
The article presents a certain investment strategy based on the difference between two moving averages, modified to allow the extraction of patterns. The strategy concept dropped the traditionally considered intersections of two averages and opening positions just after those intersections. Based on the observation of changes happening in the moving averages difference, it has been noticed that for some values of this difference and some values of additional strategy parameters, an interesting pattern appears that allows short-term prediction. These patterns also depended on the first derivative of the moving averages difference and the location of the current price relative to certain thresholds of the difference. Therefore, the strategy uses five parameters, including Stop Loss, adapted to the properties of the time series through machine learning. The importance of machine learning is highlighted by comparing simulation results with and without it. The strategy effectiveness was tested in the Matlab environment on the time series of the WIG20 (primary index of the Warsaw Stock Exchange) historical data. Satisfactory results were obtained considered in terms of minimizing investment risk measured by the Calmar indicator.
Citation: Antoni Wilinski, Mateusz Sochanowski, Wojciech Nowicki. An investment strategy based on the first derivative of the moving averages difference with parameters adapted by machine learning[J]. Data Science in Finance and Economics, 2022, 2(2): 96-116. doi: 10.3934/DSFE.2022005
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The article presents a certain investment strategy based on the difference between two moving averages, modified to allow the extraction of patterns. The strategy concept dropped the traditionally considered intersections of two averages and opening positions just after those intersections. Based on the observation of changes happening in the moving averages difference, it has been noticed that for some values of this difference and some values of additional strategy parameters, an interesting pattern appears that allows short-term prediction. These patterns also depended on the first derivative of the moving averages difference and the location of the current price relative to certain thresholds of the difference. Therefore, the strategy uses five parameters, including Stop Loss, adapted to the properties of the time series through machine learning. The importance of machine learning is highlighted by comparing simulation results with and without it. The strategy effectiveness was tested in the Matlab environment on the time series of the WIG20 (primary index of the Warsaw Stock Exchange) historical data. Satisfactory results were obtained considered in terms of minimizing investment risk measured by the Calmar indicator.
Great progress has been made in the field of statistics and probability theory for interdisciplinary research. New techniques and methods have been developed to meet the challenges of data analysis. Statistical methods are becoming increasingly important in various areas of science. The increasing complexity of scientific problems requires the development of new and suitable statistical methods for interdisciplinary research. Current challenges include, ecology: Quantifying biodiversity; the epidemiology of infectious diseases: Disease outbreak detection; financial mathematics: stock option valuation; industrial engineering: Stochastic optimization; and genomics: Personalized medicine, to name a few.
For a random lifetime T, the conditional inactivity time is defined as Tt=t−T|T≤t, and t>0. An important measure developed based on the conditional inactivity time is the α-quantile inactivity time (α-QIT), which is the α-quantile of Tt. Assuming that the distribution function of T is denoted by F, the α-QIT can be expressed by the following relationship:
qα(t)=t−F−1(−αF(t)),t>0, |
where −α=1−α and F−1(p)=inf{x:F(x)=p} is the inverse function of F. Let T be the event time referring to the instances of a species. Among all instances experienced the event at a time before t, we expect 100α% of these instances to have experienced the event after time t−qα(t). In this sense, a smaller qα(t) means larger T. The α-QIT is a competitor for the mean inactivity time (MIT) function. The MIT has been intensively studied by researchers in the field of reliability theory and survival analysis, e.g., refer to Finkelstein [1] and Kayid and Izadkhah [2]. However, when the moments of the underlying model are infinite or heavily skewed to right, α-QIT is preferred over MIT (see Schmittlein and Morrison [3] for a detailed justification of quantile-based than moment-based measures). The α-QIT concept was formally defined and studied by Unnikrishnan and Vineshkumar [4]. Shafaei [5] showd that how the underlying model can be characterized by α-QIT function. Shafaei and Izadkhah [6] stated some properties of a parallel system in terms of the α-QIT measure. For a sample T1, T2, ..., Tn of iid lifetimes, the α-QIT can be estimated by
qα,n(t)=t−F−1n(−αFn(t)),t≥T(1), |
where Fn(t) is the empirical distribution function, i.e.,
Fn(t)=1n∑ni=1I(Ti≤t), |
and
F−1n(p)=inf{x:Fn(x)≥p}={0p=0,T(1)0<p≤1n,T(2)1n<p≤2n,…T(n)1−1n<p≤1. |
Then, qα,n can be written as in the following.
qα,n(t)={t0≤t<t1,t−T(1)t1≤t<t2,t−T(2)t2≤t<t3,…t−T(n−1)tk−1≤t<tk,t−T(n)t≥tk, | (1) |
where
ti=inf{y:−αFn(y)>i−1n}=inf{y:F−1n(−αFn(y))=T(i)}. |
Note that F−1n(−αFn(ti))=T(i), for every i=1,2,…,k for some k≤n t1=T(1) and ti≥T(i) for i=2…...,k. The expression (1) shows that qα,n(t) consists of line segments with slope 1 on intervals (ti,ti+1), i…,2,...,k−1 and falls at each point ti by T(i)−T(i−1), i=1,2,...,k where T(0)=0. Figure 1 shows a schematic plot of qα,n(t).
