Maximum Likelihood Estimation Examples

Suppose that we have a random sample from a population of interest.  We may have a theoretical model for the way that the population is distributed.  However, there may be several population parameters of which we do not know the values.  Maximum likelihood estimation is one way to determine these unknown parameters.   The basic idea behind maximum likelihood estimation is that we determine the values of these unknown parameters.  We do this in such a way to maximize an associated joint probability density function or probability mass function.  We will see this in more detail in what follows.  Then we will calculate some examples of maximum likelihood estimation. Steps for Maximum Likelihood Estimation The above discussion can be summarized by the following steps: Start with a sample of independent random variables X1, X2, . . . Xn from a common distribution each with probability density function f(x;ÃŽ ¸1, . . .ÃŽ ¸k).  The thetas are unknown parameters.Since our sample is independent, the probability of obtaining the specific sample that we observe is found by multiplying our probabilities together.  This gives us a likelihood function L(ÃŽ ¸1, . . .ÃŽ ¸k)   f( x1 ;ÃŽ ¸1, . . .ÃŽ ¸k) f( x2 ;ÃŽ ¸1, . . .ÃŽ ¸k) . . .  f( xn ;ÃŽ ¸1, . . .ÃŽ ¸k) ÃŽ   f( xi ;ÃŽ ¸1, . . .ÃŽ ¸k).Next, we use Calculus to find the values of theta that maximize our likelihood function L.  More specifically, we differentiate the likelihood function L with respect to ÃŽ ¸ if there is a single parameter.  If there are multiple parameters we calculate partial derivatives of L with respect to each of the theta parameters.To continue the process of maximization, set the derivative of L (or partial derivatives) equal to zero and solve for theta.We can then use o ther techniques (such as a second derivative test) to verify that we have found a maximum for our likelihood function. Example Suppose we have a package of seeds, each of which has a constant probability p of success of germination.  We plant n of these and count the number of those that sprout.  Assume that each seed sprouts independently of the others.  How do we determine the maximum likelihood estimator of the parameter p? We begin by noting that each seed is modeled by a Bernoulli distribution with a success of p. We let X be either 0 or 1, and the probability mass function for a single seed is f( x ; p ) px (1 - p)1 - x.   Our sample consists of n  Ã‚  different Xi, each of with has a Bernoulli distribution.  The  seeds that sprout have Xi 1 and the seeds that fail to sprout have Xi 0.   The likelihood function is given by: L ( p ) ÃŽ   pxi (1 - p)1 - xi We see that it is possible to rewrite the likelihood function by using the laws of exponents.   L ( p )   pÃŽ £ xi (1 - p)n - ÃŽ £ xi Next we differentiate this function with respect to p.  We assume that the values for all of the Xi are known, and hence are constant.  To differentiate the likelihood function we need to use the product rule along with the power rule: L ( p )   ÃŽ £ xip-1 ÃŽ £ xi (1 - p)n - ÃŽ £ xi - (n - ÃŽ £ xi )pÃŽ £ xi (1 - p)n-1 - ÃŽ £ xi We rewrite some of the negative exponents and have: L ( p ) (1/p) ÃŽ £ xipÃŽ £ xi (1 - p)n - ÃŽ £ xi - 1/(1 - p) (n - ÃŽ £ xi )pÃŽ £ xi (1 - p)n - ÃŽ £ xi [(1/p) ÃŽ £ xi  - 1/(1 - p) (n - ÃŽ £ xi)]ipÃŽ £ xi (1 - p)n - ÃŽ £ xi Now, in order to continue the process of maximization, we set this derivative equal to zero and solve for p: 0 [(1/p) ÃŽ £ xi  - 1/(1 - p) (n - ÃŽ £ xi)]ipÃŽ £ xi (1 - p)n - ÃŽ £ xi Since p and (1- p) are nonzero we have that 0 (1/p) ÃŽ £ xi  - 1/(1 - p) (n - ÃŽ £ xi). Multiplying both sides of the equation by p(1- p) gives us: 0 (1 - p) ÃŽ £ xi  - p (n - ÃŽ £ xi). We expand the right hand side and see: 0   ÃŽ £ xi  - p ÃŽ £ xi  - p n pÃŽ £ xi   ÃŽ £ xi - p n. Thus ÃŽ £ xi p n and (1/n)ÃŽ £ xi   p.  This means that the maximum likelihood estimator of p is a sample mean.  More specifically this is the sample proportion of the seeds that germinated.  This is perfectly in line with what intuition would tell us.  In order to determine the proportion of seeds that will germinate, first consider a sample from the population of interest. Modifications to the Steps There are some modifications to the above list of steps.  For example, as we have seen above, is typically worthwhile to spend some time using some algebra to simplify the expression of the likelihood function.  The reason for this is to make the differentiation easier to carry out. Another change to the above list of steps is to consider natural logarithms. The maximum for the function L will occur at the same point as it will for the natural logarithm of L.  Thus maximizing ln L is equivalent to maximizing the function L. Many times, due to the presence of exponential functions in L, taking the natural logarithm of L will greatly simplify some of our work. Example We see how to use the natural logarithm by revisiting the example from above.  