Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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Complete statistic for joint distribution of independent variables [closed]

If $X_1, ..., X_m\sim U(0,\theta)$ and $Y_1, ..., Y_n\sim U(0,\theta ')$, then $X_{(m)}$ and $Y_{(n)}$ (the last order statistics) are complete sufficient for $X$ and $Y$. Is there a straightforward ...
Charlie Roth's user avatar
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How to interpret a noninformative joint prior?

I am currently working on a homework assignment and have the following question: $\theta_1$ and $\theta_2$ are parameters of interest and $y_1$ and $y_2$ are the likelihood functions which are $\text{...
ak_mng's user avatar
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3 votes
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Cramer-Rao / Wolfowitz bound with nuisance parameter

Let $F$ be a distribution with two parameters, $\theta$ and $\phi$, whose values are non-random but unknown. Consider a sampling procedure in which $N$ samples $x_1, \ldots x_N$ are obtained from i.i....
Luis Mendo's user avatar
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Expectation and variance of bivariate skew normal distribution

I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of ...
Kasthuri's user avatar
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Running a Regression and Unclear How to Think About the Problem

I am wanting to run a linear regression to better understand how force is a function of position in terms of delaminating/warping face frames. This is with respect to a product manufactured related to ...
Joseph H.'s user avatar
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14 views

Expected Variance of EM Estimator in GMM with Respect to Observations

Title: Variance of EM Estimator in GMM with Respect to Observations Body: I'm estimating a parameter S from observations X and <...
Alireza Ghazavi's user avatar
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24 views

What type of statistical test could I use in this case? [closed]

I measured the rxn (reaction) time of a certain experiment at 2 temperatures (20C and 30C) and at 3 pH levels (5, 7, 9). Therefore, there were a total of 12 treatments. However, for some treatments I ...
Sp2221's user avatar
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1 answer
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Optimization of fault diagnosis sequence using probability and cost [closed]

I am developping the algorithm to optimize the fault diagnosis sequence using probability and cost. For exemple, I have 3 diagnosis actions possibles : option 1 : probability which can detect the root ...
stat_man's user avatar
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22 views

Proving how scaling of predictor variable in linear regression, affects the fitted coefficient [duplicate]

In linear regression the OLS solution is given by: $$ \hat{\beta} = (X^TX)^{-1}X^TY $$ I want to show that if you scale the $i$th predictor variable by a constant, then the corresponding $i$th ...
Dylan Dijk's user avatar
1 vote
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Distribution of $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ [closed]

Given $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$, find the distribution of the sample inter quartile range, $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ in terms of $F$ where, $F_n$ is the emperical distribution ...
reyna's user avatar
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Understanding proof of variance of mean of random effects model and relationship to E[MS_treatments]

My textbook makes the following proof: But I cannot understand how it arrives. The variance of $y_{ij} = \sigma^2 + \sigma_t^2$ by assumption this leads to: $MS_e = \sigma^2$ $MS_{treatments} = \...
Estimate the estimators's user avatar
4 votes
1 answer
101 views

Probability mass function of sample median (Bootstrap)

Consider a sample $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$. Let $T_n=F_n^{-1}(1/2)$ be the sample median where, $F^{-1}(x)=\inf\{t:F(t)\ge x\}$ and $F_n(y)=\frac{1}{n}\sum_{i=1}^n\mathbb{I}(X_i\le ...
reyna's user avatar
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Is it okay to report zero eigen values for some PC components if the first few components are greater than one?

I have a dataset which collected household responses over 3 years. It is unbalanced since not all households participated during the three surveys. I run multilevel PCA in stata to account for within ...
Zainab Hassan's user avatar
2 votes
1 answer
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Combinatorics: create a 3-digit number out of 2 digits?

We have a set of two numbers {1,2} an we wish to create a 3-digit number out of the two numbers without order mattering. Here is my approach since there are 2 values for each spot then it's 2^3 since ...
manylya's user avatar
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2 votes
1 answer
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Time series model selection based on demand

I am working on a time series forecasting project in my office. We have to forecast the demand for 10 products at each store level (no hierarchy). Just at store level is enough..We have 20 stores.So, ...
The Great's user avatar
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1 answer
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Simple vs. Composite Hypothesis Question

I am looking at one of the questions in the qualifying exam, where a multiple logistic regression model is being fitted, where the response is about dying. The question says that there are multiple ...
Texas2022's user avatar
1 vote
1 answer
29 views

Derivation of Closed-Form Gompertz-Makeham Life Expectancy

I am trying to replicate the derivation of the life expectancy formulas from Missov & Lenart (2013). They propose that the following: Theorem 2. Life expectancy of the Gompertz–Makeham mortality ...
Martin Georg Haas's user avatar
1 vote
0 answers
63 views

How to deal with a indicator function in a likelihood? [duplicate]

Question: A panel study followed 25 married couples over five years. One item of interest is the relationship between divorce rates and the various characteristics of the couples. For example, the ...
Irohas's user avatar
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2 votes
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Order of an exponential family possibly less than the dimension of the convex support and parameter space

I am going though Brown's book on the Fundamentals of Exponential Families. I am puzzled by the statement that he makes on page 16 where it is possible for $\text{Order of the exponential family} <...
amc's user avatar
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ECM Specification of ARDL model

I have a question regarding the model reparametrization of an ARDL model. Consider the following ARDL$(p,q,q,\ldots,q)$ model: \begin{equation} y_{it} = \alpha_i + \sum_{j=1}^{p} \lambda_{ij} y_{i,t-j}...
Maximilian's user avatar
2 votes
1 answer
140 views

Since $f(x)$ is a pdf, we must have that $(1 / \sigma)f((x - \mu)/\sigma) \ge 0$ for all values of $x$, $\mu$, and $\sigma$? Incorrect reasoning?