For the univariate case, Mahdy [7] proposed the estimator (2) for the α-QIT function and investigated its asymptotic properties. Balmert and Jeong [8] created a nonparametric inference of the median inactivity time function for right-censored data. Balmert et al. [9] applied a log-linear quantile regression model to the inactivity time for right-censored data. Kayid [10] applied the Kaplan-Meiere survival estimator to the α-QIT function for estimation and inference.
We can have two or more dependent events. For example, if successive events of the same person/instance are tracked, the event times depend on each other. Another example is that researchers are interested in determining the effect of a treatment on specific event times related to the eyes, ears, hands or legs. One organ was randomly selected for treatment and the other was a control organ. The events associated with these organs depend on their progression. In such cases, we need to extend the measures in question to bivariate or multivariate settings. In the following section, I refer to the authors who have implemented this idea. Basu [11] and Johnson and Kotz [12] examined the multivariate hazard rate function as a gradient vector. The mean residual lifetime was extended by Nair and Nair [13] to obtain a vector of dependent lifetimes. Shaked and Shanthikumar [14] introduced the dynamic multivariate MRL concept. Kayid [15] developed the multivariate MIT concept. Navarro [16] characterized the basic model by the bivariate hazard rate function. The concept of the α-quantile residual lifetime (α-QRL) was extended to the multivariate context by Shafaei and Kayid [17]. Shafaei et al. [18] discussed the multivariate α-QRL concept in a dynamic way. Buono et al. [19] applied multivariate RHR for discussing reliability attributes of systems. Kayid [20] extended the α-QIT concept to bivariate context and discussed its estimation.
Let F be the distribution function of a random pair T=(T1,T2). Then, the α-QIT vector at point t=(t1,t2) is defined to be (qα,1(t),qα,2(t)). The first element of this vector is
qα,1(t)=sup{x:P(t1−T1>x|T≤t)=−α}=sup{x:F(t1−x,t2)=−αF(t)}=inf{t1−z:F(z,t2)=−αF(t)}=t1−F−11(−αF(t);t2), |
where −α=1−α and
F−11(p;t2)=inf{z:F(z,t2)=p}, |
is the partial inverse of F in terms of the T1. The second element of the α-QIT vector is defined similarly.
qα,2(t)=t2−F−12(−αF(t);t1), |
where F−12(p;t1)=inf{z:F(t1,z)=p} is the partial inverse of F in terms of the second element. The reversed hazard rate vector of T is (r1(t),r2(t)) and
ri(t)=∂∂tilogF(t),i=1,2. |
The RHR satisfies the following relation.
{∂∂t1F(t1,t2)=r1(t1,t2)F(t1,t2),∂∂t2F(t1,t2)=r2(t1,t2)F(t1,t2). |
Kayid [20] showed that if ri(t) is decreasing (increasing) in ti, then qα,i(t) is increasing (decreasing) in ti. It is a surprising fact that for most of the standard bivariate models, ri(t) is decreasing in ti (Finkelstein [1]). For example, bivariate Gumbel, Pareto, normal, and gamma models have decreasing reversed hazard rate functions. This implies that qα,i(t) is increasing in ti. For some examples of such models, refer to Kayid [20]. This motivates me to introduce a new estimator of qα,1(t) and qα,2(t) under the assumption that they are increasing with respect to t1 and t2, respectively. It is expected that applying this knowledge, I have a more accurate estimator than the usual estimator defined by Kayid [20]. Such monotone estimators are defined and studied by Kochar et al. [21], Franco Pereira and Una-Alvarez [22], and Shafaei and Franco Pereira [23].
The rest of this paper is structured as follows. In Section 2, the promised increasing estimator of the bivariate α-QIT function is proposed and its asymptotic properties are discussed. Then, the performance of the new estimator is compared with that of the usual estimator in a simulation study. In Section 4, the proposed estimator is applied to investigate the effect of laser treatment on the time to blindness. In Section 5, I summarize the results.
Let …,...,Tn be an iid random sample from bivariate distribution F. The empirical distribution function is defined by
Fn(t1,t2)=n−1∑ni=1I(T1i≤t1,T2i≤t2), |
and the partial inverse of Fn, with respect to the first and second elements, are as in the following respectively:
F−11,n(p;t2)=inf{x:Fn(x,t2)≥p}, |
and
F−12,n(p;t1)=inf{x:Fn(t1,x)≥p}. |
Kayid [20] proposed the following estimator of the bivariate α-QIT vector.
qα,n(t)=(qα,1,n(t),qα,2,n(t)), |
where
{qα,1,n(t)=t1−F−11,n(−αFn(t);t2),qα,2,n(t)=t2−F−12,n(−αFn(t);t1), |
with the knowledge of increasing bivariate α-IQT, we define the natural estimator
iqα,n(t)=(iqα,1,n(t),iqα,2,n(t)), |
where
{iqα,1,n(t)=supy≤t1qα,1,n(y,t2),iqα,2,n(t)=supy≤t2qα,2,n(t1,y), |
Let t2>0 be fixed and define T1[t2]=T1|T2≤t2, then the distribution function of T1[t2] is
F∗1(x;t2)=P(T1[t2]≤x)=F(x,t2)F2(t2), |
where F2(t2)=P(T2≤t2). Denote α-QIT of T1[t2] by q∗α,1(t1;t2), then it can be shown that
q∗α,1(t1;t2)=qα,1(t1,t2). | (2) |
Similarly, for every fixed t1>0, we define T2[t1]=T2|T1≤t1 following distribution F∗2(.;t1). Let q∗α,2(t2;t1) be the α-QIT of T2[t1], then we can investigate that
q∗α,2(t2;t1)=qα,2(t1,t2). | (3) |
Given a bivariate iid random sample (T1i,T2i), i=1,2,…,n from distribution F, and for every fixed t2, consider the following univariate random sample which follows from F∗1(.;t2).
