For questions about estimation and how and when to estimate correctly

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20 views

Is this estimation of an double integral right?

I'm in the setting, where $\Omega$ is a bounded domain, $t_0 > 0$ is fix and I have $g, \ h \in L^\infty((0,t_0);L^1(\mathbb{R}^n \setminus \Omega))$. So this should say, that $g$ and $h$ are ...
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0answers
31 views

Big Oh notation problem

Retaining the three terms in the series, estimate the remaining series using "Big Oh" notation with the best integer value possible, as $x\to 0$. The series is $$\ln (\tan (x)) =\ln(x)+ ...
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0answers
24 views

Variable value estimation for given product/fracture values

I have a data set (time series) with given values for certain fractions xy = x/y (where x,y are not constant over time) Thus, there are following fractions: AB = A/B CB = C/B AD = A/D CD = C/D AE = ...
2
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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 ...
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0answers
9 views

How to interpret the Quantified properties of estimator?

https://en.wikipedia.org/wiki/Estimator The link provides a very good explanation of the estimator. I am beginner to statistics and inference , so i have some confusion about the quantified properties ...
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1answer
26 views

Sufficient statistics problem

$X_1, X_2, \ldots, X_n$ are iid $N(0,\theta), 0 < \theta < \infty$ Show $$\sum_{i=1}^{n} X_i^2$$ is a sufficient statistic for $\theta$. My attempt at this is $S = (X_1^2 + ...
4
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1answer
40 views

Cramer-Rao lower bound for any unbiased estimator

The first part of a question I am trying to solve asked to find the maximum likelihood estimator for $\theta$ for a pdf $f_X(x)=\frac{2x}{\theta^2}$, $0 < x \le \theta$ , $0$ otherwise. ($X_1, ...
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0answers
21 views

Minimum mean square estimator for closing value of the NYSE

I am attempting to develop an estimator for the closing value of the NYSE $x(n)$ based on previous $N$ closing values, $x(n-1), x(n-2), ... x(n-N)$. I want to find the Minimum Mean-Square estimator ...
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1answer
119 views

Cost Function of Neural Network (Forward Propagation)

This question is related to Andrew Ng's machine learning course on Coursera. Basically, when I calculate the cost function of a neural network, I use the following formula that was described by Ng: $$ ...
1
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1answer
68 views

Determining the MVUE of $ f(x;\theta) = \theta^x (1-\theta)$.

The Statement of the Problem: Let $X_1, X_2, ... , X_n$ be a random sample from $$ f(x;\theta) = \theta^x (1-\theta) \quad x = 0,1,2,... $$ (a) Find the ML estimator of $\theta$. (b) Show that $T ...
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1answer
43 views

How many bathtubs in a min?

During the summer about $750,000$ gallons of water fall over the edge of Niagara Falls every second. If an Olympic sized swimming pool holds about $660,000$ gallons of water, how many Olympic sized ...
2
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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 ...
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2answers
26 views

How to estimate magnitude of expontent?

When given an exponent, such as 6^12, is there a simple way to approximate how large(magnitude) the result is, without performing the calculation? Is this method accurate for large exponents?
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0answers
181 views

Confidence Interval for Incidence Rates

I have a huge sample of patients followed up for a certain Event. I would like to calculate the following crude incident rates: #{Events}/(1000*PersonYear). My sample is big enough to assume that this ...
2
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2answers
97 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a fixed positive integer ($m>1)$ and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series. My question here is : Is $\lim S_{n,m} <\infty $ as $ n ...
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1answer
82 views

How to calculate entropy from a set of samples?

entropy (information content) is defined as: $$ H(X) = \sum_{i} {\mathrm{P}(x_i)\,\mathrm{I}(x_i)} = -\sum_{i} {\mathrm{P}(x_i) \log_b \mathrm{P}(x_i)} $$ This allows to calculate the entropy of a ...
0
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1answer
64 views

In terms of $a, b,$ and $\theta$, what is the biased $b(\hat \theta)$?

The Statement of the Problem: Let $\{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Suppose that $\hat \theta$ is an estimator for a parameter $\theta$ and $E_{\theta}(\hat \theta) = ...
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2answers
41 views

Expected value and the standard simple regression model

Given the standard simple regression model: $y_i = β_0 + β_1 x_i + u_i$ What is the expected value of the estimator $\hat\beta_1$in terms of $x_i, \beta_0$ and $\beta_1$ when $\hat\beta_1=\sum x_i ...
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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 ...
3
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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 $\displaystyle a_n \approx \sum_{k=1}^{n}\frac{1}{k^5}+R(n)$ ...
6
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1answer
79 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
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0answers
23 views

Establishing consistency

I need to establish the (weak) consistency of an estimator of the mean, $T=a+b\bar{X}$. I tried to apply Chebyshev's inequality, but I couldn't do much because the parameter that subtract in the ...
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3answers
61 views

