find the MLE estimator for $\theta.$ The likelihood function is ...
1
vote
0answers
12 views

Unbiased estimator for maximum

Assume $n$ independent random variables with unknown distributions $\{X_1,X_2,...,X_n\}$. Multiple "samples" or observations for each of these variables are given (not necessarily with the same ...
0
votes
0answers
24 views

Estimating the mean internal distance between borders of an irregular shape

I have two overlapping (not matching) irregular shapes ($X$ and $Y$), and I would like to estimate the mean distance between their limits. What I've been trying so far is obtaining the irregular ...
1
vote
1answer
25 views

Distribution of the sample variance of n iid exponential variables

I have to check some properties of an estimator, but I can't find its distribution. Let $X_1,...,X_n $ be independent identically distributed exponential variables with parameter $ \theta $, i.e. ...
0
votes
1answer
43 views

How many terms are required to get $D$ digits of Riemann zeta prime function?

How many terms are required to get $D$ digits of Riemann zeta prime function $\zeta_p(s) = \sum_p \frac{1}{p^s}$? Sebah & Gourdon mentions that finding $\zeta_p(2)$ to 20 digits by using $\sum_p ...
0
votes
0answers
17 views

density of statistic

Let $X^1$ ... $X^n$ be a random sample coming from a distribution with density : $p(x)=\begin{cases}2\theta^2/x^3, \text{if} \ x\geq 0 \\ \\ 0, \text{if} \ \ x<0 \end{cases}$ Let us denote the ...
1
vote
0answers
20 views

Statistics and Some Information Challenge

relation between two attribute x,y is $y=\alpha\beta^{-x}$. According to 8 experiments these information were gained. what is the estimation of ( $\alpha, \beta$) using Least Square Error? it's 2010 ...
0
votes
0answers
25 views

How to use Richardson extrapolation

In comment section in the question "Convergence of $\sum\limits_{k=1}^{\infty} \frac{1}{p_{k^2}}$, where $p_k$ is the $k$th prime" it is suggested that one first calculate $$ f(n) = \sum_{k=1}^{n} ...
0
votes
0answers
6 views

Prove recursive form of linear least squares estimate.

I have a zero-mean stochastic vector $\mathbf{x}$ which I estimate using observations of a random process $y_i$ where $i=0,1,...$ I denote the linear least squares estimate of $\mathbf{x}$ using all ...
1
vote
1answer
15 views

Estimating the remainder for Mac Laurin's series

I'm practicing Taylor's series and i found some old task. Calculate value of function $f(x) = e^x + e^{-x}$ at point $x = \frac{1}{\sqrt 2}$ with error not greater than $d=\frac{1}{20}$ So here's ...
0
votes
0answers
23 views

Estimate on the difference of quotients

The following is supposedly true (found it in a paper), however I fail to see why. Let $L(x)$ be a function that goes to $0$ as $x\rightarrow\infty$, $g(n)$ a sequence which goes to $\infty$ as ...
0
votes
1answer
29 views

How to estimate the standard deviation in this normal distribution?

There is this simple looking basic statistics question that asks to estimate its mean and standard deviation. I have some doubts and just want to make sure whether my working is correct. For part ...
0
votes
2answers
38 views

Numerical integration tolerance pitfalls

Consider that we want to estimate $$\int_{\pi/2}^{\pi/2+8\pi}sin(x)dx$$ (the value of this integrate is obviously zero) with the Midpoint rule. We start with the endpoints $a=\pi/2$ and $b=\pi/2+8\pi$ ...
16
votes
3answers
269 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
0
votes
0answers
8 views

Using EM versus estimating all parameters directly

My question is really a request for a clarification, and pertains to whether or not there are questions for which it is necessary to use EM or if it's more of a convenience. Let's presume we have n ...
0
votes
0answers
9 views

Two stage GMM estimator in Matlab

I am trying to create a simple GMM estimator for the mean of a normally distributed random variable using the first three odd central moments of a normal distribution (all of which should be zero ...
1
vote
1answer
61 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
0
votes
2answers
74 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
1
vote
0answers
27 views

Statistics question: Estimating mean when standard deviation is known

I am reading a textbook to learn more about statistics. This section is about estimating the mean of a population when standard deviation of the population is known. My simple question is this: How ...
1
vote
0answers
28 views

When does l1 regularisation give a sparse solution?

I was maximising a likelihood function, which is convex. I know that the system has a K-sparse solution. I wanted to know the conditions (or some sufficient conditions) on the likelihood function ...
1
vote
0answers
17 views

Bias of the MLEs for the two-parameter Weibull distribution

Is it possible to obtain a formula for or an equation on the exact bias of the MLE-vector for the two-parameter Weibull distribution (both parameters unknown). I've read papers offering Monte Carlo ...
2
votes
1answer
60 views

Sufficient statistic

Let $\mathbf{X}=(X_1,\ldots,X_n)$ with joint frequency function $f(\mathbf{x};\theta_1,\theta_2)$ where $\theta_1,\theta_2$ vary independently. The set ...
0
votes
0answers
23 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
0
votes
1answer
55 views

How to estimate $\sum_{x=1:n}{xf(x)}$ having $\tilde{f}$

I have an estimator $\tilde{f}(x)$ whose error is at most $\epsilon$, i.e., $\frac{|f(x)-\tilde{f}(x)|}{|f(x)|} \leq \epsilon$. I want to estimate $\sum_{i=1:n}i.f(i)$ with a small error. But if I ...
3
votes
1answer
33 views

Oscillations about equilibrium for coupled differentail equations

I have the following system of equations: $$\begin{align} \frac{dX}{dt} &= 2Y-2\\ \frac{dY}{dt} &= 9X-X^3 \end{align}$$ I would like to study the property of solutions to this function about ...
0
votes
0answers
36 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable?