χ(1,t2)={T1ij: whenT2ij≤t2,j=1…...,k1(t2)}. |
I can apply this sample to estimate q∗α,1(t1;t2), as in the following.
q∗α,1,n(t1;t2)=t1−F∗−11,n(−αF∗1,n(t1;t2)), |
where
F∗1,n(t1;t2)=#(T1ij≤t1)k1(t2), |
and
F∗−11,n(p)=inf{x:F∗1,n(x;t2)≥p}. |
Applying the knowledge of increasing q∗α,1(t1;t2) in terms of t1, it is natural to use the following estimator.
iq∗α,1,n(t1;t2)=supy≤t1 q∗α,1,n(y;t2). |
Again, for a bivariate iid random sample (T1i,T2i), i…,2,...,n from distribution F, and for every fixed t1, consider the following sample.
χ(2,t1)={T2ij: whenT1ij≤t1⋯=1,2,...,k2(t1)}, |
which follows from F∗2(.;t1). Then, the estimator of q∗α,2(t2;t1) is defined by
q∗α,2,n(t2;t1)=t2−F∗−12,n(−αF∗2,n(t2;t1)), |
where
F∗2,n(t2;t1)=#(T2ij≤t2)k2(t1), |
and
F∗−12,n(p)=inf{x:F∗2,n(x;t1)≥p}. |
In an increasing context,
iq∗α,2,n(t2;t1)=supy≤t2 q∗α,2,n(y;t1). |
It is clear that
{iqα,1,n(t)=iq∗α,1,n(t1;t2),iqα,2,n(t)=iq∗α,2,n(t2;t1). | (4) |
Theorem 1. Let us assume that the following two conditions are fulfilled.
(C1). F(t1,t2) be twice differentiable with respect each element.
(C2). ∂∂t1F(t) and ∂∂t2F(t) are bounded from zero on the intervals (0,F−11(−α;t2)) and (0,F−12(−α;t1)), respectively, for every t1>0 and t2>0.
Then, (iqα,1,n(t),iqα,2,n(t)) is consistent for (iqα,1(t),iqα,2(t)).
Proof. By Theorem 7 from Kayid [24], we have
|iq∗α,1,n(t1;t2)−iq∗α,1(t1;t2)|→0, almost every where, |
and
|iq∗α,2,n(t2;t1)−iq∗α,2(t2;t1)|→0, almost every where. |
Thus, the result follows from (2)–(4).
To state the next theorem, I need two following conditions.
(C3) ∂∂t1qα,1(t) and ∂∂t2qα,2(t) exist and there are c1>0 and c2>0 such that ∂∂t1qα,1(t)>c1 and ∂∂t2qα,2(t)>c2 for all 0<t1<b1 and 0<t2<b2 for some positive b1 and b2.
(C4) ∂2∂t1∂t1qα,1(t) and ∂2∂t2∂t2qα,2(t) exist and
sup0<t1<b1|∂2∂t1∂t1qα,1(t)|≤c3<∞andsup0<t2<b2|∂2∂t2∂t2qα,2(t)|≤c4<∞. |
Theorem 2. Assume that C1–C4 are satisfied. Then, we have
√n|(iqα,1,n(t),iqα,2,n(t))−(qα,1,n(t),qα,2,n(t))|→0, in probability. |
Proof. By Theorem 5 from Kayid [21], we have
sup0<t<b1|iq∗α,1,n(t1;t2)−q∗α,1,n(t1;t2)|→0, inprobability, |
and
sup0<t<b2|iq∗α,2,n(t2;t1)−q∗α,2,n(t2;t1)|→0, inprobability. |
Thus, the result follows from relations (2)–(4) and the concept of convergence in probability in bivariate setting.
The following lemma, which is the result of the well-known Slutsky theorem, is used in the proof of the next theorem (see Van der Vaart [25] for Slutsky's theorem and related results).
Lemma 1. If √n(Xn−Yn)→0 in probability and √nXn converges, in distribution, to a random variable X with distribution F, then √nYn converges, in distribution, to a random variable Y with the same distribution F.
Theorem 3. Under the conditions C1–C4, we have
√n|(iqα,1,n(t),iqα,2,n(t))−(qα,1(t),qα,2(t))|→N(0,CΣC), in distribution, |
where
C=−[∂∂pF−11(p;t2)|p=−αF(t)00∂∂pF−12(p;t1)|p=−αF(t)], |
and elements of Σ are
σ11=σ22=α−αF(t), |
and
σ12=σ21=F(F−11(−αF(t);t2),F−12(−αF(t);t1))−−α2F(t). |
Proof. Theorem 7 of Kayid [20] states that under some mild conditions:
√n|(qα,1,n(t),qα,2,n(t))−(qα,1(t),qα,2(t))|→N(0,CΣC), in distribution, |
where C and Σ are defined in this theorem. Thus, applying Lemma 1, the result follows immediately.
