For questions about estimation and how and when to estimate correctly

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6
votes
0answers
56 views

Estimation of the order of torsion in $\mathrm{GL}(n,\mathbb Z)$

Let $A \in \mathrm{GL}(n,\mathbb Z)$ be a torsion, I would like to prove that $\mathrm{order}(A)\leq K\exp (cn^{\alpha})$, with $0<\alpha <1$, for $n$ "large enough". I know that if ...
4
votes
0answers
87 views

Can the limit of the MSE of an estimator be infinity?

Is it ever possible for the limit of the MSE of an estimator be infinity? I was doing an exercise and it turns out that the estimator is consistent but the limit of the MSE is infinity, so I am ...
4
votes
0answers
260 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
3
votes
0answers
39 views

Estimating the sum of a series within arbitrary certainty.

Find the sum of the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^5} = a_n$ within three decimal places. The sum is estimated by $\displaystyle a_n \approx \sum_{k=1}^{n}\frac{1}{k^5}+R(n)$ ...
3
votes
0answers
82 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...
3
votes
0answers
45 views

Rao-Cramer lower bound regularity condition and dominated convergence

Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is ...
3
votes
0answers
98 views

Upper bound for area of polygons

is there a formula for an upper bound for the area of a polygon, knowing the length of its edges? In the ideal situation, the answer would be a function $f_n(l_1,l_2,\dots,l_n)$ of the edges lengths ...
3
votes
0answers
67 views

Asymptotic stability

we know from the theory of ODE that $\left\|\exp(tA)\right\|\leq Ke^{-\delta t}$ for $K,\delta >0$ and $t\in\mathbb{R}^+$ if the real part of all eigenvalues are strict non-positiv. My question is: ...
2
votes
0answers
120 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
2
votes
0answers
46 views

Is Stirling's Approximation used here, to prove the asymptotic inequalities?

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$. Then prove, for large $n$, ...
2
votes
0answers
46 views

For what $r,s$ exist unbiased estimation of $f(p) = p^{r}(1 - p)^{s}$ for binomial distribution?

We have sample $x_1, ..., x_n$ generated by independent binomial random variables $\xi_1, ..., \xi_n$. We know parameter $k$ but don't know probability $p$. k is number of tests: $\xi_i \sim ...
2
votes
0answers
56 views

Source estimation for identification of anomalous events

I’m stuck on the following problem. There are two sources $S_A$ and $S_B$ at the ends of a channel. Both are made up of a white noise component $W_i$ plus an impulsive component $I_i$: $S_A = W_A + ...
2
votes
0answers
43 views

Ratio estimator in sampling

Let the population $U=(1,2,3)$. We want to estimate $R=\frac{\mu_y}{\mu_x}$.Consider the estimators $$\hat{R_1}=\frac{\overline{y}}{\overline{x}},\hat{R_2}=\frac{\overline{y}}{\mu_x}$$ where ...
2
votes
0answers
38 views

Method of moment estimator. Deconvolution

The distributions $Y, X, Z$ and $W$ are related as follows: $$Y_1 = X + Z$$ $$Y_2 = X + W,$$ that is $X$ (random variable) is a common factor to the random variables $Y_1$ and $Y_2$, which ...
2
votes
0answers
31 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
2
votes
0answers
38 views

MLE for CTMC parameters

Let the data set be $$D = \{(s_0, t_0), (s_1, t_1), ..., (s_{N-1}, t_{N-1})\}$$ where $N=|D|$. Each $s_i$ is a state from the state space $S$ and during the time $[t_i,t_{i+1}]$ the chain is in state ...
2
votes
0answers
29 views

Can I adjust linear growth of a a subpopulation to a linear decay of the general population?

I need to estimate the amount of CF patients in Poland in the next four years. I have: estimations of the Polish population for the future years a CF patients' register for the last couple of years ...
2
votes
0answers
51 views

Sampling with no duplicates

I am sampling a population of unknown size and unknown distribution. The sample will be taken over distinct time intervals, but I have to reject any duplicates in the given time interval. The sample ...
2
votes
0answers
19 views

Decay of reciprocal gamma function and similar functions

It is known that the reciprocal gamma function $1/\Gamma$ is entire of order 1 meaning, in particular, that for any $\varepsilon > 1$ there exists $C_\varepsilon > 0$ such that $$ \left| ...
2
votes
0answers
50 views

Maximum likelihood estimator(MLE)

Consider a sample from a distribution with PDF $$f(x) = \begin{cases} \frac{1}{2}(1+\theta x), & -1 \leq x \leq 1\\ 0, & otherwise \end{cases} $$ find the maximum likelihood estimator of ...
2
votes
0answers
50 views

How to estimate parameters of exponential functions?

I am a new bird to math! I have an expression as follows, $$e^{-ux}+e^{-uy}=z$$ I have at least ten pairs of $(x,y,z)$, so how can I estimate the value of $u$? And how to evaluate my results? Hand ...
2
votes
0answers
72 views

Simulation Velocity of a harmonic oscillator system

I am write a simulation for get true Velocity of a harmonic oscillator system as Where P=[p1 p2;p2 p3] can find using Rung-Kutta Integration method with P(0)=[1 0; 0 1] This is code to find p Now, ...
2
votes
0answers
287 views

Maximum Likelihood Estimation with Laplace Distribution

I want to estimate the parameters $a$ and $b$ of the model $y_i = ax_i + b + \varepsilon_i, i=1,...,n $ via Maximum Likelihood. The $\varepsilon_i$ are assumed to be Laplace-distributed with density ...
2
votes
0answers
53 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
2
votes
0answers
222 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 ...
2
votes
0answers
106 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} = ...
2
votes
0answers
52 views