We begin with the likelihood function: L ( p )   pÃŽ £ xi (1 - p)n - ÃŽ £ xi . We then use our logarithm laws and see that: R( p ) ln L( p ) ÃŽ £ xi ln p (n - ÃŽ £ xi) ln(1 - p). We already see that the derivative is much easier to calculate: R( p ) (1/p)ÃŽ £ xi - 1/(1 - p)(n - ÃŽ £ xi) . Now, as before, we set this derivative equal to zero and multiply both sides by p (1 - p): 0 (1- p ) ÃŽ £ xi -  p(n - ÃŽ £ xi) . We solve for p and find the same result as before. The use of the natural logarithm of L(p) is helpful in another way.  It is much easier to calculate a second derivative of R(p) to verify that we truly do have a maximum at the point (1/n)ÃŽ £ xi   p. Example For another example, suppose that we have a random sample X1, X2, . . . Xn from a population that we are modelling with an exponential distribution.  The probability density function for one random variable is of the form f( x ) ÃŽ ¸-1 e -x/ÃŽ ¸ The likelihood function is given by the joint probability density function. This is a product of several of these density functions: L(ÃŽ ¸) ÃŽ   ÃŽ ¸-1 e -xi/ÃŽ ¸   ÃŽ ¸-n e -ÃŽ £ xi/ÃŽ ¸    Once again it is helpful to consider the natural logarithm of the likelihood function.  Differentiating this will require less work than differentiating the likelihood function: R(ÃŽ ¸) ln L(ÃŽ ¸) ln [ÃŽ ¸-n e -ÃŽ £ xi/ÃŽ ¸] We use our laws of logarithms and obtain: R(ÃŽ ¸) ln L(ÃŽ ¸) - n ln ÃŽ ¸Ã‚   -ÃŽ £xi/ÃŽ ¸ We differentiate with respect to ÃŽ ¸ and have: R(ÃŽ ¸)   - n / ÃŽ ¸Ã‚   ÃŽ £xi/ÃŽ ¸2 Set this derivative equal to zero and we see that: 0 - n / ÃŽ ¸Ã‚   ÃŽ £xi/ÃŽ ¸2. Multiply both sides by ÃŽ ¸2 and the result is: 0 - n ÃŽ ¸Ã‚   ÃŽ £xi. Now use algebra to solve for ÃŽ ¸: ÃŽ ¸ (1/n)ÃŽ £xi. We see from this that the sample mean is what maximizes the likelihood function.  The parameter ÃŽ ¸ to fit our model should simply be the mean of all of our observations. Connections There are other types of estimators.  One alternate type of estimation is called an unbiased estimator.  For this type, we must calculate the expected value of our statistic and determine if it matches a corresponding parameter.

Essay about The Spirit In Context - 1872 Words

What are the possible points of agreement and points of tension between different religious schools of thought on the body, in relation to the Holy Spirit? How might Christians respond to these? Introduction Scripture seems to contradict itself with regard to the human body. Sometimes the body is seen as precious, for instance in 1 Cor 6:19-30, and sometimes as a hinderance to unity with Christ, 2 Cor 5:6-8. This seeming ambiguity within scripture has created distinct and conflicting schools of thought among Christian thinkers regarding the body and spirituality. Some advocate that the body is an essential part of our spirituality and should be embraced; others that the body is a hinderance to it, and should be approached with suspicion†¦show more content†¦Anthony of the Desert, who is considered to be the father of Christian monasticism, is said to have experienced overwhelming shame at the prospect of food, rest and other bodily needs, so that he would not be seen taking care of his physical needs by others.5 St Francis of Assisi was equally undermining of his bodily needs. Even when close to death, he feared that accepting necessary treatment was an indulgence. It was only at the last minute that he was successfully persuaded to serve his body by caring for it in the same way that it had cared for him, by which point it was too late. Dualism continued through the ages in various forms, but was specifically revived during the Age of Enlightenment, which emphasized science and reason over faith and tradition. 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The AIDS Epidemic Outbreak Free Essays

Who would have ever thought that a disease, possibly brought to America by infected African monkeys, would affect the country forever? This is exactly what happened in the late nineteenth century when the Acquired Immune Deficiency Syndrome (AIDS) was introduced to America. The unpredictable epidemic caused a huge outrage for years to come. The outbreak of the HIV virus, AIDS, in the early 1980†³s resulted in medical research, public misconceptions, and ultimately growing awareness. We will write a custom essay sample on The AIDS Epidemic Outbreak or any similar topic only for you Order Now Appearing first only in homosexual men, AIDS was an unfamiliar virus to the entire United States. Reports of unknown and unexplainable symptoms caused much confusion among patients and even doctors. In 1981, the first reports explained that 41 homosexual men in the San Fransisco area had â€Å"†¦ a rare and often rapidly fatal form of cancer† (Altman n. p. ). After years of researching these cases and millions of others appearing later in the decade, scientists discovered that it was not cancer they were dealing with. They called it HIV (Human Immune Virus) which leads to AIDS (Acquired Immune deficiency Syndrome) that â€Å"†¦ rogressively destroys the body†s ability to fight infections and certain cancers† (ARIC n. p. ). With