If Z1 and Z2 are 2 independent standard normal random variables, then the characteristic function of (Z1+Z2) is:
Exp(-t)
Exp(-2t)
Exp(-t/2)
None of the above
The "Risk" associated with any decision rule is:
The expected loss, where the expectation is taken with respect to the uncertainty associated with the parameters
The risk of a scalar estimator is gnerally less than its variance
The risk of a vector estimator is the trace of its matrix mean squared error
The risk of a vector estimator is just its matrix mean squared error
If the loss function is quadratic, then:
The risk of a scalar estimator is just its variance
The risk of a scalar estimator is gnerally less than its variance
The risk of a vector estimator is the trace of its matrix mean squared error
The risk of a vector estimator is just its matrix mean squared error
If an estimator is "inadmissible", then:
There is at least one other estimator whose loss is less than or equal to the loss of this estimator everywhere in the parameter space, and strictly less somewhere in the parameter space
There is at least one other estimator whose risk is less than or equal to the risk of this estimator everywhere in the parameter space, and strictly less somewhere in the parameter space.
There is at least one other estimator whose risk is strictly less than the risk of this estimator everywhere in the parameter space
It cannot be weakly consistent
If an estimator is "Mini-Max", then:
It must be admissible
It cannot be admissible
It may be admissible or inadmissible
Its risk function must "cross" the risk function of at least one other estimatar
If an scalar statistic is "sufficient", then:
It will be an admissible estimator of the population parameter
It will be an efficient estimator of the population parameter
It will be an unbiased estimator of the population parameter
It contains all of the sample information that is needed to estimate the population parameter
The Newton-Raphson algorithm:
May yield multiple solutions, all of which will be local maxima, and one of which will be the global maximum
May yield multiple solutions. some of which relate to local maxima and some of which relate to local minima
Will always converge to a global extremum in a finite number of iterations
Will converge in 3 steps if the underlying function is a cubic polynomial
The "Invariance" property of MLE's implies that:
Their variance approaches zero as the sample size increases without limit
Their variance achieves the Cramer-Rao lower bound
Any monotonic function of an MLE is the MLE for that function of the parameter(s)
Any continuous function of an MLE is the MLE for that function of the parameter(s)
If X follows a uniform distribution on [0 , 1], and Y = 5X, then:
The Jacobian for the mapping from X to Y is 0.2, and Y is uniform on [0 , 5]
The Jacobian for the mapping from X to Y is 5, and Y is uniform on [0 , 0.2]
The Jacobian for the mapping from X to Y is 0.2, and Y is uniform on [0 , 0.2]
The Jacobian for the mapping from X to Y is 5, and Y is uniform on [0 , 5]
When we evaluate the Jacobian associated with a transformation from one probability distribution to another:
We use the absolute value because a density function cannot take negative values
We must be dealing with scalar random variables, not random vectors
The intention is make sure that the support of the new random variable is the full real line
The intention is make sure that the support of the new random variable is the positive half of the real line
If our random data are statistically independent, then:
The likelihood function is just the sum of the marginal data densities, viewed as a function of the parameter(s)
The log-likelihood function is just the product of the logarithms of the marginal data densities, viewed as a function of the parameter(s)
The log-likelihood function is just the sum of the logarithms of the marginal data densities, viewed as a function of the parameter(s)
The likelihood function will have a unique turning point, and this will be a maximum (not a minimum) if the sample size is large enough
The "Likelihood Equations" are:
The same as the "normal equations" associated with least squares estimation of the multiple linear regression model
Guaranteed to have a unique solution if the sample data are independent
Obtained by getting the second derivatives of the log-likelihood function with respect to each of the parameters, and setting these equal to zero
The first-order conditions that we have to solve in order to maximize the likelihood function
When we "concentrate" the likelihood function, the objective is to:
Focus attention on just the important parameters by conditioning on the 'nuisance parameters' in the problem
Reduce the dimension of that part of the optimization problem that has to be solved numerically
Take a monotonic transformation of the likelihood function so that it is easier to find the global maximum
Convert what would be a non-linear optimization problem into one that is approximately linear
Suppose that Y follows a Binomial distribution with parameter 'p' equal to the probability of a 'success', and 'n' repetitions. Then the MLE of the standard deviation of Y is:
The square root of np(1-p)
The square root of y(n-y)/n, where y is the observed number of 'successes' in the sample
The square root of n(y-n)/y, where y is the observed number of 'successes' in the sample
The square root of ny, where y is the observed number of 'successes' in the sample
The connection between a sufficient statistic and an MLE is:
A sufficient statistic is always an MLE
There is no connection in general
All MLE's are linear combinations of sufficient statistics
If an MLE is unique, then it must be a function of a sufficient statistic