Abstract
Let X be an observable random variable with unknown distribution function \(F(x) = \mathbb{P}(X \leq x)\), \(- \infty< x < \infty\), and let
We call θ the power of moments of the random variable X. Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size n drawn from \(F(\cdot)\). In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties:
where \(\log x = \ln(e \vee x)\), \(- \infty< x < \infty\). In particular, we show that
This means that, under very reasonable conditions on \(F(\cdot)\), \(\hat {\theta}_{n}\) is actually a consistent estimator of θ.
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1 Motivation
The motivation of the current work arises from the following problem concerning parameter estimation. Let X be an observable random variable with unknown distribution function \(F(x) = \mathbb{P}(X \leq x)\), \(- \infty< x < \infty\), and let
We call θ the power of moments of the random variable X. Clearly θ is a parameter of the distribution of the random variable X. Now let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size n drawn from the random variable X; i.e., \(X_{1}, X_{2}, \ldots, X_{n}\) are independent and identically distributed (i.i.d.) random variables whose common distribution function is \(F(\cdot)\). It is natural to pose the following question: Can we estimate the parameter θ based on the random sample \(X_{1}\), …, \(X_{n}\)?
This is a serious and important problem. For example, if \(\theta> 2\) and if the distribution of X is nondegenerate, then it is clear that \(0 < \operatorname{Var} X < \infty \) and so by the classical Lévy central limit theorem, the distribution of
is approximately normal (for all sufficiently large n) with mean 0 and variance \(\sigma^{2} = \operatorname{Var}X = \mathbb{E}(X - \mu)^{2}\) where \(\mu = \mathbb{E}X\). Thus the problem that we are facing is how can we conclude with a high degree of confidence that \(\theta> 2\).
In this paper we propose the following point estimator of θ and will investigate its asymptotic properties:
Here and below \(\log x = \ln(e \vee x)\), \(- \infty< x < \infty\).
Our main results will be stated in Sect. 2 and they all pertain to a sequence of i.i.d. random variables \(\{X_{n}; n \geq1\}\) drawn from the distribution function \(F(\cdot)\) of the random variable X. The proofs of our main results will be provided in Sect. 3.
2 Statement of the main results
Throughout, X is a random variable with unknown distribution \(F(x) = \mathbb{P}(X \leq x)\), \(-\infty< x < \infty\) and write
Clearly, just as θ as defined in Sect. 1 is a parameter of the distribution \(F(\cdot)\) of the random variable X, so are \(\rho_{1}\) and \(\rho_{2}\). These parameters satisfy
The main results of this paper are Theorems 2.1–2.5.
Theorem 2.1
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of the random variable X. Then
and there exists an increasing positive integer sequence \(\{l_{n}; n \geq1 \}\) (which depends on the probability distribution of X when \(\rho_{1} < \infty\)) such that
Theorem 2.2
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of the random variable X. Then
and there exists an increasing positive integer sequence \(\{m_{n}; n \geq1 \}\) (which depends on the probability distribution of X when \(\rho_{2} > 0\)) such that
Remark 2.1
We must point out that (2.2) and (2.4) are two interesting conclusions. To see this, let \(\{U_{n}; n \geq1 \}\) be a sequence of independent random variables with
Since
it follows from the Borel–Cantelli lemma that
However, for any sequences \(\{l_{n}; n \geq1\}\) and \(\{m_{n}; n \geq1 \}\) of increasing positive integers,
Remark 2.2
For an observable random variable X, it is often the case that \(\rho _{1} = \rho_{2}\). However, for any given constants \(\rho_{1}\) and \(\rho_{2}\) with \(0 \leq\rho_{1} < \rho_{2} \leq\infty\), one can construct a random variable X such that
For example, if \(0 < \rho_{1} < \rho_{2} < \infty\), a random variable X can be constructed having probability distribution given by
where \(d_{n} = 2^{ (\rho_{2}/\rho_{1} )^{n}}\), \(n \geq1\) and
Combining Theorems 2.1 and 2.2, we establish a law of large numbers for \(\log\max_{1 \leq k \leq n} X_{k}\), \(n \geq1\) as follows.
Theorem 2.3
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of the random variable X and let \(\rho\in[0, \infty]\). Then the following four statements are equivalent:
If \(0 \leq\rho< \infty\), then anyone of (2.5)–(2.8) holds if and only if there exists a function \(L(\cdot): (0, \infty) \rightarrow(0, \infty)\) such that
The following result concerns convergence in distribution for \(\log\max _{1 \leq k \leq n}X_{k}\), \(n \geq1\).
