The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Textbooks with worked examples on identifiability and regularity

I'm looking for textbooks that would have worked examples on identifiability and regularity, something similar to a Schaum's outline. Are there any textbooks that have worked examples on these ...
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45 views

Find the maximum of an integral function with respect to another function

I'm facing this statistical data analysis problem, where I have to maximize a certain statistic in order to find the optimal filtering function. I'm a little bit out of practice with the mathematics ...
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75 views

Classification problem: admissible rule is a Bayes rule for some prior $\pi$

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$. Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in ...
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39 views

Comparison of two error distributions to determine “goodness of fit”

I am a physicist who is a few years out of doing his last course in statistics, so I am hoping to get some advice when comparing some data I recently generated. The context is as follows. I have two ...
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63 views

Hypotesis test: $X_i | \theta \sim Exp(\theta)$ (Likelihood Ratio Test)

Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$ P.S: ...
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72 views

Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
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75 views

Hypothesis testing with poisson distribution

At a nuclear plant great care is taken to measure the employees health.These are the number of visits made by each of the 10 employees to the doctor during a calender year. 3,6,5,7,4,2,3,5,1,4 ...
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470 views

Exponential integral approximation

I have an equation that contain exponential integral of the form: $$ \begin{equation} E_k\left(\frac{a+b ~x}{c}\right) \end{equation} $$ Where $k\geq 0$ ($k=0,1,2,...$), $a$, $b$, and $c$ are ...
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90 views

Properties of almost sure convergence

If, $\Sigma$ is the population covariance matrix and $S$ is the sample covariance matrix, $p$ is the number of variables, $\frac{p}{n} \rightarrow c$ as $n \rightarrow 0$, $\frac{1}{p}|S|_{F}^{2} ...
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85 views

Nondimensionalize an equation with logs

I have an equation with logs in it. Specifically the equation takes the form: $log(Y)=\alpha+\beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3$ Where $X_1$ through $X_3$ are variables. The trouble is that Y ...
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21 views

How to test that a a time series follows a Markov process?

I am searching for a statistical test that tells wether a finite-alphabet time series is a Markov process of a given order or not.
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59 views

Interval overlap maximisation problem

Consider a line of equally spaced sensors and a disturbance which travels unidirectionally along the line at a fixed speed such that the disturbance takes time $\tau$ to travel between adjacent ...
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26 views

Point Estimator.

When the point estimator under consideration has a pdf , the $P[T=\tau(\theta)]=0 $ where $\tau(.)$ is some function of parameter $\theta$ T is an estimator of $\tau(\theta)$ But i did many ...
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75 views

Calculation of conditional joint probability given certain conditionals for data which aren't independent

The context of this problem is the estimation of the distribution of a parameter $v$ given sets of data $A$ and $B$, where $A$ and $B$ are not independent. Suppose I know $P(v | A)$ and $P(v | B)$. ...
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25 views

Choosing an appropriate part of an unreliable dataset

I have a dataset of ~2000 entries (for example, model of car). For each car model I know the weight of the car, and the power output. I don't know the price or age, which is likely to affect the ...
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42 views

Minimum distance estimation approach in inference

I am having a problem with a very basic concept in the minimum distance estimation approach in statistical inference. I've read a paper which uses, for a parametric model family of discrete ...
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42 views

Run test statistics

I'm currently implementing the Wald-Wolfowitz runs test on a variable $\epsilon \in \{0;1\} $. According to the original paper, the number of runs, $V_n$ converges to the normal distribution with: ...
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73 views

Survival Analysis

I have some survival times which are exponentially distributed for two groups G1 (treatment) and G2 (control). The data are censored with a censoring distribution given by h(c), so I only observe: 1) ...
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1k views

Beta Distribution Sufficient Statistic

So I have this homework problem that I am struggling a little bit with coming to a solid answer on. The problem goes like this: Suppose X~Beta($\theta,\theta), (\theta>0)$, and let $\{X_1, X_2 , ...
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36 views

