Questions tagged [consistency]

Refers generally to a property of a statistical procedure to go to the "right" place as the sample size tends to infinity, primarily referring to estimators converging to the true parameter value as the sample sizes diverges. Use also for Fisher consistency, the property that an estimator when applied to the complete population gives the right answer.

Filter by
Sorted by
Tagged with
0 votes
1 answer
29 views

Assumptions needed for consistency of plug-in estimator

Assume $X,Z$ are random variables and let $x_0$ be a fixed number. I want to estimate $A =\mathbb{E}_{X,Z}[\frac{X}{P(X=x_0|Z)}]$. If $P(X=x_0|Z=z)$ is known for all $z$ we can apply the LLN and ...
0 votes
2 answers
26 views

Example of non-consistency of M-estimators in case of pointwise converging criterion functions

When one wants to establish consistency of an M-estimator $\widehat{\theta}_n$, one typically requires uniform convergence of the criterion function $\theta \mapsto M_n(\theta)$. That is, one requires ...
3 votes
1 answer
38 views

What's the relationship between "bias-variance tradeoff" and "consistent model selection"?

I'm very confused about the relationship between "bias-variance tradeoff" and "consistent model selection". Based on my current interpretation, the ultimate goal of taking care of ...
3 votes
1 answer
392 views

Theoretical justification of Parametric bootstrap?

I've been reading about bootstrap, and while it's relatively easy to find theoretical results (consistency and higher-order correctness) for the nonparametric bootstrap (e.g., Asymptotic Statistics by ...
1 vote
0 answers
71 views

Maximum likelihood estimation when the model is misspecified (and the true data generating process is a mixture model)

I'm interested in the properties of maximum likelihood estimators under a particular form of model misspecification: We observe data $\left\{X_i\right\}$ generated from a finite mixture model Let $\...
3 votes
1 answer
306 views

Consistency function in FSelector

I am new in this field and I read some articles on Feature Selection. What does the "consistency" function do in R's FSelector package? For instance, ...
1 vote
3 answers
91 views

Asymptotic normality implies consistency

I'm trying without success to solve the following exercise in my econometric textbook: Show that $\sqrt{N}\left(\widehat{\beta_1} - \beta_1 \right) \xrightarrow{d} \mathcal{N}(0,a^2)$, where $a^2$ is ...
0 votes
2 answers
42 views

How to find a consistency index for a binary variable?

I am working on a project where there are 2 variables to monitor, say sales (actual) and projected sales (target). The sales ...
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 ...
1 vote
1 answer
108 views

Are sample quantiles consistent with population quantiles?

The Wikipedia page about quantiles describes two approaches to the definition of quantiles: population quantiles, and sample quantiles. The section on sample quantiles lists nine different flavors of ...
0 votes
1 answer
54 views

Bias vs consistency in instrumental variable estimation

So in Mostly Harmless Econometrics, page 154, they analyse the bias of instrumental variables: They consider the case of one endogenous variable $x$, multiple instruments $Z$, and $\eta$ is the ...
2 votes
1 answer
87 views

Mathematical Step for consistency

Let me state my problem from the beginning: Let $i$ be an index representing countries ($i = {1,2,\ldots,N }$), and $t$ represent time, denoted as available data for country $i$ ($t = {1,2,\ldots,T_i }...
12 votes
3 answers
16k views

How to prove $s^2$ is a consistent estimator of $\sigma^2$?

I am trying to prove that $s^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2$ is a consistent estimator of $\sigma^2$ (variance), meaning that as the sample size $n$ approaches $\infty$ , $\text{var}(s^2)...
6 votes
2 answers
490 views

"Consistent estimator" or "consistent estimate"?

Question: Are both expressions "consistent estimator" and "consistent estimate" meaningful? The quote below is intended to be illustrative; however, I am interested in the question above in a general ...
1 vote
0 answers
90 views

Sequence of probability measures is consistent

Denote $\varphi (y|x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-(y-x)^2}{2}}$. Show that the sequence of probabilit measures $\\$ $P_n(B) = \int...\int_B\varphi(x_1|0)\varphi(x_2|x_1)...\varphi(x_n|x_{n-1})d(...
1 vote
1 answer
67 views

Does increasing number of observations lead to the decreasing of Mean Square Error of consistent estimators?

I know that not all weakly consistent estimators exhibit MSE-consistency : https://stats.stackexchange.com/a/610835/397467. Anyway, does increasing the sample size leads to a reduction in their mean ...
1 vote
1 answer
96 views

Difference between consistent and unbiased estimator [duplicate]

I have a problem where I have to think of an example to explain a practical example of consistency and unbiased. The example I thought of is the sample mean. Consistency is when the estimator (sample ...
2 votes
1 answer
2k views

Does asymptotically unbiased mean convergence in probability?

I'm arguing with a friend and he doesn't have a solid argument for his statement. I claim that if a random variable is asymptotically unbiased then it is consistent, and that implies convergence in ...
1 vote
1 answer
132 views

Is convergence in probability implied by consistency of an estimator?