Confidence interval for sample

I have a sample of size $n=19593$ of count data ...
1
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1answer
31 views

construct confidence interval from proportions

Suppose you have a population of count data, i.e., $1,2,3, \dots, k$, you have a sample of the population of size $n$, and you have a confidence interval for the proportion of $1$'s , $2$'s,\dots$n$'s ...
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0answers
100 views

Gauss-Legendre quadrature error

I'm trying to evaluate the error in Gauss-Legendre quadrature formulae on $[a,b]$. So far I have that the error is less or equal to $$ \frac{f^{(2n)}(\xi)}{(2n)!}\langle p_n,p_n \rangle, \enspace ...
2
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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 ...
0
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1answer
24 views

Solve the system of equations by variable estimation

Solve the system of equations: $\left\{\begin{array}{l}(x-1)\sqrt{x-y^2}=y(x-2y+1)\\y\sqrt{x-1}+3\sqrt{x-y^2}=2x+y-1\end{array}\right.$ I guess there is only one solution $(x;y)=(2;1)$. This is my ...
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2answers
156 views

Estimate how many times to flip a coin to get at least 30 heads with probability of 80%

Im completely stumped by this problem. It goes as follows: Estimate how many times a fair coin must be thrown in order to obtain at least 30 heads with a probability of 0.80. Ive tried playing with ...
6
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1answer
212 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
0
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1answer
42 views

Bayes estimator under squared error loss

Consider one random variable X from the Bernoulli distribution with parameter θ. Let p, the prior density, be equal to 6θ(1 − θ), for θ ∈ (0, 1). Under squared error loss, L(t, θ) = (t − θ)$^2$, the ...
2
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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 ...
0
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1answer
40 views

Functions of polynomial growth and the Schwartz space

A smooth function $m \in \mathcal C^\infty(\mathbb R^n)$ is said to be slowly increasing if for all $\alpha \in \mathbb N^n_0$ there exists $C_\alpha, k_\alpha$ such that $|\partial_\alpha f(x)| \leq ...
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2answers
289 views

Maximum a Posteriori (MAP) Estimator of Exponential Random Variable with Uniform Prior

What would be the Maximum a Posteriori (MAP) estimator for $ \lambda $ for IID $ \left\{ {x}_{i} \right\}_{i = 1}^{N} $ where $ {x}_{i} \sim \exp \left( \lambda \right), \; \lambda \sim U \left[ ...
3
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1answer
62 views

What are the bounds (upper and lower) for $|A+A|$?

Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is ...
2
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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 ...
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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. ...
3
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3answers
62 views

Bounding $\sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$

I'm looking for a bound depending on $N$ of $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}}$. The following holds $\displaystyle \sum_{k=1}^N \frac{1}{1-\frac{1}{2^k}} = \sum_{k=1}^N ...
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1answer
71 views

Variance of sample estimator [closed]

I have a sample estimator that I want to calculate the variance of. The estimator is \begin{align*} \hat\sigma_1=\sqrt{\frac{1}{n}\sum_{i=1}^nx_i^2}\\ \end{align*} How do I calculate ...
2
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0answers
18 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| ...
0
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1answer
102 views

Deriving Point Estimate Based on Sample Mean with λ

This is a review question I'm trying to solve. I didn't receive a direct answer, only a few tips from my professor, and I want to see if I'm moving in the right direction. It's a very general ...
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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 ...
2
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2answers
101 views

Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
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0answers
21 views

Estimating a population in multiple locations

I have a set of sequential data of unknown size randomly spread across $n$ locations. I am trying to estimate the population size of all $n$ locations and provide a confidence interval as well. This ...
3
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1answer
61 views

Estimate number of songs a radio station has [duplicate]

Imagine the following problem: You listen to a radio station and take notes how often was each song played. How can you estimate based on your notes (e.g. 30 songs played once, 2 played twice, one ...
3
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2answers
48 views

Difficult to understand difference between the estimates on E(X) and V(X) and the estimates on variance and std.dev. on lambda-hat

I'm having a very hard time to separate estimates on population values versus estimates on sample values. I'm struggling with this exercise (not homework, self-study for my exam in introductionary ...
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0answers
37 views

estimations in the birthday paradox?

The birthday paradox is the famous following problem: What is the probability $p_n$ that at least $2$ persons amongst $n$ persons chosen at random have the same birthday? Leap years are not taken ...
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0answers
9 views

Deriving conditional distributions for a normally distributed change point problem

Considering the change point problem of $y_i \left\{ \begin{array}{ll} y_i \tilde{~} N(u_1, \sigma) & i=1,..,t \\ y_i \tilde{~} N(u_2,\sigma) & i= t+1,...,n \\ \end{array} ...
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0answers
20 views

multivariate interval estimation

I have several samples of probabilistic vectors, i.e, each sample is of the form $(x_1, \cdots, x_n)$ such that $\sum_{i=1}^n x_i\leq 1$ (they are sub-probabilistic vectors), how can I obtain a ...
1
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1answer
214 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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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) ...