In the real world, lifetime random pairs T1,T2,...,Tn may be censored by a random censorship Ci, in the sense that the observations are T1i=T1i∧Ci, T2i=T2i∧Ci, δ1i=I(T1i>Ci) and δ2i=I(T2i>Ci). Note that a∧b=min{a,b}. Let censorship random variable Ci be independent from desired lifetimes and follows from distribution G and the reliability function −G=1−G, i.e., −G(t)=P(Ci>t). Also, let R(t1,t2)=P( T1i>t1, T2i>t2) and R(t1,t2)=P(T1i>t1,T2i>t2). Then, we have
R(t1,t2)= R(t1,t2)−G(t1∨t2), |
where t1∨t2=max{t1,t2}. So, we can estimate the reliability function R by
Rn(t1,t2)=1n∑ni=1I( T1i>t1, T2i>t2)−Gn(t1∨t2). |
Under this censoring scheme, Lin and Ying [26] showed that Rn(t1,t2) is strongly consistent and weakly converges to a Gaussian process. Thus, when we have such censored data, the empirical distribution function could be replaced by the following estimate:
Fn(t1,t2)=1−Rn(t1,0)−Rn(0,t2)+Rn(t1,t2). |
To investigate the performance of the proposed (increasing) estimator and comparing it with the usual estimator, a simulation study is conducted. The bivariate Gumbel and Pareto distributions with respectively the following reliability functions are selected for the baseline models:
−F(t1,t2)=exp{−t1−t2−βt1t2},β>0,t1≥0,t2≥0, |
and
−F(t1,t2)=(t1+t2−1)−c,c>0,t1≥1,t2≥1. |
Both models are important from practical and theoretical points of view. The Gumbel distribution was introduced by Gumbel [27], and the Pareto model was used by Jupp and Mardia [28] to analyze income data for consecutive years. Some proper values for β and c were selected. In each simulation run, r=1000 replicates of bivariate samples of size n were generated, where n was set to 25, 50 or 100. For each sample, q0.5,1,n() and its increasing version, iq0.5,1,n(), are calculated at four appropriate time points t1, t2, t3 and t4 according to the following rules: Let F1 be the marginal distribution of the first element and ti=(t1i,t2i). The equations F1(t11)=0.25, F1(t12)=0.40, F1(t13)=0.50 and F1(t14)=0.75 are solved to find t11 to t14 and given them, the equations F(t11,t21)=0.2, F(t12,t22)=0.3, F(t13,t23)=0.4, and F(t14,t24)=0.6 are solved for t21 to t24. After calculating the objective functions for r replicates, the bias (B) and mean squared error (MSE) were calculated and are shown in Tables 1 and 2 for the Gumbel and Pareto models, respectively. All simulations and calculations were performed in R (statistical programming language). The results show small values for B and MSE for both the conventional estimator and the proposed increasing estimator. As expected, the MSE increases with F(t) (see Theorem 3). The MSE values for the increasing estimator are smaller in all cases, indicating that the increasing estimator performs better than the conventional estimator. See Figures 2 and 3 for a graphicall representaion of the ratio of MSE values related to the ususal to the increasing estimator.
β | ||||||||
0.8 | 1 | 1.4 | ||||||
Estimator | n | point | B | MSE | B | MSE | B | MSE |
Usual | 25 | t1 | 0.0133 | 0.0036 | 0.0120 | 0.0037 | 0.0115 | 0.0035 |
t2 | 0.0146 | 0.0075 | 0.0096 | 0.0075 | 0.0144 | 0.0074 | ||
t3 | 0.0113 | 0.0105 | 0.0086 | 0.0099 | 0.0157 | 0.0100 | ||
t4 | 0.0118 | 0.0242 | 0.0150 | 0.0212 | 0.0203 | 0.0232 | ||
50 | t1 | 0.0071 | 0.0019 | 0.0068 | 0.0019 | 0.0062 | 0.0018 | |
t2 | 0.0093 | 0.0040 | 0.0074 | 0.0039 | 0.0071 | 0.0039 | ||
t3 | 0.0055 | 0.0053 | 0.0069 | 0.0055 | 0.0050 | 0.0054 | ||
t4 | 0.0070 | 0.0110 | 0.0050 | 0.0115 | 0.0059 | 0.0118 | ||
