Questions tagged [mathematical-statistics]
Mathematical theory of statistics, concerned with formal definitions and general results.
4,689
questions
0
votes
1answer
30 views
An easy way to find the PDF of $XY/Z$
Suppose that we know the joint distribution $f_{X,Y,Z}$. I want to find the PDF of $W = XY/Z$ in a way that does not involve too much calculation.
I have thought about first finding the Marginalized ...
0
votes
0answers
7 views
What is Importance Value?
I am reading this paper that talks about Principle Components analysis and Importance Values(IV) here
Is this Importance Value a statistical concept? I could not find any information about it in ...
0
votes
1answer
36 views
Statistical test to verify if a population has a average of k times another
Consider the situation where there are 2 positive distributions with unknown expected values $\mu_1$ and $\mu_2$,
We have samples $S_1 = \{x_1, x_2, ..., x_n\}$ and $S_2 = \{y_1, y_2, ..., y_n\}$.
...
0
votes
0answers
21 views
What is the intuition behind pi in the PDF of a Normal Distribution ? Is it related to some sort to a circle / sphere
The PDF of a Normal distribution is given as below
I am aware of the various properties of Normal distribution and how the two parameters mu and sigma affect the shape of the distribution.
What is ...
0
votes
0answers
20 views
Compute the information matries related to normal distribution
This is a problem that I have trouble with.
Suppose that we have $X_{1}, \ldots, X_{m}$ are iid
$N\left(\mu, \sigma^{2}\right), Y_{1}, \ldots, Y_{n}$ are iid $N\left(0, \sigma^{2}\right),$ the $X$ 's ...
7
votes
0answers
126 views
Random variables $X, Y$ such that $X$, $Y$ and $\sqrt{X + \sqrt{Y}}$ belongs to the same family of distributions?
Is there a family of positive distributions such that if $X$ has the distribution in question, then $\sqrt{X}$ also has a distribution from the same family. Ideally, it would be great if $X+Y$ also ...
-2
votes
0answers
21 views
Probability, Statistics [on hold]
1.Probability of getting two before three, 2. probability of getting 2 before 3 or 5 or 6, 3.probability of getting 2 before 3 or 4 or 5 or 6
1
vote
2answers
50 views
Outlier removal, extremes on both ends
A list of numbers and I want to remove the extremes on both ends.
The standard deviation is calculated: 26.3 (rounded to 1 decimal)
...
1
vote
0answers
14 views
Land Method for Lognormal Distribution
The Land Method is explain here.
Anyone knows how I can look up the H*1-α*
It says "Tables of these values are given by Gilbert (1987, Tables A-10 and
A-12) and Land (1975). " Gilbert (1987) is not ...
-1
votes
0answers
9 views
Product/SKU level time series forecasting [on hold]
Predict demand of product at each outlet for next 6 months
5 – 7 years of sales data at outlet level for each and every brand is available
As it is a time series problem we need to design per outlet ...
0
votes
1answer
14 views
Direction of a one sample Wilcoxon signed rank test
I am comparing a selection of values which come from a non-normal distribution to 0. I've done a wilcoxon in python:
result = scipy.stats.wilcoxon(values)
my W ...
0
votes
0answers
4 views
Demonstrate the internal laws of the circumstance when the inequalities of Sauer's Lemma are strict [on hold]
We proved Sauer’s lemma by proving that for every class H of finite VC-dimension d, and every subset A of the domain,
$$
\left|\mathcal{H}_{A}\right| \leq |\{B \subseteq A: \mathcal{H} \text { ...
0
votes
1answer
24 views
Data Mining/Statistical Methods to find trends, peaks, etc
currently I am working on a project for my final exam. The data is coming from a streaming plattform. The data I am using are some logging data (data when customers have problems with the streaming ...
0
votes
0answers
17 views
Basic results on convergence in distribution
Let $\{X_i\}_{i=1}^n$ be independent zero mean random variables with finite variance and $\{r_n\},\{d_n\}$ positive monotone increasing real sequences. Assume that
$$\frac{\sum_{i=1}^{r_n} X_i}{Var(\...
1
vote
1answer
25 views
Range of integration for joint and conditional densities
Did I mess up the range of integration in my solution to the following problem ?
Consider an experiment for which, conditioned on $\theta,$ the density of $X$ is
\begin{align*}
f_{\theta}(x) = \...
1
vote
0answers
10 views
Is there an example that a probability distribution is having an increasing mean residual lifetime(IMRL) but not a Dreasing Failure Rate(DFR)
It is well known in risk theory that a random variable having a Decreasing Failure rate implies the Random Variable has an increasing mean residual lifetime. However, the converse is not true. Is ...
0
votes
0answers
8 views
Which model is best using the hettest for heteroscedasticity test?