Theorem 3.5.1 of Statistical Inference, second edition, by Casella and Berger, says the following: Theorem 3.5.1 Let $f(x)$ be any pdf and let $\mu$ and $\sigma > 0$ be any given constants. Then ...
The Pointer's user avatar
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Most powerful test when the level of the test is 0

Suppose we have continuous-time distributions with densities $f_0(x)$ and $f_1(x)$ under the null and alternative hypotheses, respectively. Show that if the level of the test is $\alpha=0$ then $\phi(...
Cyno Benette's user avatar
1 vote
1 answer
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Deriving Sample version of Anderson Darling test statistic from the theoretical version

In literature, I have seen two types of Anderson-Darling test statistic. One is expressed as $A_T^2 = n\int_{-\infty}^{\infty}\frac{(F_n(x)-F(x))^2}{F(x)(1-F(x))}dF(x)$ and the other is given by $A_s^...
DevD's user avatar
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4 votes
1 answer
201 views

Is there a mathematical function with a shape like an inverted sigmoid pdf but X values that go beyond 1?

I am looking for a mathematical function that can produce a shape like the one below: The above curve has probabilities ranging between 0 and 1, with the x values being able to take any positive ...
André.B's user avatar
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Skew vs one sided standard deviation [closed]

If a unimodal dataset x with mean 0 E(x) = 0 is positively skewed E(x^3) > 0, does it imply that mean squared value of positive data (y: x/ x>0) is more than negative data (z: x/ x<0) ...
dayum's user avatar
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Concentration around the median implies concentration around the mean [duplicate]

Let $M$ denote the median of a function $f(X)$ that is Lipschitz continuous with $\left \| f \right \|_{Lip}=1$. I am trying to show that if $\left \| f(X)-M \right \|_{\psi_{2}}\leq C$, then $\left \|...
Shawn Kemp's user avatar
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Numerical quadrature for Pareto distribution

I would like to numerically evaluate an integral of the following type, when evaluating $f(x)$ at any given point is numerically costly: $$ \int_{x_m}^\infty x^{-\alpha}f(x) \, dx, \quad \alpha >1, ...
spellard's user avatar
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0 answers
8 views

Best method for building models using cumulative autocorrelated data

context: As an example, say I am trying to find the causes of maintenance on a vehicle using its use. With an ultimate aim of understanding the relationship between the two. If I only have variables ...
Pooman19's user avatar
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0 answers
53 views

Given a defect rate, what is the minimum number of samples need to be tested to confirm that the issue is fixed with 90% confidence? [duplicate]

For example, we find 13 defects out of 1000 random samples tested. Now the issue is fixed. What is the minimum number of samples we need to test to confirm that the defect rate is reduced to 0.1% with ...
TreeCity's user avatar
15 votes
2 answers
956 views

The Bing Tibetan Glitch Emoji Problem

Here's a very interesting mathematical statistics puzzle that I randomly stumbled into while using Bing. This turned out to be deep enough that I spent quite some time thinking how one could solve it! ...
Mike Battaglia's user avatar
1 vote
0 answers
26 views

Width of Confidence Intervals for Variance Estimates in Contrast to Point Estimates

We are conducting a variance decomposition using a hierarchical linear random effects Bayesian model to investigate the variance in a DV that is affected by three nested layers. We estimate credible (...
james_westfield's user avatar
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1 answer
33 views

Correlation Metric for Investigating Relationship Between two independent variables

I'm working on a study in computer engineering where I'm analyzing the relationship between the dynamic execution count (frequency) of instructions and the corresponding data error rate (SDC) in a ...
husin alhaj ahmade's user avatar
2 votes
1 answer
119 views

Karlin-Rubin theorem: relationship between test statistic having the MLR property vs being sufficient

Let's suppose we are trying to compare two hypotheses for a single parameter $\theta$. The null hypothesis $H_0$ is that $\theta = \theta_0$, and the alternative is that $\theta ≥ \theta_0$. The ...
Mike Battaglia's user avatar
3 votes
0 answers
64 views

If $A^2$, and $B^2$ are DEPENDENT random variables, will $A$, and $B$ be necessarily DEPENDENT too?

I know that if $A$, and $B$ are independent, the independence is preserved for $A^c$, and $B^c$, where $c$ is a constant. I am wondering if the same applies to the case where the random variables are ...
Roberto's user avatar
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6 votes
3 answers
377 views

Is it meaningful to test the interaction effect if there is no significant effect in the main effect model?