100 | t1 | 0.0057 | 0.0010 | 0.0049 | 0.0009 | 0.0043 | 0.0010 | |
t2 | 0.0033 | 0.0021 | 0.0035 | 0.0022 | 0.0035 | 0.0020 | ||
t3 | 0.0017 | 0.0028 | 0.0058 | 0.0027 | 0.0022 | 0.0028 | ||
t4 | -0.0010 | 0.0064 | 0.0027 | 0.0059 | 0.0009 | 0.0058 | ||
Increasing | 25 | t1 | 0.0395 | 0.0032 | 0.0388 | 0.0031 | 0.0386 | 0.0030 |
t2 | 0.0480 | 0.0064 | 0.0466 | 0.0065 | 0.0503 | 0.0065 | ||
t3 | 0.0440 | 0.0082 | 0.0435 | 0.0079 | 0.0473 | 0.0086 | ||
t4 | 0.0370 | 0.0204 | 0.0392 | 0.0182 | 0.0455 | 0.0196 | ||
50 | t1 | 0.0224 | 0.0016 | 0.0232 | 0.0016 | 0.0238 | 0.0017 | |
t2 | 0.0286 | 0.0035 | 0.0282 | 0.0034 | 0.0268 | 0.0033 | ||
t3 | 0.0244 | 0.0043 | 0.0252 | 0.0047 | 0.0238 | 0.0045 | ||
t4 | 0.0192 | 0.0105 | 0.0175 | 0.0108 | 0.0163 | 0.0110 | ||
100 | t1 | 0.0145 | 0.0009 | 0.0141 | 0.0008 | 0.0139 | 0.0009 | |
t2 | 0.0140 | 0.0019 | 0.0141 | 0.0019 | 0.0148 | 0.0019 | ||
t3 | 0.0103 | 0.0026 | 0.0137 | 0.0026 | 0.0115 | 0.0025 | ||
t4 | 0.0063 | 0.0060 | 0.0058 | 0.0057 | 0.0082 | 0.0055 |
c | ||||||||
0.5 | 0.7 | 1.1 | ||||||
Estimator | n | point | B | MSE | B | MSE | B | MSE |
Usual | 25 | t1 | -0.0040 | 0.1866 | 0.0149 | 0.0686 | 0.0018 | 0.0242 |
t2 | 0.0030 | 0.2704 | -0.0067 | 0.1025 | -0.0042 | 0.0346 | ||
t3 | -0.0412 | 0.4076 | -0.0285 | 0.1508 | -0.0082 | 0.0469 | ||
t4 | -0.0591 | 0.5858 | -0.0349 | 0.2085 | -0.0156 | 0.0620 | ||
50 | t1 | 0.0017 | 0.0903 | -0.0100 | 0.0407 | 0.0040 | 0.0113 | |
t2 | -0.0152 | 0.1273 | -0.0178 | 0.0554 | 0.0005 | 0.0159 | ||
t3 | -0.0312 | 0.1913 | -0.0296 | 0.0816 | -0.0025 | 0.0221 | ||
t4 | -0.0534 | 0.3003 | -0.0345 | 0.1103 | -0.0004 | 0.0273 | ||
100 | t1 | -0.0086 | 0.0461 | -0.0017 | 0.0172 | 0.0008 | 0.0057 | |
t2 | -0.0089 | 0.0577 | -0.0030 | 0.0253 | -0.0010 | 0.0080 | ||
t3 | -0.0137 | 0.0823 | -0.0057 | 0.0343 | -0.0037 | 0.0107 | ||
t4 | -0.0222 | 0.1243 | -0.0086 | 0.0495 | -0.0035 | 0.0141 | ||
Increasing | 25 | t1 | 0.0004 | 0.1832 | 0.0271 | 0.0603 | 0.1125 | 0.0197 |
t2 | 0.0032 | 0.2702 | -0.0018 | 0.1008 | 0.0049 | 0.0304 | ||
t3 | -0.0411 | 0.4076 | -0.0285 | 0.1508 | -0.0043 | 0.0438 | ||
t4 | -0.0591 | 0.5858 | -0.0349 | 0.2085 | -0.0155 | 0.0620 | ||
50 | t1 | 0.0057 | 0.0897 | 0.0009 | 0.0373 | 0.0830 | 0.0087 | |
t2 | -0.0135 | 0.1261 | -0.0142 | 0.0539 | 0.0048 | 0.0151 | ||
t3 | -0.0307 | 0.1906 | -0.0286 | 0.0802 | -0.0012 | 0.0218 | ||
t4 | -0.0534 | 0.3002 | -0.0343 | 0.1103 | -0.0004 | 0.0273 | ||
100 | t1 | -0.0046 | 0.0447 | 0.0014 | 0.0166 | 0.0631 | 0.0046 | |
t2 | -0.0072 | 0.0570 | -0.0017 | 0.0252 | 0.0008 | 0.0077 | ||
t3 | -0.0137 | 0.0823 | -0.0042 | 0.0339 | -0.0029 | 0.0105 | ||
t4 | -0.0222 | 0.1243 | -0.0085 | 0.0495 | -0.0033 | 0.0140 |
In a study that began in 1971, researchers were interested in the effect of laser photocoagulation on delaying blindness in patients with DR. Patients with visual acuity ≥ 20/100 in both eyes were selected for the study. One eye of each patient was randomly selected for laser photocoagulation (treatment) and the other eye was observed without treatment (control). The time from the start of treatment to blindness is given in months. Blindness means that visual acuity fell below 5/200 on two consecutive visits. The data for this study is available in the "diabetic" dataset in the "survival" package in R. Table 3 shows part of the dataset relating to adolescents (under 20 years of age). For patient i, T1i and T2iindicate the observed time to blindness in the control and treated eyes, respectively.