Lower bound of $\sum_{k = 1}^{N}1/(x + k)$

Let $f(x) := \sum_{k = 1}^{N}1/|x + k|$ for $x \in [0, N]$. Why is $f(x) \geq C\log N$ for all $x \in [0, N]$ where $C$ is an absolute constant. My work is: Since $x \in [0, N]$, we can remove the ...
2
votes
0answers
31 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
2
votes
0answers
52 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
2
votes
0answers
2k views

95% confidence interval around sum of random variables

Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
2
votes
0answers
73 views

Estimating a sub-population characteristic based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ...
2
votes
0answers
34 views

How do I compute the variance (or confidence interval) of a Maximum Spacing estimator?

I am trying to solve a problem using a Maximum Possible Spacing estimator (see Maximum spacing estimation on wikipedia for links). Details on what I am trying to do can be found in the following ...
2
votes
0answers
106 views

Approximability of continuous sampling by discrete sampling - definitions?

Are there standard definitions that express the notion that sampling from a given continuous random variable can be approximated "to any desired degree of accuracy" by sampling from an appropriately ...
2
votes
0answers
587 views

How can you calculate actual values when all you have is rolling averages?

Let's say you have a set of data that is rolling 6 month averages of the actual monthly data. Good data collection would mean you saved the actual values and then calculated the rolling averages, but ...
1
vote
0answers
11 views

How to calculate a time estimate and plot it?

I recently made a website for a game's community. The game basically has an online dragon which people can fight. Each fight chips off a little bit health. On the next encounter with the dragon, ...
1
vote
0answers
13 views

Maximum Likelihood Estimation with 2 parameters for a Poisson distribution

I have two observations from a Poisson distribution. The first one ($N_r$) come with a Poisson distribution with mean $k_1$. For the second one ($N_e$) I know that $N_e - M$ also come from the same ...
1
vote
0answers
82 views

Why is there no unbiased estimator for $\frac{1}{\theta}$ for Poisson Distribution?

Suppose that $X_1,\dots,X_n$ is an iid random sample from a Poisson distribution with mean $\theta$. I would like to prove that there exists no unbiased estimator of $\frac{1}{\theta}$. To do ...
1
vote
0answers
22 views

Compute the limit of the error term of Taylor polynomial of $f(x)=e^{3x}-1$

I wonder how to compute the limit of error term in Taylor polynomial of exponential functions (i.e. $f(x)=e^{3x}-1$). First, I found the Taylor polynomial of $f(x)$, which is $$T_n ...
1
vote
0answers
31 views

Estimate on the exponential integral of a complex argument (a reference needed)

Consider the exponential integral of the complex argument defined by $$ Ei( z ) = \gamma + \ln(-z) +\sum\limits_{ n = 1 }^{ \infty } \frac{ z^n }{n n!}, $$ where $ z \in \mathbb{C} \backslash ( ...
1
vote
0answers
13 views

Least Square solution for approximating a sequence

Suppose I have a sequence of length N $a_1,...,a_N$ I want to approximate this sequence by $k^1,...,k^N$ where $k$ is my variable. What is the least square solution of this? is there a closed ...
1
vote
0answers
21 views

Proof of differentialbility in mean square calculus?

let $x_t$ be a mean squared Riemann integrable over $[a, t]$ for every $t\in[a,b]$. Then $y_t=\int\limits_a^t x_\tau d\tau\ $ is mean squared continuous on $[a, b]$. Furthermore, if $x_t $ is mean ...
1
vote
0answers
57 views

Polynomial roots finding algorithm

My initial problem is a parameter estimation problem that is solved by minimining a least-square criterion with the Gauss-Newton algorithm. However finding a good initial iterate is very tedious. ...
1
vote
0answers
44 views

Is there an exact solution for this resampling (synchronization) problem?

I want to know if there is an exact solution for the following problem and how to approach solving it: I have a discrete-time signal where the Nyquist theorem is satisfied: $$ r_k = \sum_i ...
1
vote
0answers
25 views

Estimating compound growth

I have a compound interest function with the following parameters: Value at time 0 = 13.8 Interest rate = 0.05 time interval = 10 I need to check quickly, (without a calculator, only pen and paper) ...
1
vote
0answers
37 views

Estimating the size of my population

I have a following problem: Imagine you have a hat with many different balls in it and you want to estimate, how many balls are totally in the hat. The only think you are allowed to do is to take one ...
1
vote
0answers
50 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
1
vote
0answers
21 views

Squared error consistent is asymptotically unbiased lim$_{n→ \infty} E[(\hat\theta_n - \theta)^2] = 0$

An estimator $\hat\theta_n$ is said to be squared error consistent for $\theta$ if lim$_{n→ \infty} E[(\hat\theta_n - \theta)^2] = 0$ a) Show that any squared error consistent $\theta_n$ is ...
1
vote
0answers
39 views

Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
1
vote
0answers
43 views

Estimate uncertainty of a function

I have a question about estimate the uncertainty of a function. I have a variable $x$. I assume that I can predict the variable $x$ is followed by a function $f(x)$ such as $$f(x)=\exp(-x^2)$$ Now, ...
1
vote
0answers
10 views

How to do density estimation restricted to a linear model (multidimensional plane)?

I would like to fit a plane to sampling data. So I would like to do a density estimation but restricted to linear models on the main domain. Is there any standard method? This is also a problem in ...