Theorem 2.4
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of the random variable X. Suppose that there exist constants \(0 < \rho< \infty\) and \(-\infty< \tau< \infty\) and a monotone function \(h(\cdot): [0, \infty) \rightarrow(0, \infty)\) with \(\lim_{x \rightarrow\infty}h(x^{2})/h(x) = 1\) such that
Then
Also, by Theorems 2.1–2.3, we have the following result for the point estimator \(\hat{\theta}_{n}\).
Theorem 2.5
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of the random variable X. Let
Then we have
and the following three statements are equivalent:
If \(0 \leq\theta< \infty\), then anyone of (2.12)–(2.14) holds if and only if there exists a function \(L(\cdot): (0, \infty) \rightarrow(0, \infty)\) such that
Remark 2.3
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of some nonnegative random variable X. For each \(n \geq1\), let \(X_{n,1} \leq X_{n,2} \leq\cdots\leq X_{n,n}\) denote the order statistics based on \(X_{1}, X_{2}, \ldots, X_{n}\). To estimate the tail index of \(F(\cdot)\), the well-known Hill estimator, proposed by Hill [1], is defined by
where \(\{k_{n}; n \geq1 \}\) is a sequence of positive integers satisfying
Mason [2, Theorem 2] showed that, for some constant \(\theta\in (0, \infty)\),
if and only if
Since \(L(\cdot)\) defined in (2.17) is a slowly varying function,
is always true and hence (2.15) follows from (2.17). However, the following example shows that (2.15) does not imply (2.17). Thus condtion (2.15) is weaker than (2.17).
Example 2.1
Let \(\{X_{n}; n \geq1\}\) be a sequence of i.i.d. random variables drawn from the distribution function \(F(\cdot)\) of some nonnegative random variable X given by
where \(\theta\in(0, \infty)\) is the tail index of the distribution and \([t]\) denotes the integer part of t. Then (2.15) holds but (2.17) is not satisfied. To see this, let
Then
Since, for \(x \geq e\), \(0 \leq\ln x -[\ln x] \leq1\), we have
and hence (2.15) holds. However, for \(1 < a < e\) and \(x_{n} = e^{n}\), \(n \geq1\), we have
Thus, for \(\theta\in(0, \infty)\),
and hence
i.e., \(L(\cdot)\) is not a slowly varying function. Thus (2.17) is not satisfied and hence, for this example, the well-known Hill estimator cannot be used to estimate the tail index θ.
3 Proofs of the main results
Let \(\{A_{n}; n \geq1 \}\) be a sequence of events. As usual the abbreviation \(\{A_{n} \mbox{ i.o.} \}\) stands for the case that the events \(A_{n}\) occur infinitely often. That is,
For events A and B, we say \(A = B\) a.s. if \(\mathbb{P}(A \mathrel{\Delta} B) = 0\) where \(A \mathrel{\Delta}B = (A \setminus B) \cup(B \setminus A)\). To prove Theorem 2.1, we use the following preliminary result, which can be found in Chandra [3, Example 1.6.25(a), p. 48].
Lemma 3.1
Let \(\{b_{n}; n \geq1 \}\) be a nondecreasing sequence of positive real numbers such that
and let \(\{V_{n}; n \geq1 \}\) be a sequence of random variables. Then
Proof of Theorem 2.1
Case I: \(0 < \rho_{1} < \infty\). For given \(\epsilon> 0\), let \(r(\epsilon) = (\frac{1}{\rho_{1}} + \epsilon )^{-1}\). Then
and hence
By the Borel–Cantelli lemma, (3.1) implies that
By Lemma 3.1, we have
and hence
Thus
Letting \(\epsilon\searrow0\), we get
By the definition of \(\rho_{1}\), we have
which is equivalent to
Then, inductively, we can choose positive integers \(l_{n}\), \(n \geq1\) such that
Note that, for any \(0 \leq z \leq1\), \(1 - z \leq e^{-z}\). Thus, for all sufficiently large n, we have
Since \(\sum_{n=1}^{\infty} n^{-2} < \infty\), by the Borel–Cantelli lemma, we get
which ensures that
Clearly, (2.1) and (2.2) follow from (3.2) and (3.3).
Case II: \(\rho_{1} = \infty\). Using the same argument used in the first half of the proof for Case I, we get
and hence
Note that
We thus have
It thus follows from (3.4) and (3.5) that
proving (2.1) and (2.2) (with \(l_{n} = n\), \(n \geq1\)).