minimal research there was nothing to initially â€Å"combat† the virus; but, thanks to time and funding, there were some drugs that seemed to help stop the spread of the virus in ones body. These anti-retroviral drugs still don’t constitute as cures ! for the virus, but have definitely helped and continue to help patients live longer (ARIC n. p. ). Since a vaccine to â€Å"†¦ evoke an immune system response that will prevent infection or disease development† still has not been found, other treatments have been tested. Accupuncture, stress management, hypnosis, exercise, good nutrition, and an overall positive attitude all seem to alleviate symptoms even if they are not proven cures of the virus (Packer 78-88). Although AIDS is a life threatening disease, there were many people living with it by keeping healthy and staying safe. Scott Fried had not tried any of the aniviral drugs; however, practices such as taking 80 herbs and vitamins a day, staying physically fit, visiting the doctor every three months, and being happy have kept him alive with the virus for thirteen years (Fried speach). Most victims have not lived as long as he which is why AIDS has lead to some extensive research. Investigators supported by private and public funds continued to search for a cure or even an explanation of HIV/AIDS for two decades because like the â€Å"b! lack plague†, AIDS has killed off millions of people in a short time. Also like the black plague did, AIDS carried with it a long string of misconceptions. In the Middle Ages anyone carrying the â€Å"black plague† was to be avoided. Similarly, the public would often avoid people with AIDS. There were fabrications that AIDS could be caught by another human being from sneezing, coughing, hugging, kissing or even any â€Å"casual contact† ; so, people stayed away from those who were infected by AIDS. Another lie was that AIDS was caused by something magical or mystical (Taylor 23-24). Though some were overly careful of â€Å"catching† the virus, others were not careful enough. Since the initial outbreak was among gay men it was simply assumed that only gay men could be affected. Then reports of IV drug users having the disease still seemed to eliminate the chance of the common person being infected. Next, hemophiliacs and people who had had blood transfusions were reported followed by blacks, Hispanics, lesbians, straight women and finally white, heterosexual, males. Still everyone said â€Å"It can†t happen to me,† until it did (Fried speech). This assumption that AIDS only affected few amounts of people and only minorities was the basis of all misconceptions about that virus. People did not believe the virus could have any impact on their lives; but, it did impact everyones lives directly and or indirectly. Now that â€Å"AIDS is the second leading cause of death in the United States among people aged 25 to 44†³ (ARIC n. p. ), the misconceptions have been proven to be false. Although it was statistically correct that 66% of people infected by AIDS were homosexual males, 24% were IV drug users, and only 4% were heterosexual males or females (Packer 17) it was still a fact that AIDS can affect anyone. After years of research and statistical reports there were finally people relaying these messages to the public. Motivational speakers, like Scott Fried, continue to reach out to teens and adults about AIDS and sex. Information about these subjects can also be found easily, not only at hospitals or doctors† offices; but in books, magazines, pamphlets, and even on the internet. Conducting a simple search on http://www. yahoo. com came up with 164 â€Å"category matches,† 1,206 â€Å"web site matches,† and 237,000 web page matches. This means that HIV/AIDS appears at least 238,370 times on th! e world wide web and it shows that AIDS in an important component of modern life. Not only are people becoming aware of the AIDS epidemic, but they are becoming aware of sexuality through learning about AIDS. Before the outbreak of this fatal virus sex was only spoken about discretely. Now even kids are being made aware of the dangers of sex and spreading HIV. In this way HIV/AIDS had a positive affect on the country. People are aware of the facts and probably make better decisions. Some experts have said that AIDS will remain the way it is now and others predict it growing into a â€Å"huge epidemic† (Taylor 28). Hopefully due to this expansion of knowledge and recognition, he AIDS virus will not spread as quickly and infect as many people as it has in the past. HIV positive, homosexual, male, Scott Fried, said, â€Å"Ironically one of the blessings that HIV/AIDS has brought me is the abundance of love. . . † and perhaps that is true. Pertaining to the eighties and the early onset of the virus, AIDS caused much more commotion than love. However, every cloud has a silver lining and the hysteria has finally cleared up some myths. It has opened up the public to not only HIV/AIDS awareness, but sexual cognizance as well. How to cite The AIDS Epidemic Outbreak, Essay examples