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Let the random variable $X$ have a uniform density given by $$ f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]} $$

where $-\infty\leq\theta\leq\infty $

the likelihood function for a sample of size $n$ is

$$L(\theta;x_1,\ldots,x_n)=\prod_{i=1}^nf(x_i;\theta)=f(x_1;\theta)\ldots f(x_n;\theta)$$

Why the likelihood function for a sample of size $n$ is the "joint density" of the $n$ random variables?

  • I am asking the question because I knew that we have to select $\hat \theta$ in such a way that the random variables $X_1,\ldots,X_n$ assume a particular value $x\prime_1,\ldots,x\prime_n$ such that $f_{X_1,\ldots,X_n}(x_1,\ldots,x_n;\theta) $ is a maximum.

But here, for the Uniform density function, there is no hint that the value $x_1$ of the random variable $X_1$ maximizes the function $f_{X_1}(x;\theta),\ldots,$ the value $x_n$ of the random variable $X_n$ maximizes the function $f_{X_n}(x;\theta).$ Then how $x_1,\ldots,x_n$ maximizes $f_{X_1,\ldots,X_n}(x_1,\ldots,x_n;\theta) $ for the given density function , ie, for the Uniform density function?

Again,

$$L(\theta;x_1,\ldots,x_n)=\prod_{i=1}^nf(x_i;\theta)=\prod_{i=1}^n I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]} (x_i)=I_{[y_n-\frac{1}{2},y_1+\frac{1}{2}]} (\theta)$$

where $Y_1$ is the smallest of the observations & $Y_n$ is the largest.

  • Why have we changed the range? Is it so as Likelihood-function is a function of $\theta$ and here $x_1,\ldots,x_n$ fixed?

  • Again i have not understand the point: if we subtract $\frac{1}{2}$ from the largest value $y_n$ then it will be the lower limit of $\theta$ and if we add $\frac{1}{2}$ to the smallest value $y_1$ then it will be the upper limit of $\theta$. Can you please give me a numerical example of the situation?

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  • $\begingroup$ The first question asked "Why the likelihood ..." was also asked here where the OP has already accepted an answer. $\endgroup$ – Dilip Sarwate Jun 13 '13 at 11:43
  • $\begingroup$ @DilipSarwate The first question of here is about why is it "joint density?" not why is it a "function of $\theta$"? $\endgroup$ – time Jun 13 '13 at 12:38
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    $\begingroup$ Should one relate this nearly auto-replicating phenomenon to the nearly instantaneous acceptation of answers? $\endgroup$ – Did Jun 13 '13 at 12:38
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1) If n rv's (not one) are independent, then their joint density is the product of their individual densities (this extends the notion of the probability of intersection of n independent events in the case of rv's). If, moreover, they are (assumed) identically distributed, then $$ L(X_1,\ldots,X_n| \theta)=\prod_{i=1}^nf(X_i|\theta)=f(X_1|\theta)\ldots f(X_n|\theta)$$ is the joint density. If we view this expression as a function of $\theta$, then it becomes

$$ L(\theta|x_1,\ldots,x_n)=\prod_{i=1}^nf(\theta|x_i)=f(\theta|x_1)\ldots f(\theta|x_n)$$ and now it is the likelihood function of the sample. Is there a difference? Big one: In the joint density, the argument/variable of the function is the X's while the $\theta$'s are treated as constants. In the likelihood function, the arguments/variables are the $\theta$'s while the x's are treated as constants (changing from uppercase to lowercase for the x's is a usual -and good- mnemonic) So the likelihood function is not the joint density - they may have the same functional form, but they are functions of different variables.

2) In view of 1), we do not select the $\theta$'s "so as the rv's assume a particular value...etc" as you write. We select the $\theta$'s given the specific values that the r.v's have already taken (values that constitute our sample), so as the resulting value of the function is maximum. Then we flip our point of view again and say (very loosely speaking), that this value of $\theta$ "maximizes the probability that the sample comes from the assumed distribution".

3) The uniform distribution is special in that the variable itself does not appear in the density function. The specific case we study is even more special, because not even the limits of the interval into which the rv ranges appear in it: the density is equal to unity for the specified range, and zero elsewhere (even though the range of the rv is not [0,1]). Instead of writing a piece-wise density we write compactly $$ f_U (X_i)=I_{[\theta-\frac{1}{2},\;\theta+\frac{1}{2}]} (X_i)$$ Then the joint density for the case of n i.i.d rv's is $$ L(X_1,\ldots,X_n| \theta)=\prod_{i=1}^nf(X_i|\theta)=\prod_{i=1}^n I_{[\theta-\frac{1}{2},\;\theta+\frac{1}{2}]} (X_i) =\min\{I_{[\theta-\frac{1}{2}\le X_1\le\theta+\frac{1}{2}]} ,...,I_{[\theta-\frac{1}{2}\le X_n\le\theta+\frac{1}{2}]}\}$$ where the last equality comes from basic properties of indicator functions. Consider now the joint density as a likelihood function of $\theta$, with given x's. Then we should maximize this likelihood function? No. We are applying here maximum likelihood estimation, and the ML estimators are M-estimators, which maximize the expected value of the likelihood function (or its sample analogue, whatever is feasible). So our estimation problem here can be stated as $$ \max_{\theta}E\min\{I_{[\theta-\frac{1}{2}\le x_1\le\theta+\frac{1}{2}]} ,...,I_{[\theta-\frac{1}{2}\le x_n\le\theta+\frac{1}{2}]}\}$$