Patient (i) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
T1i | 6.9 | 1.63 | 13.83 | 35.53 | 14.8 | 6.2 | 22 | 1.7 | 43.03 |
T2i | 20.17 | 10.27 | 5.67 | 5.90 | 33.9 | 1.73 | 30.2 | 1.7 | 1.77 |
Patient (i) | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
T1i | 6.53 | 42.17 | 48.43 | 9.6 | 7.6 | 1.8 | 9.9 | 13.77 | 0.83 |
T2i | 18.7 | 42.17 | 14.3 | 13.33 | 14.27 | 34.57 | 21.57 | 13.77 | 10.33 |
Patient (i) | 19 | 20 | 21 | 22 | 23 | 24 | |||
T1i | 1.97 | 11.3 | 30.4 | 19 | 5.43 | 46.63 | |||
T2i | 11.07 | 2.1 | 13.97 | 13.80 | 13.57 | 42.43 |
Figures 4 and 5 draw the bivariate median inactivity time functions and their increasing versions, iqn,0.5,1(t) and iqn,0.5,2(t), respectively.
The comparison of the proposed increasing median inactivity time functions iqn,0.5,1(t) and iqn,0.5,2(t) at different points is informative in investigating the treatment effect. To provide a simple and powerful statistics, we can consider the points on the identity line and use the following statistics
dn(t)=iqn,0.5,1(t,t)−iqn,0.5,2(t,t),t≥0. | (5) |
If I assume that the treatment dose not effect the time length to blindness, dn(t) should be positive or negitive values near zero, reflecting some random errors. However, if the treatment causes longer time to blindness, I expect relatively larger values for iqn,0.5,1(t,t) than iqn,0.5,2(t,t), i.e., positive values for dn(t). Figure 6 plots dn(t) in all points of the observed T1 or T2. The plot shows positive values that increase with t and indicates that the treatment causes longer time to blindness. The effect of treatment also increases with time. The bivariate median inactivity functions are on the right side of Figure 6 to provide a better comparison of these functions.
Assuming an increasing α-QIT function, I define a new estimator for this function. It is proven that the proposed estimator is consistent. It is asymptotically close to the usual estimator in the sense that the difference to the usual estimator converges to zero with high probability. It is also shown that the proposed estimator converges weakly to a Gaussian process when normalized. Interestingly, none of the asymptotic results assume that the true α-QIT function increases, which increases the applicability of the estimator in general. The simulation results show that the MSE for the proposed increasing estimator is smaller than that of the conventional estimator. When using the proposed estimator, it was found that the laser treatment causes a delay in glare.
The author thanks the two anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.
This work was supported by Researchers Supporting Project number (RSP2024R392), King Saud University, Riyadh, Saudi Arabia.
The data for this study is available in the "diabetic" dataset in the "survival" package in R.
The author declares that there is no conflict of interest.
[1] |
Anderson BD, Deistler M, Felsenstein E, et al. (2016) The structure of multivariate AR and ARMA systems: Regular and singular systems; the single and the mixed frequency case. J Econometrics 192: 366–373. https://doi.org/10.1016/j.jeconom.2016.02.004 doi: 10.1016/j.jeconom.2016.02.004
![]() |
[2] |
Babu CN, Reddy BE (2014) A moving-average filter based hybrid ARIMA–ANN model for forecasting time series data. Appl Soft Comput 23: 27–38. https://doi.org/10.1016/j.asoc.2014.05.028 doi: 10.1016/j.asoc.2014.05.028
![]() |
[3] |
Brock W, Lakonishok J, LeBaron B (1992). Simple technical trading rules and the stochastic properties of stock returns. J Financ 47: 1731–1764. https://doi.org/10.1111/j.1540-6261.1992.tb04681.x doi: 10.1111/j.1540-6261.1992.tb04681.x
![]() |
[4] |
Chan JC (2013) Moving average stochastic volatility models with application to inflation forecast. J Econometrics 176: 162–172. https://doi.org/10.1016/j.jeconom.2013.05.003 doi: 10.1016/j.jeconom.2013.05.003
![]() |
[5] |
