We have introduced likelihood functions as a useful tool for parameter estimation. They also play a role in hypothesis testing and lead to the so-called likelihood ratio test (LRT). The test is based on the likelihood ratio statistic
A LRT is a test whose rejection region is of the form
for a suitable constant c < 1. The rationale of the approach is rather intuitive. If the statistic λ(x) is small, the restriction of θ to Θ0 does not seem justified.
Example 9.40 Let us consider again the one-sided test on the expected value of a normal population
For simplicity, let us assume that the variance σ2 is known. We have considered a one-tail test, with a somewhat heuristic justification. Let us see if we can find a more formal justification based on a LRT. The likelihood function, for a sample of size n, is
Note that this is a function of μ, for given x and σ2. In the likelihood ratio, we have Θ = (−∞, +∞) and Θ0 = (−∞, μ0). We know from Example 9.38 that the denominator of the likelihood ratio in Eq. (9.41) is maximized for 
. As to the numerator, there are two cases:
. In this case, the constraint defining Θ0 is nonbinding27 in the maximization of the likelihood function at the numerator, and λ(x) = 1. This case is not interesting.
. In this case, the constraint defining Θ0 is binding and in fact the statistic seems to contradict the null hypothesis; the likelihood function at the numerator is maximized for μ = μ0.
Assuming that the interesting case applies, the likelihood ratio may be rewritten as follows:
Now, we should reject when λ(x) < c, which implies rejection when the statistic
is large. This is exactly what we do in standard test procedures; the rejection region is the left tail of a standard normal distribution. Furthermore, the probability of error of type I is set to a by selecting the critical value z1−α. The case with σ2 unknown is dealt with by a similar token, even though the calculations are a bit more involved.
The LRT approach is not only a systematic way to devise testing procedures, but it also plays a key role in establishing theoretical results on most powerful tests; we refer the interested reader to the advanced references on mathematical statistics.
Problems
9.1 The director of a Masters’ program wants to assess the average IQ of her students. A sample of 18 students yields the following results:
- Build a 95% confidence interval for the average IQ.
- Assuming that IQ is normally distributed, how would you estimate the probability that IQ is larger than 130? What if you do not want to assume normality? Compare the two approaches.
9.2 You have to compute a confidence interval for the expected value of a random variable. Using a standard procedure, you take a random sample of size N = 20, and the sample statistics are 
and S = 4.58.
- Compute a confidence interval at level 95%.
- If you take additional observations, raising sample size to N = 50, how large do you expect the new confidence interval be? What hypothesis are you making in your reasoning?
9.3 Find the 97% confidence interval, given a sample mean of 128.37, sample standard deviation of 37.3, and sample size of 50. What is the width of the confidence interval? Suppose that you want to cut the confidence interval by 50% (i.e., the new width should be half the previous one). How many additional observations would you use?
9.4 In standard confidence intervals, you use the sample mean as an estimator of expected value. Now suppose that a friend of yours suggests the following alternative estimator:
where we assume that N is an even number. Prove that
where E[X] = μ and Var(X) = σ2. Would you use this estimator? Why or why not?
9.5 TakeItEasy produces special shoes for runners, whose average life is 1250 km. In order to improve the product, they experiment with a new design, and test prototypes with a sample of 30 runners. The sample mean of product life is 1315 km, with a standard deviation of 70 km. Can we say that TakeItEasy has actually improved their product?
9.6 Air quality is measured by the concentration of a dangerous pollutant. The mayor of a city has engaged in a program to improve traffic conditions in order to decrease the concentration of that pollutant. Of course, there is a lot of day-to-day variability in measures. In the past, the average concentration was 29 (measured in some units). In a sample of 20 days after completion of the program, the sample mean has been 26.9, with a standard deviation of 8. Can we say that the mayor’s program has been effective?
9.7 You want to compare the reliability of two machines that insert chips onto electronic cards. The main problem is the occurrence of jams in the feeding mechanism, as this requires stopping production to fix the trouble. To this aim, you observe the number of jams when producing standard batches of electronic cards, resulting in the following table:
Is there a significant difference between the two machines?
9.8 A study was done to measure the impact of fatigue on human performance when carrying out a certain task. The performance is measured by an appropriate index, the larger the better, which is measured at the beginning of the shift and after 3 hours of work. Ten workers are observed, resulting in the following table:
Is there a significant effect due to weariness?
9.9 The following dataset is a random sample from a normal distribution:
Find a 95% confidence interval for variance.
9.10 In order to estimate the fraction of defective parts, you take a sample of size 1000 and find that 63 are not acceptable. Find a 99% confidence interval for the fraction of defective parts.
9.11 In one-way ANOVA we define the sum of squares SSb and SSw. Prove the identity
9.12 Apply one-way ANOVA to check equality of means for the following sample:
9.13 A m-Erlang distribution with rate λ is obtained when summing m independent exponential random variables with rate λ. This distribution may be used to model more realistic random service times in queueing systems. Devise an efficient method to generate a sample of pseudorandom variables from the m-Erlang distribution.
9.14 Define an algorithm to generate pseudorandom variables characterized by the following density function:
9.15 Consider the simulation of a continuous review (Q, R) inventory control policy. Define the relevant events for the system, and outline a procedure for the management of each event. To deal with a specific case, assume that customers arrive according to a Poisson process and order a random amount of items drawn from a given probability distribution. Lead time is assumed to be deterministic. First assume that customers are impatient, i.e., if their order cannot be satisfied immediately from stock, they are lost. Then, adapt the event management procedures to the case of patient customers (allowing for backlogged demand).
9.16 Consider a sequence of random variables
Does this sequence converge in probability to a number? What about convergence in quadratic mean?
9.17 Consider an exponential distribution with rate λ. On the basis of a random sample of size n, apply the method of moments to estimate λ.
9.18 Apply the method of maximum likelihood to estimate the parameters of a uniform distribution on the interval [a, b].
9.19 Prove the result of Eq. (9.40). You should use the result established in Problem 7.11 to find the expected value of X(n), given a sample of n independent observations from the uniform distribution on the interval [0, θ].
Leave a Reply