Test the hypothesis that $B_1=0$ at the $5 \%$ significance level

I am told to test the hypothesis and this is what I did: $H_{0}:\beta_{1}=0$ $H_{a}:\beta_{1}\not=0$ So then I have $$t^{*}=\dfrac{\hat{\beta_{1}}-\beta_{1}}{\dfrac{s_{\beta_1}}{\sqrt n}}$$ ...
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112 views

Need help deriving plug-in (functional of distribution) estimator

I need help with homework exercise, have no idea how to approach it. Assume we have i.i.d. observations $x_1,\ldots,x_n$ of a continuous random variable $X$, taking values in $\mathbb R^+$. Define ...
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227 views

Classification: Why k-Nearest Neighbor method is more appropriate for a Mixture of Gaussians?

I'm reading a book named "The Elements of Statistical Learning" in which it states 2 scenarios when we are trying to predict the class label: Scenario 1: The training data in each class were ...
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36 views

What is a good way to deal with finite trains from random variables?

EDIT: Disregard this question if it is formulated too confusing. This link provides you with the updated version of the same question What is the most powerful test for process discrimination based ...
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173 views

What statistical hypothesis test to use for comparing results of two equations?

Given a function $f\left(x\right)$, I have two formulas to compute the coefficients of the same harmonic series approximation to $f\left(x\right)$. Call the results of each formula $^1c_k$ and $^2c_k$ ...
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49 views

Estimate the size of a set given random sub sets.

Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible. I'm not going to quibble ...
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15 views

Confusing on some concepts of sufficient principle

Im reading Chapter6 of Casella Berger's statistical inference that talks about sufficiency principle. I've been confused a lot by the definition of sufficient statistics, here it is: Basically, ...
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18 views

Determining the weights of known parameters in a formula

I have a formula of the following form: $a_1*w + a_2*x + a_3*y + a_4*z$ In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the ...
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8 views

Approximation of objective based on statistical distance

I am a computer science researcher (mostly theoretical) currently in midst of statistics and not able to figure out how to proceed. At an abstract level, I have a hypothesis for an unknown ...
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9 views

testing variance composite vs composite hypothesis normal distribution

I have a random sample from $\mathrm N(\mu,\sigma^2)$ with $\mu,\sigma^2$ unknown and I'd like to test the following hypotesis $ \mathrm H_0: \sigma^2 \le0.6$ vs $ \mathrm H_1: \sigma^2 >0.6$ ...
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17 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
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Transformation of the LR-test for normal data

Given an iid normal sample $X_1,...,X_n$ with unknown mean $\mu$ and unknown variance $\sigma^2$, we want to test $H_0:\sigma^2=\sigma_0^2$ vs. $H_1:\sigma^2\not=\sigma_0^2$ using the likelihood-ratio ...
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22 views

Interaction term (second order term) always implie the linear term (first order term)? (GLM)

I am doing a GLM model with interaction terms, but the same question can be asked with an ANOVA model. Suppose I have two independent variables $X_1,X_2$ and $Y$ the dependent one. I notice that, ...
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19 views

Prediction Intervals for Discrete Data

I would like to generate a prediction interval as described [here]. I suspect that my data comes from a normal distribution, but it was discretized into bins. How would one construct prediction ...
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20 views

Application of Multivariate Analysis

The following situation is proven valuable where multivariate analysis can be applied. This example is taken from the book Applied Multivariate Statistical Analysis ...
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Hypothesis testing, find $n$ to limit type II errors

I'm trying to solve the following (basic) problem, but I don't seem to get the last part 3). The average lifespan of a certain mechanical part is set to be at least 2 years by experts. The ...
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11 views

Standard deviation of a sampling distribution

I have a question regarding the following video: https://www.khanacademy.org/math/probability/statistics-inferential/hypothesis-testing/v/hypothesis-testing-and-p-values From time 4:13 in the video. ...
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14 views

