Elements of Statistical Inference

Statistical inference is the process of...?

• Process of generalizing or drawing conclusions. 

• Process of generalizing. 

• Process of sampling and drawing conclusion.

• Process of sampling. 

Answer explanation

Statistical inference involves generalizing from a sample to a population, allowing us to draw conclusions based on data. The correct choice captures both aspects: generalizing and drawing conclusions.

What is the main goal of statistical inference?  

• To predict data based on known models. 

• To make conclusions about unknown population properties from sample data. 

• To ensure a perfect representation of a population using samples. 

• To compute probabilities of future outcomes without any uncertainty. 

Answer explanation

The main goal of statistical inference is to make conclusions about unknown population properties based on sample data. This allows researchers to estimate and draw insights about a larger group from a smaller subset.

What does the Maximum Likelihood Estimator (MLE) seek to achieve? 

• Maximize the likelihood function for a given parameter to estimate sample data. 

• Minimize the error margin of sample predictions. 

• Ensure the lowest possible variance for any estimate. 

• Determine the probability of an event given past data. 

Answer explanation

The Maximum Likelihood Estimator (MLE) aims to maximize the likelihood function, which measures how well a given parameter explains the observed sample data. This makes the first choice the correct answer.

Why is Fisher Information important in statistical inference? 

• It measures the total variance in the data sample. 

• It verifies whether a sample is representative of a population. 

• It helps approximate the variability of MLEs and derive confidence intervals. 

• It establishes the significance of statistical tests. 

Answer explanation

Fisher Information is crucial as it quantifies the amount of information that an observable random variable carries about an unknown parameter, aiding in approximating the variability of Maximum Likelihood Estimates (MLEs) and deriving confidence intervals.

What is one of the properties of MLE under regular conditions? 

• MLE is always unbiased, regardless of sample size. 

• MLE is not affected by the choice of the statistical model.

• 1. MLE minimizes the bias-variance tradeoff by default. 

   

• MLE is consistent, converging to the true value as the sample size grows. 

Answer explanation

The correct choice is that MLE is consistent, meaning it converges to the true parameter value as the sample size increases. This property ensures that with enough data, MLE provides accurate estimates.

What is the significance of a likelihood set in statistical inference? 

• It shows the range of sample data that maximizes prediction intervals. 

• It includes all parameter values where relative likelihood exceeds a threshold. 

• It guarantees the accuracy of an estimate. 

• 1. It determines the outcome of hypothesis tests. 

   

Answer explanation

The correct choice highlights that a likelihood set includes all parameter values where the relative likelihood exceeds a certain threshold, which is crucial for identifying plausible parameter estimates in statistical inference.

Which statement best describes the difference between probability and statistical inference?

• Probability uses observed data to infer the model, while statistical inference predicts data from a given model. 

• Both probability and statistical inference predict future data based on past observations. 

• Probability predicts data from a given model, while statistical inference uses observed data to infer the model. 

• Statistical inference always requires known probability distributions, while probability does not involve data. 

Answer explanation

The correct choice highlights that probability predicts data based on a model, while statistical inference uses actual data to deduce or infer the underlying model. This distinction clarifies their roles in data analysis.

Consider the Likelihood function. What can we say?

• Likelihood resembles the generic formula of a probability density function, with due modifications.

• It is a density function, in fact it specifies the probability of the parameter as we aim to estimate it.

•  We can compute the likelihood function assuming the function itself is twice derivable w.r.t. the parameter we are looking at.

• We can calculate it if all the variables we consider in our calculations are independent and identically distributed. If both condition do not hold, we do not matter, because we can compute it with no change in result, in fact individual distributions approach the standard normal distribution for a giant sample size, according to CLT.

Answer explanation

The Likelihood Function can be computed under the assumption of twice differentiability with respect to the parameter. It also resembles a continuous density function, using multiplication for occurrences in the discrete case.

Consider the Likelihood function. What can we say?

• The relative likelihood function can assume values in the same set of the standard likelihood function.

• The more realizations we consider, the more the variability raises. The best estimator is collected through a small sample. 

• all the answers are correct. 

• The Relative Likelihood Function is useful as we can compare different values of the parameter, with the aim of choosing the one which is preferable. 

Answer explanation

The correct choice highlights that the Relative Likelihood Function allows for comparison of different parameter values, aiding in selecting the most preferable one, which is a key aspect of likelihood analysis.

Consider the parameter. What is the wrong statement?

• A parameter is a random variable as it is based on random variables. 

• A better estimation of a parameter is gotten by minimizing the variance and so the bias. 

• According to the Cramér-Rao Lower Bound, the variance should not be more to the inverse of the fisher information. 

• Consistency is one of the properties of the estimators.

Answer explanation

The first choice is incorrect; a parameter is a fixed value, not a random variable. The other statements about estimation, Cramér-Rao Lower Bound, and consistency are correct regarding properties of estimators.