Let's start the optimization procedure. Note that if we choose some $\hat \theta$ such that some of the realized values in the sample fall outside the specified range, the min{} expression will be equal to zero. But if the chosen $\hat \theta$ has a value that leaves all realized values of the rv's inside the resulting common interval, then the min{} expression will be equal to unity. I adopt the notation $x_1=x_{min}$ and $x_n=x_{max}$. So a first conclusion is that we must have $$ x_1\ge \theta-\frac 12 \text{and} \ x_n\le \theta+\frac 12 \\ \Rightarrow x_n-\frac 12\le \theta\le x_1+\frac 12$$ (Parenthesis: For $\hat\theta$ to be able to obtain a value in the reals this interval must exist. So we must have $$x_n-\frac 12\le x_1+\frac 12 \Rightarrow x_n-x_1 \le 1$$ In other words, if in the sample at hand the difference between the maximum realized value and the minimum realized value is greater than unity, then one of the initial assumptions is wrong: either the n rv's do not have the postulated distribution, or they are not identically distributed). Assume that $x_n-x_1 \le 1$ holds. Then our maximization problem has been transformed (after this first step of optimization) as $$ \max_{\theta}EI_{[x_n-\frac 12\ \, \le \, \theta \, \le \, x_1+\frac 12]}$$

The $\min$ operator has disappeared because all indicator functions have become identical. Now by the properties of indicator functions, $$ EI_{[x_n-\frac 12\ \, \le \, \theta \, \le \, x_1+\frac 12]} = P\left(\left[x_n-\frac 12\ \, \le \, \theta \, \le \, x_1+\frac 12\right]\right)$$ where P() is the probability of the interval. Next we can write the upper bound for $\theta$ as a constraint and solve the constrained maximization problem

$$ \max_{\theta}P\left(\left[x_n-\frac 12\ \, , \, \theta\right]\right) \qquad \text{s.t.}\ \theta\le x_1+\frac 12$$

Obviously this probability increases with the length of the interval, and so $$ \hat \theta : [x_n-\frac 12\ , \hat \theta] = \text{maximum}$$ which obviously results in $$ \hat \theta_{\text{ML}} = x_1+\frac 12$$

Then the maximum likelihood estimate of the common distribution of the n rv's is $$U\left(x_1\, , \, x_1+1\right).$$

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  • $\begingroup$ Sorry but, starting from and including "the expected value of the likelihood function", the exposition is a terrible mishmash. I wonder where you saw that one should solve $\max\limits_\theta E[L(X_1,\ldots,X_n\mid\theta)$... $\endgroup$ – Did Jul 31 '13 at 20:47
  • $\begingroup$ Totally standard. To give just one reference, Hayashi "Econometrics"(2000), Chapter 7 Extremum Estimators. Page 447. The MLE maximizes the average log-likelihood, but we tend to forget that, because usually the (1/n) is a multiplicative factor that does not affect the results. We take the average log-likelihood as the natural sample analogue of its expected value, because usually we cannot do better than that. But in the specific model, formulating the problem as it should be formulated theoretically (with the exp val), gives us the result immediately. $\endgroup$ – Alecos Papadopoulos Aug 1 '13 at 0:43
  • $\begingroup$ So ("totally standardly"), the MLE estimator $\hat\theta$ of the parameter $\theta$ does not depend on the sample $(X_1,\ldots,X_n)$? For example, if each $P_\theta$ is the uniform distribution on the interval $(0,\theta)$, then for every sample you suggest the value $\hat\theta=0$... $\endgroup$ – Did Aug 1 '13 at 6:07
  • $\begingroup$ @Did Where did you see in my formulas that the MLE does not depend on the sample? In all the expressions in my answer that have to do with "max something" the sample is present (like $x_1$ and $x_n$) -just look. If the $P_\theta$ is the uniform distribution in the interval $(0,\theta)$, then my results suggest that the MLE is the maximum observed value, not zero, i.e. the value that maximizes the interval, which is the correct answer. I don't understand how you figured that my results would suggest the value zero in such a case... $\endgroup$ – Alecos Papadopoulos Aug 1 '13 at 9:21
  • $\begingroup$ Just look yourself: what is written after the first $\max\limits_\theta$? A deterministic function of $\theta$ (otherwise, why the expectation symbol? and why mentioning "the expected value of the likelihood function"?). Hence the $\arg\max$ over $\theta$ gives a value $\theta$ independent of the sample $(X_1,X_2,\ldots,X_n)$. Not at all what you describe in your comment (and not at all what people do "totally standardly"). $\endgroup$ – Did Aug 1 '13 at 9:29

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