MAS-I T/F

A counting process is said to possess independent increments if the number of events that occur between time s and t is independent of the number of events that occur between time s and t+u for all u>0

false. A counting process is said to possess independent increments if the number of events that occur between time s and t is independent of the number of events that occur between time t and t+u for all u>0

All Poisson processes have stationary and independent increments."

false. This is because non-homogeneous Poisson processes do not have stationary increments

The assumption of stationary and independent increments is essentially equivalent to asserting that at any point in time the process probabilistically restarts itself.

True

The leverage for each observation in a linear model must be between 1/n and 1

True

The leverage for each observation in a linear model must sum to the number of explanatory variables

FALSE. The leverage must sum to p, which is the number of parameters including the intercept. The number of variables does not necessarily equal to the number of parameters

If an explanatory variable is uncorrelated with all other explanatory variables, the corresponding variance inflation factor would be zero

FALSE. If an explanatory variable is uncorrelated with all other explanatory variables, the corresponding variance inflation factor would be 1

The standardized deviance residuals should have an approximately standard normal distribution, provided the numbers of observations for each covariate pattern are not too small

TRUE; however, it should be noted that only the standardized deviance residuals have an approximately standard normal distribution, whereas the raw residuals in general do not

Deviance residuals can be plotted in observation order to assess serial correlation

TRUE

Problems with qualitative predictors are referred to as classification problems

false. Problems with a qualitative response are referred to as classification problems

Problems with a quantitative response are referred to as regression problems

True

Categorical variables are variables that take on one of a limited number of different values

True

In general, performing LOOCV requires fitting a model for a total of n times

True

With least squares regression, the LOOCV estimate for the test MSE can be calculated by fitting a model once

True

Performing k-fold cross validation requires fitting a model for a total of k times

True

The proportion of variance explained by an additional principal component increases as more principal components are added.

False. The proportion of variance explained by an additional principal component decreases as more principal components are added.

The cumulative proportion of variance explained increases as more principal components are added.

True. The cumulative proportion of variance explained increases as more principal components are added.

Using all possible principal components provides the best understanding of the data.

False. Using the first few principal components is often sufficient to get a good understanding of the data.

A scree plot provides a method for determining the number of principal components to use

False. Using the first few principal components is often sufficient to get a good understanding of the data.:True. A scree plot provides a method for determining the number of principal components to use. A scree plot graphs the proportion of variance explained against the principal components.

In Lasso, as λ increases, the number of predictors in the chosen model will increase.

FALSE. Larger values of the tuning parameter λ correspond to a greater shrinkage penalty, which will tend to decrease the number of parameters in the model

In Lasso, as λ increases, the bias of the parameters in the chosen model will increase.

TRUE. Larger values of the tuning parameter λ will tend to shrink the parameters more (compared to their least squares estimates), leading to greater bias

In Lasso, as λ increases, the variance of the predictions made by the chosen model will increase.

FALSE. Larger values of the tuning parameter λ will tend to shrink the parameters more (compared to their least squares estimates), which decreases the variance of the predictions made by the model. This is an example of the so-called "bias-variance tradeoff"

Stochastic trends are characterized by explainable changes in direction.

false. Stochastic trends are characterized by unexplainable changes in direction

Deterministic trends are better suited to extrapolation than stochastic trends.

True

Deterministic trends are typically attributed to high serial correlation with random error.

false. Stochastic trends are typically attributed to high serial correlation with random error

Deterministic trends typically arise from high serial correlation in the underlying data.

FALSE. Deterministic trends typically arise from some underlying physical or seasonal process; stochastic trends, on the other hand, are due to high serial correlation in the data.

For a binary response variable with a continuous explanatory variable, logistic regression is an inappropriate method of statistical analysis.

FALSE. For a binary response variable, logistic regression is an appropriate method of statistical analysis.

Ordinal variables are a type of continuous explanatory variable.

FALSE. Ordinal variables are a type of categorical explanatory variable

ANOVA is a useful approach for analyzing the means of groups of continuous response variables, where the groups are categorical.

TRUE

Deviance can be used to assess the quality of fit for nested models.

TRUE

A small deviance indicates a poor fit for a model.

FALSE

A saturated model has a deviance of zero.

TRUE

Linear models for time series are stationary when they include functions of time.

FALSE. Linear models for time series are non-stationary when they include functions of time.

All moving average processes are stationary.

TRUE

All random walk processes are non-stationary.

TRUE

All states in an irreducible Markov chain are recurrent.

FALSE. All states in an irreducible Markov chain can either be recurrent or transient. All states in a finite irreducible Markov chain are recurrent.