Every definition of consistency I see mentions something convergence in probability-like in its explanation. From Wikipedia's definition of consistent estimators: having the property that as the ...
1 vote
0 answers
30 views

Strong consistency of kernel density estimator

I am studying the book Nonparametric and Semiparametric Models written by Wolfgang Hardle and have difficulty with the following exercise: $\textbf{Exercise 3.13}$ Show that $\hat{f_h}^{(n)}(x) \...
0 votes
0 answers
32 views

Calculating MLE under restriction on coefficients

Consider the following simple linear regression model: $y_i = a + b \cdot x_i + \epsilon_i \space\space\space\space\space\space\space where \space i= 1, 2, 3, \cdots , n$ here $\epsilon_i \space's$ ...
0 votes
0 answers
12 views

Metric for run-to-run consistency of time series data

If I run $n$ samples of a physical experiment, I expect to see roughly similar time vs. position plots but with slight variations run-to-run. What are good statistical metrics to quantify the ...
1 vote
1 answer
76 views

Unbiased and constistency of OLS

For the linear regression $y_t = Bx_t+e_t$ where we have the assumptions: $E(e_t)=0$, $E(e_t^2) = \sigma^2$, $E(e_t e_s)= 0$ for $s\neq t $ ...
4 votes
1 answer
553 views

What is an example of a weakly consistent but not strongly consistent estimator?

I just can't think of any example. I am using definitions: weakly consistent: $\forall \varepsilon > 0 \lim_{n\rightarrow \infty} P(|\hat{\theta}_n - \theta| \geq \varepsilon) = 0$; strongly ...
1 vote
0 answers
35 views

Proof of Variable Selection Consistency for LASSO in Zhao & Yu, 2006

I'm going through the proof of Proposition 1 in Zhao & Yu, 2006 (https://www.jmlr.org/papers/volume7/zhao06a/zhao06a.pdf), titled On Model Selection Consistency of LASSO. The proof is in Appendix ...
1 vote
6 answers
331 views

How to create optimal cut-off scores for a test placing students into different courses

Edit: Shared my solution as an answer here Our goal is to determine optimal cut-off test scores for course placement. The course placement has already been manually assigned to each test-taker. The ...
0 votes
0 answers
10 views

Questions about efficiency of estimators with longitudinal data

I'm modelling data with repeated observations; I'm reading up on options and pitfalls, and have a few questions. Coefficient estimates are still unbiased and consistent in the presence of ...
1 vote
1 answer
88 views

Proving consistent/inconsistency of a fusion of KF estimates

I have a distributed fusion scenario with a single target where two sensor nodes $i,j$ estimate the true state $\mathbf{x}$ using a local Kalman filter. The (linear, Gaussian) measurement errors of ...
5 votes
1 answer
346 views

Identifiable but has no consistent estimator

Let $P_\theta$ denote the distribution of the random variable $X$. The distribution depends on the parameter $\theta$ that lies in some parameter space $\Theta$. Consider a function $f(\theta)$ of $\...
2 votes
0 answers
33 views

Why are lags of the dependent variable a no-no in traditional random effects models?

This post says: Lagged versions of the dependent variable are a no-no in traditional random effects models. The problem is that they are correlated with the random intercept and produce inconsistent ...
1 vote
2 answers
93 views

Consistency of the pooled standard deviation estimate

Suppose that $X_{ik}\sim\mathcal N(0,\sigma^2)$ for $k = 1,2,\dots, n_i$ are independent and identically distributed for each $i \in\{ 1,2\}$. Note that I assume equal means ($0$) and variances ($\...
0 votes
0 answers
18 views

Can you use inter-rater reliability to measure reliability of a whole measure with items measuring different things? (Krippendorff's Alpha or similar)

Let's say that you have one case study, where 30 raters are asked to apply a measure that has 5 items. Can you compute a single Krippendorff's Alpha (kalpha) value across all 5 items (that are the ...
0 votes
0 answers
15 views

How to decompose the scaled estimation error of the debiased machine learning estimator? [Chernozhukov et al 2018]

In page C4 of Chernozhukov et al (2018) the authors introduce a debiased ML estimator $\check{\theta}_{0}$ for scalar parameter $\theta_{0}$: $$ Y = D\theta_{0} + g_{0}(X) + U, \quad \mathbb{E}[U|X,D] ...
0 votes
1 answer
57 views

Example consistency of functions

I know that as long as the population moments exist and the data are identically distributed the raw sample moments, the central sample moments and the sample quantiles are consistent. This concept of ...
3 votes
0 answers
83 views

Conditions needed for the convergence of Bayesian posterior distribution to point mass (posterior consistency)?