I have calculated hettest using STATA
and i have found two models have no Heteroscedasticity problem. but i can not identify which one is the best model for my data. can you please help me out?
I ...
0
votes
2answers
35 views
What test should i use to prove A is better than B [closed]
I have sets of data that have number of goods produced by 2 machines, A & B for 2 months. Each machines has two numbers that is plan value and actual value. I'm not really familiar with hypothesis ...
2
votes
2answers
69 views
Computing the confidence interval for two samples but getting slightly different answers
Consider two samples $X_1,..,X_k$ and $Y_1,..,Y_m$ where $X_i \sim \mathcal{N}(\mu_x,\,\sigma^{2})\,$ and $Y_i \sim \mathcal{N}(\mu_y,\,\sigma^{2})\,.$ Say $k=m=100$ and $k+m=n$. Say that the ...
6
votes
2answers
87 views
Using all Metropolis-Hastings proposals to estimate an integral
Suppose we run the Metropolis-Hastings with target distribution $\mu$ to compute the integral $\int f\:{\rm d}\mu$. Usually, we use the estimator $$A_n:=\frac1n\sum_{i=0}^{n-1}f(X_i).$$ However, ...
0
votes
1answer
15 views
How to compute P values for Breusch pagan test?
I got bptest() output for my OLS model as follows;
studentized Breusch-Pagan test
data: modelfit
BP = 3.3201, df = 3, p-value = 0.3449
I read that BP test is ...
1
vote
0answers
11 views
Probability of random population value being higher than sample maximum
Considering a small sample size (n < 10) from a population, I'm trying to find how likely a random population value would be greater than the maximum of the sample.
Hoping ye could help me with ...
3
votes
1answer
49 views
linear combination and univariate normal
Show that $(X_1,X_2)$ has a bivariate normal distribution with means $\mu_1, \mu_2$, variances $\sigma _1^2 $ and $\sigma _2^2$, and correlation coefficient $\rho $ if and only if every linear ...
0
votes
0answers
26 views
What distribution can be predicted more accurately?
I am working on simple statistical prediction. However, whatever i did, i come up with a high range. I have used standard deviation and also percentile. Yes they work well in the given data but the ...
2
votes
0answers
28 views
Determining between using Discrete vs Continuous Survival Analysis
This is a follow up question based on what has been explained here: Survival analysis: continuous vs discrete time but I wish to further confirm it for my case.
I have a dataset collected over a ...
1
vote
0answers
5 views
Suggesting multiple labels prediction for multi-class training data
I have a dataset of tickets the machine gets for some machine components failure. The ticket are in text form.
For each failure we have around 8-10 diagnosis labels. This tells what the issue might ...
2
votes
1answer
26 views
Network analysis: Formal definition of the number of 1st order neighbours of degree 1
I am looking for a formal definition of a network metric I am using in a scientific article. Let $i$ be a vertex in a graph $G$ and $N(i)$ are the first order neighbors of vertex $i$. I am interested ...
0
votes
2answers
115 views
+50
Finding $\mathsf P(\textbf{Y}\mid\textbf{Z})$ where $Z_1,Z_2,\ldots,Z_n$ are conditionally independent
As an extension of my previous question which is described below...
Assume $Y_1$, $Y_2$, $\ldots$ ,$Y_n$ are random variables over a
regular lattice indexed by $i= 1,2,\ldots,n$ where
$Y_i\in\{...
0
votes
0answers
11 views
Finding the expected value of Laplace distribution truncated to interval [closed]
Hello I am experiencing difficulties finding the solution for the following Problem: Find the expectation of X where X has a Laplace distribution truncated to interval [5, 9].
0
votes
0answers
15 views
Finding the expected value of Laplace distribution for specific interval
Hello I am experiencing difficulties finding the solution for the following Problem:
Find the expectation of X where X has a Laplace distribution on [5, 9].
I understand how to find the expectation ...
6
votes
1answer
210 views
+50
Example of a non-measurable maximum likelihood estimator
If you have a measurable parameter space $(\Theta, \mathcal{F})$ and a parametric family of probability measures $(P_\theta)_{\theta \in \Theta}$ on a measurable space $(\mathcal{X}, \mathcal{B})$ ...
0
votes
0answers
8 views
Prove that in such cases, it is possible to find an ERM hypothesis for $H_n$ in the unrealizable case in time $O(mnm^{O(n)})$
Let $H_1$ , $H_2$ ,... be a sequence of hypothesis classes for binary classification.
Assume that there is a learning algorithm that implements the ERM rule in the realizable case such that the ...
3
votes
1answer
66 views
Finding conditional probability of an individual component of a joint distribution
Assume $Y_1$, $Y_2$, $\ldots$ ,$Y_n$ are random variables over a
regular lattice indexed by $i= 1,2,\ldots,n$ where
$Y_i\in\{1,2,...,K\}$. Let the probability of a particular
configuration $\...