Suppose we have a linear regression model with two covariates, $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2$. There are three possible scenarios: Both $\beta_1$ and $\beta_2$ are significant. Either $\...
zjppdozen's user avatar
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0 answers
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Maths Questions - Probability distribution of choices

I need help with a question I am trying to work out: Individuals have 5 choices: Choice 1: utility1 = alpha * R1 + beta * C1 + random_shock_1(mu=0,sig=1) Choice 2: utility2 = alpha * R2 + beta * C2 + ...
Laurence_jj's user avatar
4 votes
1 answer
108 views

Do we ever use the prior predictive distributions of Bayesian Statistics?

As my question states, I am wondering if there is any chance we use the prior predictive distribution. I am studying Bayesian Statistics and have understood what it is. It is a must to go through in ...
mathccino's user avatar
12 votes
3 answers
1k views

Why is method of moments (MoM) not unique? What is uniqueness?

I just learned that a MoM estimator is not unique... What does that mean? What does being unique mean in general? For example, if we were to find the mean using a MoM estimator for a uniform ...
chunguc1004's user avatar
6 votes
1 answer
111 views

Shattering with halfspaces

I came across the result that, for the concept class of linear predictors where the weight vector has zero-norm $k$, I am able to shatter logarithmically many points with respect to the class where $k=...
redbull_nowings's user avatar
1 vote
0 answers
15 views

Formula to find percentage of winning between two different numbered dice

Lets say I have a dice with the numbers from 1 to 10 and another one numbered from 1 to 7. I roll both dices once. Whats the chances in percentage that the 10 numbbered dice will win in terms of ...
Flo's user avatar
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2 votes
2 answers
65 views

How can I get the probability of a predicted outcome conditional on the posterior in bayesian regression?

Assume that I am running the following regression: $\hat{y_t} = \beta_0 + \beta_1 \cdot x_t$ where as $\hat{y_t}$ is a continuous variable. Lets assume a gaussian likelihood and nonconjugate priors ...
karl henriksson's user avatar
0 votes
0 answers
23 views

Compare Root Mean Square Values

I'm trying to compare a regression neural network to a commonly used equation. I have an 80:20 split for my training:test, and I get the root mean square error on the test set from the neural network ...
Jack789's user avatar
4 votes
2 answers
174 views

Proof that posterior does not depend on data breakdown

Suppose one calculates a posterior distribution on some parameter $P(\theta\mid x)$ based on some prior. Now suppose we take that data and divide it into two parts, such that $x = x_1 \cup x_2$. ...
Nathaniel Bubis's user avatar
0 votes
0 answers
27 views

Demonstration of Convergence in Probability of the Average Prediction Error for a Consistent Machine Learning Algorithm

I'm quite new to this topic, but I've set myself the task of understanding how to demonstrate that the average of prediction errors in the sample for a machine learning algorithm, which consistently ...
Tomás Rubio's user avatar
3 votes
0 answers
44 views

$p$-hacking and nested models

I have a model $\vec{y}=X\vec{\beta}+\vec{\varepsilon}$ with guassian independent errors $\vec{\varepsilon}\sim N(\vec{0},\sigma^2I_n)$. Let $n$ denote the number of rows of $X$ and $d$ denote the ...
Terence C's user avatar
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1 vote
0 answers
51 views

The logic behind the derivation of the autocovariance generating function?

I am working with complicated sets of time series, where an observed system output is the sum of a number of individual ARMA, error and noise series. I am therefore looking at ways to identify these ...
hydrologist's user avatar
1 vote
0 answers
20 views

After the mutiple imputation (MICE package in R), I still found that some variables are still with missing values. How to deal with it?

I have a relatively large data set with around 12000 samples with 550 variables. Originally, I have around 800 variables, I used a rule that if missing rate in each variable is larger than 80% I will ...
Steven Xu's user avatar
4 votes
1 answer
48 views

Conditional distribution of $X_i|\overline{X}$

Suppose $x_1 \cdots x_n$ are iid random variables with normal distribution of mean $\mu$ and variance $1$, and $\overline{X}=\frac{1}{n}\sum_i^n x_i$ What is the conditional distribution of $\begin{...
user404474's user avatar
0 votes
0 answers
8 views

Overdispersion Mixed generalized linear model

I am running a mixed generalized linear model to analyze insect capture in a baited trap. The experiment consisted of 3 separate cages, in each one one treatment (C+, C- or T) and 10 insects were ...
Andrea88's user avatar
1 vote
0 answers
45 views

UMP size $\alpha$ hypothesis test for discrete distribution

Let $\left(Z_1, \ldots Z_n\right)$ be an i.i.d. sample with common probability mass function $$ p_Z(j)=\vartheta^2 I_{[j=1]}+2 \vartheta(1-\vartheta) I_{[j=2]}+(1-\vartheta)^2 I_{[j=3]} $$ Find a UMP ...
Stats_Rock's user avatar

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