If all states in a finite Markov chain are recurrent, the Markov chain is irreducible.

FALSE. If all states in a finite Markov chain are recurrent, it does not mean that the Markov chain has one class. An irreducible Markov chain has only one class.

If a one-dimensional random walk has a transition probability of 0.5 in either direction, all states in the Markov chain are recurrent.

TRUE. One-dimensional and two-dimensional symmetric random walk are recurrent and irreducible. Thus, all states in these random walks are recurrent.

Is it a property of Counting process? N(t) must be greater than or equal to zero.

T

N(t) must be an integer.

T

Is it a property of Counting process? If s

T

Is it a property of Counting process? The number of events that occur in disjoint time intervals must be independent.

F

Is it a property of Counting process? For s

T

Deterministic trends typically arise from high serial correlation in the underlying data.\

FALSE. Deterministic trends typically arise from some underlying physical or seasonal process; stochastic trends, on the other hand, are due to high serial correlation in the data.

Deterministic trends are most naturally modeled using regression models (as opposed to autoregressive models).

TRUE. We will tend to use regression models to model deterministic trends. These models allow us to specify the underlying structure of the trend as a function of time and model and forecast it appropriately; autoregressive models are more appropriate for stochastic trends, as they posit each observation as a function of past observations.

When forecasting the time series, we will extrapolate deterministic trends.

TRUE. Because there is presumed to be a continuing underlying physical process guiding the deterministic trend, we assume it will continue into the future (at least into the near future) and hence extrapolate it. This is opposed to stochastic trends, which result from high levels of serial correlation; we have no reason to think that this type of trend will continue in the same direction, and so we do not extrapolate it when forecasting.

If ε=0, the 95% confidence interval is equal to the 95% prediction interval.

True. The prediction interval accounts for the irreducible error. If the irreducible error is zero, the prediction interval will be the same as the confidence interval.

The prediction interval is always at least as wide as the confidence interval.

True. The prediction interval is almost always wider than the confidence interval.

The prediction interval quantifies the possible range for E(y∣x)

False. The confidence interval quantifies the possible range for E(y∣x). The prediction interval quantifies the possible range for y∣x.

Leave-one-out cross-validation (LOOCV) is a special case of k-fold cross-validation.

TRUE. LOOCV is just k-fold cross-validation where k=n.

k-fold cross-validation has higher variance than LOOCV when k

FALSE. LOOCV has higher variance than k-fold validation, for k

You are performing a principal components analysis on a data set with 50 observations from three independent continuous variables. The maximum number of principal components that can be extracted from this data is three.

:TRUE. In general, the number of principal components that can be extracted will be equal to the number of variables in the data set

The first principal component represents the direction along which the data vary the most.

TRUE. The first principal component is extracted by finding the direction along which the data vary the most.

The third principal component will be orthogonal to the first principal component.

TRUETRUE. Each principal component will be orthogonal to each of the others:FALSE. Validation set approach tends to overestimate the test error rate more than LOOCV, which is one of its drawbacks.

When there is positive serial correlation in a time series, the standard errors of the estimated regression parameters are likely to be over-estimated.

false. When there is a positive serial correlation in a time series, the standard errors of the estimated regression parameters are likely to be underestimated. They tend to be lower than their true value.

GLS is an improvement over ordinary least squares regression for serially correlated time series because GLS is based on maximizing the likelihood given the white noise in the data.

false. GLS is an improvement over ordinary least squares regression for serially correlated time series because GLS is based on maximizing the likelihood given the autocorrelation in the data.

GLS can be used to provide better estimates of standard errors of the regression parameters to account for autocorrelation in the residual series.

true. That's why GLS is preferred over ordinary least squares regression when dealing with time series data.

the tuning parameter λ in a smoothing spline model, Larger values of λ result in smoother splines

TRUE. The tuning parameter λ in a smoothing spline controls the penalty term for roughness; λ=0 imposes no roughness penalty, while larger values of λ impose higher roughness penalties, leading to smoother splines

the tuning parameter λ in a smoothing spline model, Larger values of λ result in greater effective degrees of freedom for the model.

FALSE. As the tuning parameter λ increases from 0 to ∞, the effective degrees of freedom decreases from n down to 2. This happens because larger λ values impose more (smoothness) constraints, leaving fewer degrees of freedom.

the tuning parameter λ in a smoothing spline model, Larger values of λ result in a more biased model.

TRUE. The tuning parameter λ controls the smoothness, which impacts the bias-variance tradeoff for the model. In particular, as λ increases, the bias will increase, but the variance will decrease.