The following 2 theorems (from Bayesian Data Analytics 3rd edition by Gellman, appendix B) show proofs for why Bayesian posteriors converge to a point mass around θ0. Where θ0 is the true parameter ...
1 vote
0 answers
66 views

Methods describe the temporal consistency of kernel density data

I am working on a spatial time series analysis project. The task is to study the spatial distribution of point features (e.g., crime events, traffic accidents) over time. I aim to find the places with ...
6 votes
4 answers
5k views

How is it that an ML estimator might not be unique or consistent?

Christian H Weiss says that: In general, it is not clear if the ML estimators (uniquely) exist and if they are consistent. Can someone explain what he means? Do we not generally know the shape of ...
2 votes
2 answers
547 views

What does the likelihood function converge to when sample size is infinite?

Let $\mathcal{L}(\theta\mid x_1,\ldots,x_n)$ be the likelihood function of parameters $\theta$ given i.i.d. samples $x_i$ with $i=1,\ldots,n$. I know that under some regularity conditions the $\theta$ ...
1 vote
0 answers
34 views

Network meta-analysis with placebo only comparison

I have a network meta-analysis where all treatment are compared to placebo. Obviously this is not properly a disconected network but consistency cannot be assessed. Is it problematic?
1 vote
1 answer
75 views

Kolmogorov Smirnov Test Consistency

I was reading that the Kolmogorov Smirnov 2 sample test is consistent, that is Probability of rejection under $H_1$ is 1 for sample size going to infinity. Say we have 2 random variables X and Y. K-S ...
5 votes
2 answers
5k views

Is the sample mean always an unbiased estimator of the expected value?

Given a random variable $x$ with a well-defined expected value $\mu$, is the mean of the set of samples $\{x_1,\ \cdots,\ x_n\}$, which we'll call $\widehat{\mu}$, always an unbiased estimator of $\mu$...
13 votes
1 answer
3k views

Why is the definition of a consistent estimator the way it is? What about alternative definitions of consistency?

Quote from wikipedia: In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter $θ^*$—having the property that as the ...
3 votes
2 answers
844 views

Interpretation of the consistency property of a loss function

I am looking for an interpretation of the consistency property of a loss function used for classification (e.g., the SVM's hinge loss: $V(t)=\max(0,1-t)$). I copy from Wikipedia: Furthermore, it ...
2 votes
1 answer
135 views

What are the minimum conditions needed for the consistency of OLS estimator in the following linear regression model?

Suppose $Y_i=X_i'\beta+\epsilon_i$ with $E(\epsilon_i|X_i)=0$. Consider the usual OLS estimator for $\beta$ using a random sample $\{X_i,Y_i\}_{i=1}^n$: $\widehat{\beta}=(\frac{1}{n}\sum_{i=1}^nX_iX_i'...
6 votes
3 answers
216 views

What 's the $(\Omega,\mathcal{F},P_{\theta} )$ those $T_{n}$ defined on?

Definition (Consistency) Let $T_1,T_2,\cdots,T_{n},\cdots$ be a sequence of estimators for the parameter $g(\theta)$ where $T_{n}=T_{n}(X_1,X_2,\cdots,X_{n})$ is a function of $X_{1},X_{2},\cdots,X_{n}...
0 votes
0 answers
40 views

Showing an estimator is consistent for a parameter

Suppose $X_1, ..., X_n$ i.i.d.~ $\mathscr{N}(0, \theta^2)$ with unknown $\theta>0$. I have an estimator $\hat{\theta}_n=\sqrt{\frac{\pi}{2}}\frac{1}{n}\Sigma_{n}^{i=1}|X_i|$ for $\theta$. How do I ...
0 votes
0 answers
44 views

If I have T observations from a binomial(n,p), how to consistently estimate n and p?

Suppose I have a dataset $\{S_t\}_{t=1}^T$, where $S_t\overset{i.i.d.}{\sim}Binomial(n,p)$, how to consistently estimate $n$ and $p$ using this dataset? It would be great if you could provide a method ...
2 votes
1 answer
125 views

I need to prove that $\hat\theta=\max\{X_1,...,X_n\}$ is a mean square consistent estimator for $\theta$

Let $X_1,...,X_n$ a i.i.d from a population with distribution $U[0,\theta]$, i.e., $f_{X_i}(x)=\frac{1}{\theta}g_{[0,\theta]}(x)$, for $i=1, \ldots, n$ where \begin{align} g_{[0,\theta]}(x) = \begin{...
0 votes
0 answers
28 views

Can I delete a dimension of a scale if this dimension reach a very low internal consistency when doing psychological network analysis

I am trying to do a psychological network analysis, and I aim to do the analysis on dimension-level (summing all the items belonging to this dimension). However, I've found that one dimension of the ...
1 vote
0 answers
27 views

Show that the MLE of $\alpha$ is consistent by definition

Suppose that $(X_1,\dots, X_n)$ is an iid random sample from $X\sim f(x;\alpha, \beta)$ and $$ f(x;\alpha, \beta)=\frac{\alpha x^{\alpha-1}}{\beta^{\alpha}}, \, 0<x\le \beta, \alpha>0, \beta>...

1
2 3 4 5
8