Homework #5 for Stat 610

Due Thursday March 29, in class.

1. An estimate theta.hat of theta is called uniformly minimum
   variance unbiased (UMVU) if it is unbiased and if no other
   unbiased estimator has smaller variance.  Show that the
   sample mean Xbar is UMVU for the mean theta in the following
   examples (all involving iid data X_1,...,X_n).

   a. Bernoulli( theta )
   b. Normal( theta, sigma^2 )
   c. Poisson( theta )

2. Consider iid Normal( mu, variance=theta ) data X_1,...,X_n
   where mu is considered known.
   Show that

     theta.hat = (1/n) sum_{i=1}^n (X_i - mu)^2

   is UMVU for theta.

3. Zero-inflated Poisson:  Data X_1,...,X_n form a random
   sample of counts from the following  model

    f(x;theta) = (1/2) 1[x=0] + (1/2) exp( -theta ) theta^x /x!

   for theta>0 and x=0,1,2,... Derive the Fisher information
   for theta.

4. Problem 3.4.11 (B&D)
   Suppose Y_1, Y_2, ..., Y_n are independent Poisson random variables with
   E(Y_i) = mu_i and log( mu_i  ) = alpha + beta z_i for constants 
   z_i and unknown parameters theta = (alpha, beta).

   a. Write the joint distribution of {Y_i} as an exponential family
      in the natural parameterization.

   b. Derive the Fisher information matrix I(theta).

5. Censored survival times.
   Patients i = 1,2, ..., n enter a study, at random entry
   time S_i according to some distribution on the interval [0,T].
   Patients are independent, and patient i survives to time S_i + X_i, 
   but we only know S_i and that the survival has been censored if 
   S_i + X_i > T.  Let C_i = T - S_i be the censoring time in these cases.
   Suppose that X_i is independent of S_i and is Exponentially distributed
   with mean theta.  In class we developed the MLE (assuming at
   least one uncensored observation)

    theta.hat  = (U+V)/m 
    where U = sum_i X_i 1[ not censored ], V = sum_i C_i 1[ censored ], 
    and m = total number of uncensored observations. 
    Derive the Fisher information for theta.

6. Mutant frequencies, revisited.
   A population of cells under study are mutant at a specific genetic 
   locus at rate theta.  For each i = 1,2,...,2n,  m_i  cells
   are grown in well i, and X_i of them are mutant; we observe
   only the Bernoulli trial Y_i = 1[X_i >= 1 ], noting that
   a reasonable model has X_i ~ Poisson( m_i theta ).  The experiment
   is considered to have two parts: 1<=i <= n in which m_i = M,
   and the second part in which m_i = N > M. 

   a. Consider each part of the experiment separately. Write
      out the likelihood, loglikelihood, and score function 
      for each part, and compute the MLE for theta from each
      part [say theta.hat.1, theta.hat.2].  
      Also derive the Fisher information from each part (say I_1, I_2).
      Note that each MLE is approximately normal with variance
      equal to the respective inverse Fisher information if
      the number of wells per part is moderately large.

   b. Although in the combined experiment the MLE is not
      available in closed form, there is an expression
      for the Fisher information in this case, and thus 
      formula for the asymptotic variance of the MLE. Carefully
      compare this variance to the variance of a simpler closed form 
      estimator

       theta.tilde = (1/2) theta.hat.1 + (1/2) theta.hat.2

      from above. Thus establish the superiority of the MLE.