Case III: \(\rho_{1} = 0\). By the definition of \(\rho_{1}\), we have
which is equivalent to
Then, inductively, we can choose positive integers \(l_{n}\), \(n \geq1\) such that
Thus, for all sufficiently large n, we have by the same argument as in Case I
and hence by the Borel–Cantelli lemma
which ensures that
Thus (2.1) and (2.2) hold. This completes the proof of Theorem 2.1. □
Proof of Theorem 2.2
Case I: \(0 < \rho_{2} < \infty\). For given \(\rho_{2} < r < \infty\), let \(r_{1} = (r + \rho_{2} )/2\) and \(\tau= 1 - (r_{1}/r)\). Then \(\rho _{2} < r_{1} < r < \infty\) and \(\tau> 0\). By the definition of \(\rho_{2}\), we have
and hence, for all sufficiently large x,
Thus, for all sufficiently large n,
and hence
Since
by the Borel–Cantelli lemma, we have
which implies that
Letting \(r \searrow\rho_{2}\), we get
Again, by the definition of \(\rho_{2}\), we have
which is equivalent to
Then, inductively, we can choose positive integers \(m_{n}\), \(n \geq1\) such that
Then we have
Thus, by the Borel–Cantelli lemma, we get
which ensures that
Clearly, (2.3) and (2.4) follow from (3.6) and (3.7).
Case II: \(\rho_{2} = \infty\). By the definition of \(\rho_{2}\), we have
which is equivalent to
Then, inductively, we can choose positive integers \(m_{n}\), \(n \geq1\) such that
Thus
and hence by the Borel–Cantelli lemma
which ensures that
It is clear that
It thus follows from (3.8) and (3.9) that
Case III: \(\rho_{2} = 0\). Using the same argument used in the first half of the proof for Case I, we get
Letting \(r \searrow0\), we get
Thus
and hence (2.3) and (2.4) hold with \(m_{n} = n\), \(n \geq1\). □
Proof of Theorem 2.3
It follows from Theorems 2.1 and 2.2 that
Since (2.6) follows from (2.5), we only need to show that (2.6) implies (2.8). It follows from (2.6) that
Since, for \(n \geq3\),
and
it follows from (3.10) that
which is equivalent to (2.8).
For \(0 \leq\rho< \infty\), note that
We thus see that, if \(0 \leq\rho< \infty\), then (2.8) is equivalent to
(We leave it to the reader to work out the details of the proof.) We thus see that (2.8) implies (2.9) with \(L(x) = x^{\rho} \mathbb{P}(X > x)\), \(x > 0\). It is easy to verify that (2.8) follows from (2.9). This completes the proof of Theorem 2.3. □
Proof of Theorem 2.4
For fixed \(x \in(-\infty, \infty)\), write
Then
Since \(h(\cdot): [0, \infty) \rightarrow(0, \infty)\) is a monotone function with \(\lim_{x \rightarrow\infty}h(x^{2})/h(x) = 1\), \(h(\cdot)\) is a slowly varying function such that \(\lim_{x \rightarrow \infty} h(x^{r})/h(x) = 1\) \(\forall r > 0\) and hence
Clearly,
It thus follows from (2.10) that, as \(n \rightarrow\infty\),
so that
i.e., (2.11) holds. □
Proof of Theorem 2.5
Since \(\hat{\theta}_{n} = \frac{\log n}{\log\max_{1 \leq k \leq n} \vert X_{k} \vert }\), \(n \geq1\), Theorem 2.5 follows immediately from Theorems 2.1–2.3. □
4 Conclusions
In this paper we propose the following simple point estimator of θ, the power of moments of the random variable X, and investigate its asymptotic properties:
In particular, we show that
This means that, under very reasonable conditions on \(F(\cdot)\), \(\hat {\theta}_{n}\) is actually a consistent estimator of θ. From Remark 2.3 and Example 2.1, we see that, for a nonnegative random variable X, \(\hat{\theta}_{n}\) is a consistent estimator of θ whenever the well-known Hill estimator \(\hat{\alpha}_{n}\) is a consistent estimator of θ. However, the converse is not true.
References
Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Mason, D.M.: Laws of large numbers for sums of extreme values. Ann. Probab. 10, 754–764 (1982)
Chandra, T.K.: The Borel–Cantelli Lemma. Springer, Heidelberg (2012)
Acknowledgements
The authors are grateful to the referee for carefully reading the manuscript and for offering helpful suggestions and constructive criticism which enabled them to improve the paper. The research of Shuhua Chang was partially supported by the National Natural Science Foundation of China (Grant #: 91430108 and 11771322) and the research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (Grant #: RGPIN-2014-05428).
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Chang, S., Li, D., Qi, Y. et al. A method for estimating the power of moments. J Inequal Appl 2018, 54 (2018). https://doi.org/10.1186/s13660-018-1645-7
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DOI: https://doi.org/10.1186/s13660-018-1645-7