Aditive Outliers in Seasonal Time Series

I have adjusted a $SARIMA(1,0,1)(0,1,1)_{12}$ for my serie. Trying to improve my model I have detected with SAS aditive outliers in MAY and JUNE. My question is how to solve this outliers because if ...
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12 views

Doubt about the degree of freedom on standard error

Once I learn that the sample variance should be $s^{2}=\sum(X_{i}-\mu)^{2}/(N-1)$, I was told that "N-1" is came from the degree of freedom, even I didn't understand well the reason. Now, I am doubt ...
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18 views

Finding variance with method of moments

In part (a), I know Var(Ni)=p(p-1) but how do I find the variance of the estimator with this result? And for part (b) of this question, I have to clue on how to tackle this question. I don't get ...
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33 views

Prove that risk function is analytic?

I'm considering the statistical minimax estimation problem of the bounded normal mean: Specifically, the problem is to find the minimax estimator of $X \sim N(\theta,1)$ where $\theta \in ...
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27 views

$100(1-\alpha)$% confidence interval

$100(1-\alpha)$% confidence interval for population mean $\bar Y$ $$\bar y\pm Z_{\frac{\alpha}{2}}\sqrt{\mathbb v(\bar y)}$$ Why is this $Z$ value depending on $\alpha$ important for constructing ...
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8 views

Calculate probability distribution $p\left(\left.X_{1:T}\right|Z_{1:T},y_{1:T}\right)$ in linear- non-Gaussian state space model.

I have a linear, non-Gaussian state space model. Observation equation: $y_{t}=a+bX_{t}+cZ_{t}+\epsilon_{t}$ $\,\,\,\,$ $\epsilon_{t}\sim\mathcal{N}\left(0,\omega^{2}\right)$ Transition equations: ...
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20 views

Relative Error of Normal Approximation

I have this math statistics assignment that I worked out and came out with a couple of graphs. How do I interpret the graphs? Task: If you consider the average of 20 iid rvs, normal distribution is ...
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8 views

How to prove that noncentral chi-square distribution is stachastically increasing?

How to prove that noncentral chi-square distribution is stachastically increasing in n (degree of freedom) and sigma (noncentrality parameter)
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11 views

How to interpret Standard Residual Deviation?

have fitted regression lines for 200 data sets where each data set contains 40 data points.As a result of that now I have 200 Standard Residual Deviation values and need to interpret that results. How ...
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12 views

Correct choice of analytical statistic method for time series problem

I want to make a statement about corruption (y) influenced by the ratification of the UN-convention-Against-Corruption "contract" (x). Luckily, most of them signed it within 3 years. So, I have ...
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15 views

Test whether the proportion of days which do not have “Good” air quality has changed.

I have the following data of AQ-value $$36,52,54,60,60,31,22,13,67,59,52,54,30,12,33,39,50,51,34,56,62,51,33,39,23,12,35,28,41,54,44,66,42,33,28,22,36,57,46,39etc.$$ I have total $364$ such values in ...
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31 views

Why does $\int f(x)(y-r(x))\;dP(y,x) = 0$?

My question is, why does: $$\int f(x)(y-r(x))\;dP(y,x) = 0,$$ where $r(x) = \int y \;dP(y|x)$ and $P$ is a probability distribution function. It was also given (in my book) that: $$\int ...
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19 views

Find the UMP test for Hypothesis testing

Let $X_1,...,X_n$ be a random sample from the $N(μ,σ^2)$. Assume $μ=0$ (a) Find the UMP test for $H_0 : σ^2=σ_0^2$ versus $H1 : σ^2 < σ_0^2$ at signicant level $\alpha$. (b) For the UMP test in ...
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8 views

Asymptotic confidence interval

Let x1, x2, ..., xn be a random sample with a density function given by $ f(x) = \frac{3}{\theta^3} x^2 I_{(0,\theta]}(x)$ where $I_{(0,\theta)}(x)$ is the indicator function and $\theta > 0$ ...