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
1
vote
1answer
50 views

Stratified random sampling without replacement

I came across this statement and can't decide if it's true or false. Statement: In a stratified random sampling without replacement, with proportional allocation to the population size, the sample ...
2
votes
2answers
65 views

unbiased estimator in a random sample

I Have a statistic statement here which I need to decide if it's true or false Statement: "When the sample size is random, there is no way to get an unbiased estimator for the population average." ...
4
votes
1answer
69 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
0
votes
0answers
32 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
0
votes
0answers
23 views

Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
0
votes
1answer
35 views

Find a 10% likelihood interval

The function is: (n choose x)$[(1-y)^{k}]^{x}[1-(1-y)^{k}]^{n-x}$ Suppose n = 100, k = 10, x = 89 I found the maximum likelihood of y-hat to be 0.0116 Now I need to find a 10% likelihood interval. ...
0
votes
0answers
20 views

How could you find the probability that the estimator is within 0.03 of the mean?

p = fraction of large population that smokes n = sample size y = # in sample that smoke The maximum likelihood estimate of p is p-hat = y/n Consider the random variable Y and estimator F = Y/n ...
0
votes
1answer
34 views

Kalman filter innovation residual inversion

I'm trying to implement a Kalman filter in a computationally efficient way. The main issue is the inversion of the innovation residual: $$S=HPH^T+R$$ $$K=PH^TS^{-1}$$ My question is, can one assume ...
0
votes
0answers
24 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
0
votes
1answer
31 views

Show that $\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$

Let $A=[-\frac1u,\frac1u]$, Show that $$\displaystyle\int_Ax^2\mu_X(dx)\le\frac{12}{11u^2}\{1-\Re(\Phi_X(u))\}$$ where $\Phi_X(u)$ is the characteristic function of the r.v. $X$ Hint: ...
1
vote
0answers
25 views

Improving Schauder estimate for a linear elliptic PDE with oblique boundary

Let $\Omega \subset \mathbb R^n$ a $C^{2,\alpha}$ domain, $f \in C^{0,\alpha}(\overline{\Omega})$, $g \in C^{1,\alpha}(\overline{\Omega})$, $h \in C^{1,\alpha}(\overline{\Omega};\mathbb{R}^n)$ such ...
0
votes
1answer
117 views

why is the answer 21,845 and not 218,450?

How can you tell whether $$\frac{(250,000)(5.47)}{6.26}$$ is closer to 21,845 or 218,450 without calculating it exactly? Thank you.
2
votes
1answer
63 views

Combining statistical distributions

I have a situation where a distribution is dependent on 2 variables, one of which follows the poisson distribution, and the other the normal distribution, and I want to establish the method of ...
0
votes
0answers
16 views

regression problem

Regression was estimated using OLS. We get y=a0 + a1x1 + a2x2 + error term. We know covariance matrix ∑ of our estimator. 1. How to get confidence interval for a1/a2 ratio? 2. In what case would ...
0
votes
1answer
51 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
0
votes
0answers
14 views

Lp estimates from Elliptic Equation

Using the theorem: Let $f \in L^{p}(\Omega)$, $1<p<\infty$, and let $w$ be the Newtonian potential of $f$, $w(x)=\int_{\Omega}\Gamma(x-y)f(y)dy$. Then $w\in W^{2,p}(\Omega), \Delta w=f$ a.e and ...
0
votes
0answers
14 views

Optimal combination of multiple estimates of a random variable

For the following estimation problem: y = hx + n, x is the sent data, y is the observation (received data), h is a scaling factor (known), n is an AWGN random variable with zero mean ...
1
vote
1answer
18 views

Probability that a sample comes from one of two distributions

Let's say I have two normal distributions with means $\mu_1$, $\mu_2$ and standard deviations $\sigma_1$, $\sigma_2$ (which I know). I am handed a random variate from one of the distributions (I don't ...
0
votes
0answers
31 views

Approximation of optimum for two linear programs

Suppose you got two linear programs. They are the same except that one has a shifted objective by a positive constant (1) $$\min c^Tx$$ (2) $$\min c^Tx + d$$ For (2) there exists a ...
2
votes
0answers
96 views

Is it compulsory to make transformation to the econometric model in order to have only diagonal elements on variance-covariance matrix of errors?

I need some sharped and advanced advices for the following issue ... Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $y_{jis} = ...
0
votes
1answer
107 views

Test if estimator is unbiased

I'm having problems with the following question for my econometrics homework. Is $\ \ \hat \beta_2 = (y_n - y_1)/(n - 1)\ $ an unbiased estimator of $\beta_2$ for $\ \ y_t\ =\ \beta_1\ +\ \beta_2 \ ...
1
vote
1answer
151 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
0
votes
0answers
28 views

Estimating “task duration” variance through questions in a questionnaire

I am working to estimate the average task duration for complex processes - a total of 19 tasks / processes (each task made of several subtasks of varying duration). I will be organising a ...
0
votes
1answer
23 views

Show that $\hat{\delta}_1=\hat{\beta}_1+(X_1^T X_1)^{-1} X_1^TX_2\hat{\beta}_2$

Let $\hat{\beta}=(\hat{\beta}_1,\hat{\beta}_2)^T$ be the least squares estimator in the regression model $Y=X_1\beta_1+X_2\beta_2+u$. Let $\hat{\delta}_1$ be the least squares estimator of the ...