Dias GF, Kapetanios, G (2018) Estimation and forecasting in vector autoregressive moving average models for rich datasets. J Econometrics 202: 75–91. https://doi.org/10.1016/j.jeconom.2017.06.022 doi: 10.1016/j.jeconom.2017.06.022
![]() |
[6] |
Ellis CA, Parbery SA (2005) Is smarter better? A comparison of adaptive, and simple moving average trading strategies. Res Int Bus Financ 19: 399–411. https://doi.org/10.1016/j.ribaf.2004.12.009 doi: 10.1016/j.ribaf.2004.12.009
![]() |
[7] |
Gencay R (1998) The predictability of security returns with simple technical trading rules. J Empir Financ 5: 347–359. https://doi.org/10.1016/j.ribaf.2004.12.009 doi: 10.1016/j.ribaf.2004.12.009
![]() |
[8] |
Gencay R (1996). Non-linear prediction of security returns with moving average rules. J Forecasting 15: 165–174. https://doi.org/10.1002/(SICI)1099-131X(199604)15:3<165::AID-FOR617>3.0.CO;2-V doi: 10.1002/(SICI)1099-131X(199604)15:3<165::AID-FOR617>3.0.CO;2-V
![]() |
[9] | Hamilton JD (1994) Time series analysis. Princeton, NJ: Princeton university press. 2: 690–696 |
[10] |
Horváth L, Rice G (2015) Testing for independence between functional time series. J Econometrics 189: 371–382. https://doi.org/10.1016/j.jeconom.2015.03.030 doi: 10.1016/j.jeconom.2015.03.030
![]() |
[11] | Johnston J, DiNardo J (1972) Econometric methods (Vol. 2). New York. |
[12] | LeBaron B (1992) Do moving average trading rule results imply nonlinearities in foreign exchange markets? Social Systems Research Institute, University of Wisconsin. |
[13] |
Ling S, McAleer M, Tong H (2015) Frontiers in time series and financial econometrics: An overview. J Econometrics 189: 145–250. https://doi.org/10.1016/j.jeconom.2015.03.019 doi: 10.1016/j.jeconom.2015.03.019
![]() |
[14] | Main R (2017) Evaluating Traders' Performers with the Calmar Ratio, www. proptradingfutures/thecalamr-ratio. Access J: www.proptradingfutures.com/the-calmar-ratio/. |
[15] |
Metghalchi M, Marcucci J, Chang YH (2012) Are moving average trading rules profitable? Evidence from the European stock markets. Appl Econ 44: 1539–1559. https://doi.org/10.1080/00036846.2010.543084 doi: 10.1080/00036846.2010.543084
![]() |
[16] |
Moon YS, Kim J (2007) Efficient moving average transform-based subsequence matching algorithms in time-series databases. Inf Sci 177: 5415–5431. https://doi.org/10.1016/j.ins.2007.05.038 doi: 10.1016/j.ins.2007.05.038
![]() |
[17] | Murphy JJ (1999) Technical analysis of the financial markets: A comprehensive guide to trading methods and applications. Penguin. |
[18] | Schwager JD (1984) A complete guide to the futures markets: fundamental analysis, technical analysis, trading, spreads, and options. John Wiley & Sons. |
[19] | Schwager JD (1995) Technical analysis. (Vol. 43). John Wiley & Sons. |
[20] | Wei WW (2006) Time series analysis. In The Oxford Handbook of Quantitative Methods in Psychology: Vol. 2. |
[21] | Wilder JW (1978) New concepts in technical trading systems. Trend Research. |
[22] | Yamane T (1973) Statistics: An introductory analysis. Researchgate.com |
[23] | Young WT (1991) Calmar Ratio: A Smoother Tool. Futures 20: 40 |
[24] |
Zhu K, Li WK (2015) A bootstrapped spectral test for adequacy in weak ARMA models. J Econometrics 187: 113–130. https://doi.org/10.1016/j.jeconom.2015.02.005 doi: 10.1016/j.jeconom.2015.02.005
![]() |
[25] |
Zhu Y, Zhou G (2009) Technical analysis: An asset allocation perspective on the use of moving averages. J Financ Econ 92: 519–544. https://doi.org/10.1016/j.jfineco.2008.07.002 doi: 10.1016/j.jfineco.2008.07.002
![]() |
[26] | Wilinski A (2011) Prediction Models of Financial Markets Based on Multiregression Algorithms, Comput Sci J Mold., vol. 19, no. 2, pp. 178–188, |
[27] | Wilinski A, Zablocki M (2015) The Investment Strategy Based on the Difference of Moving Averages with Parameters Adapted by Machine Learning. Advanced in Intelligent Systems and Computing Springer, Cham Heidelberg New York, 342: 207–227 |
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β | ||||||||
0.8 | 1 | 1.4 | ||||||