1
vote
1answer
20 views
relation between location of a sample mean in sampling distribution and the standard error
Can
someone explain the following statement with an example
We'll describe the location of the sample mean by calculating how many standard errors it is away from the center of the sampling ...
2
votes
1answer
47 views
How to compute confidence and prediction intervals for (nonlinear) regression using non squared “loss function”
Does somebody know (or can point me to a reference) how can I compute a confidence and prediction intervals for (posibly nonlinear) regression using non squared "loss functions"?
Let me add some ...
0
votes
0answers
14 views
Applying L’hôpital’s rule to an exponential function yielding different results [migrated]
I was solving problem 2.38b in Casella and Berger when I came across the below issue:
$$\lim_{p \rightarrow 0} f_{P}(p) = p^{r}.[1 - (1 - p)e^{2tp}]^{-r}$$
Applying L’hôpital’s rule to:
$$\lim_{p \...
0
votes
1answer
28 views
precision and recall for biased data
here is my assumption input data's labels are 99% true and there are only 1% false(trying to say most of the data are true).
My classifier's probability is 0.5. just like unbiased coin flip.
in this ...
0
votes
0answers
29 views
Machine Learning Features Recorded At Different Timestamps & Locations
Suppose that you are trying to model the following hypothetical situation - where each of your features is a separate time series, as is the target variable. However the feature time series occur ...
0
votes
0answers
34 views
Is it scientifically feasible to compare two samples of different sizes, while values in each sample are average numbers? [closed]
I want to compare result from my protein docking experiments to various protein structures of the same protein.
I have two samples and in Sample 1, I have ...
0
votes
0answers
10 views
Principal components and how to determine how many principal components to use
Here I've been given the X vector and the means and variances for each variable, how are you suppose to figure out how many principal components would be required to explain 75% of the variance of X, ...
0
votes
0answers
11 views
Principal Components Analysis, Reading PC Plot
For Question 2v, can someone please explain to me why each subject is located where they are in terms of the two principal components.
0
votes
0answers
7 views
Significance Calculation for Difference in Percent of two samples
I have a test and control group. Each has a current year(CY) and prior year(PY) order rate, as well as sample size. From my understanding, I can calculate the significance of the Year over Year(YOY) ...
2
votes
0answers
25 views
Confidence Interval of a bounded variable [duplicate]
Given a random variable $X$ which can take values on $[0,1]$.
I am interested in calculating the confidence values for the mean estimator.
We know that the mean is normally distributed and can ...
0
votes
1answer
15 views
How to modify the mean?
I have data collected in consecutive time periods using two sensors (i.e. sensor1 and sensor2) as follows.
...
1
vote
1answer
46 views
How to compute $Pr(a \leq Y \leq b | a \leq X \leq b)$?
In general, how to compute
$Pr(a \leq Y \leq b | a \leq X \leq b)$ if given $F_{Y|X}(y|x)$ and $f_x(x)$? Notice that $a$ and $b$ for the intervals of the two random variables are the same.
I know how ...
0
votes
0answers
7 views
Statistical error in the approximation-estimation tradeoff
Show that
$$E(g_\tau ^G(X)-g^* (X))^2 = E(X^T \hat{\beta}-X^T\beta^G)^2+E(X^T\beta^G-g^*(X))^2$$
where $g_\tau ^G(X) = X^T \hat{\beta}$ and $g^G(X) = X^T \beta^G$ where G is a class of linear ...
0
votes
0answers
21 views
Expected Optimism 0-1 Loss with 0-1 Response
Want to show that
$$ E_X op = \frac{2}{n} \sum_{i=1}^n Cov_X(g(x_i), Y_i)$$
For 0-1 loss function with 0-1 response.
Want I've done
$$op = l_{in}(g) - l(g)=\frac{1}{n}\sum_{i=1} ^n Loss(Y_i', g(...
1
vote
0answers
42 views
Which statistical test to use to compare 2 tests/measurements against a gold standard
I'd like to know how best to go about comparing 2 tests/measurements against a gold standard. Say I have 100 patients with 3D printed models and CT scans of their hands, and I'll measure a couple ...
0
votes
0answers
11 views
Derivation of variance of Odds Ratio in a 2x2 table
Assuming that the elements of a 2x2 table (a,b,c,d) are binomial in distribution with a being $Bin(n_1,p_1)$ and c being $Bin(n_2,p_2)$. How can we show that $\hat {var(logOR)} = \frac 1 a + \frac 1 b ...
0
votes
0answers
14 views
Sample covariance vs population variance of means [duplicate]
Was looking at this link and wondering why for population covariance, the denominator is n, while for sample covariance, denominator is n-1.
How does this 1/n(n-1) replace the 1/n^2 in the proof in ...