Estimator | n | point | B | MSE | B | MSE | B | MSE |
Usual | 25 | t1 | 0.0133 | 0.0036 | 0.0120 | 0.0037 | 0.0115 | 0.0035 |
t2 | 0.0146 | 0.0075 | 0.0096 | 0.0075 | 0.0144 | 0.0074 | ||
t3 | 0.0113 | 0.0105 | 0.0086 | 0.0099 | 0.0157 | 0.0100 | ||
t4 | 0.0118 | 0.0242 | 0.0150 | 0.0212 | 0.0203 | 0.0232 | ||
50 | t1 | 0.0071 | 0.0019 | 0.0068 | 0.0019 | 0.0062 | 0.0018 | |
t2 | 0.0093 | 0.0040 | 0.0074 | 0.0039 | 0.0071 | 0.0039 | ||
t3 | 0.0055 | 0.0053 | 0.0069 | 0.0055 | 0.0050 | 0.0054 | ||
t4 | 0.0070 | 0.0110 | 0.0050 | 0.0115 | 0.0059 | 0.0118 | ||
100 | t1 | 0.0057 | 0.0010 | 0.0049 | 0.0009 | 0.0043 | 0.0010 | |
t2 | 0.0033 | 0.0021 | 0.0035 | 0.0022 | 0.0035 | 0.0020 | ||
t3 | 0.0017 | 0.0028 | 0.0058 | 0.0027 | 0.0022 | 0.0028 | ||
t4 | -0.0010 | 0.0064 | 0.0027 | 0.0059 | 0.0009 | 0.0058 | ||
Increasing | 25 | t1 | 0.0395 | 0.0032 | 0.0388 | 0.0031 | 0.0386 | 0.0030 |
t2 | 0.0480 | 0.0064 | 0.0466 | 0.0065 | 0.0503 | 0.0065 | ||
t3 | 0.0440 | 0.0082 | 0.0435 | 0.0079 | 0.0473 | 0.0086 | ||
t4 | 0.0370 | 0.0204 | 0.0392 | 0.0182 | 0.0455 | 0.0196 | ||
50 | t1 | 0.0224 | 0.0016 | 0.0232 | 0.0016 | 0.0238 | 0.0017 | |
t2 | 0.0286 | 0.0035 | 0.0282 | 0.0034 | 0.0268 | 0.0033 | ||
t3 | 0.0244 | 0.0043 | 0.0252 | 0.0047 | 0.0238 | 0.0045 | ||
t4 | 0.0192 | 0.0105 | 0.0175 | 0.0108 | 0.0163 | 0.0110 | ||
100 | t1 | 0.0145 | 0.0009 | 0.0141 | 0.0008 | 0.0139 | 0.0009 | |
t2 | 0.0140 | 0.0019 | 0.0141 | 0.0019 | 0.0148 | 0.0019 | ||
t3 | 0.0103 | 0.0026 | 0.0137 | 0.0026 | 0.0115 | 0.0025 | ||
t4 | 0.0063 | 0.0060 | 0.0058 | 0.0057 | 0.0082 | 0.0055 |
c | ||||||||
0.5 | 0.7 | 1.1 | ||||||
Estimator | n | point | B | MSE | B | MSE | B | MSE |
Usual | 25 | t1 | -0.0040 | 0.1866 | 0.0149 | 0.0686 | 0.0018 | 0.0242 |
t2 | 0.0030 | 0.2704 | -0.0067 | 0.1025 | -0.0042 | 0.0346 | ||
t3 | -0.0412 | 0.4076 | -0.0285 | 0.1508 | -0.0082 | 0.0469 | ||
t4 | -0.0591 | 0.5858 | -0.0349 | 0.2085 | -0.0156 | 0.0620 | ||
50 | t1 | 0.0017 | 0.0903 | -0.0100 | 0.0407 | 0.0040 | 0.0113 | |
t2 | -0.0152 | 0.1273 | -0.0178 | 0.0554 | 0.0005 | 0.0159 | ||
t3 | -0.0312 | 0.1913 | -0.0296 | 0.0816 | -0.0025 | 0.0221 | ||
t4 | -0.0534 | 0.3003 | -0.0345 | 0.1103 | -0.0004 | 0.0273 | ||
100 | t1 | -0.0086 | 0.0461 | -0.0017 | 0.0172 | 0.0008 | 0.0057 | |
t2 | -0.0089 | 0.0577 | -0.0030 | 0.0253 | -0.0010 | 0.0080 | ||
t3 | -0.0137 | 0.0823 | -0.0057 | 0.0343 | -0.0037 | 0.0107 | ||
t4 | -0.0222 | 0.1243 | -0.0086 | 0.0495 | -0.0035 | 0.0141 | ||
Increasing | 25 | t1 | 0.0004 | 0.1832 | 0.0271 | 0.0603 | 0.1125 | 0.0197 |
t2 | 0.0032 | 0.2702 | -0.0018 | 0.1008 | 0.0049 | 0.0304 | ||
t3 | -0.0411 | 0.4076 | -0.0285 | 0.1508 | -0.0043 | 0.0438 | ||
t4 | -0.0591 | 0.5858 | -0.0349 | 0.2085 | -0.0155 | 0.0620 | ||
50 | t1 | 0.0057 | 0.0897 | 0.0009 | 0.0373 | 0.0830 | 0.0087 | |
t2 | -0.0135 | 0.1261 | -0.0142 | 0.0539 | 0.0048 | 0.0151 | ||
t3 | -0.0307 | 0.1906 | -0.0286 | 0.0802 | -0.0012 | 0.0218 | ||
t4 | -0.0534 | 0.3002 | -0.0343 | 0.1103 | -0.0004 | 0.0273 | ||
100 | t1 | -0.0046 | 0.0447 | 0.0014 | 0.0166 | 0.0631 | 0.0046 | |
t2 | -0.0072 | 0.0570 | -0.0017 | 0.0252 | 0.0008 | 0.0077 | ||
t3 | -0.0137 | 0.0823 | -0.0042 | 0.0339 | -0.0029 | 0.0105 | ||
t4 | -0.0222 | 0.1243 | -0.0085 | 0.0495 | -0.0033 | 0.0140 |
Patient (i) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
T1i | 6.9 | 1.63 | 13.83 | 35.53 | 14.8 | 6.2 | 22 | 1.7 | 43.03 |
T2i | 20.17 | 10.27 | 5.67 | 5.90 | 33.9 | 1.73 | 30.2 | 1.7 | 1.77 |
Patient (i) | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
T1i | 6.53 | 42.17 | 48.43 | 9.6 | 7.6 | 1.8 | 9.9 | 13.77 | 0.83 |
T2i | 18.7 | 42.17 | 14.3 | 13.33 | 14.27 | 34.57 | 21.57 | 13.77 | 10.33 |
Patient (i) | 19 | 20 | 21 | 22 | 23 | 24 | |||
T1i | 1.97 | 11.3 | 30.4 | 19 | 5.43 | 46.63 | |||
T2i | 11.07 | 2.1 | 13.97 | 13.80 | 13.57 | 42.43 |
β | ||||||||
0.8 | 1 | 1.4 | ||||||
Estimator | n | point | B | MSE | B | MSE | B | MSE |
Usual | 25 | t1 | 0.0133 | 0.0036 | 0.0120 | 0.0037 | 0.0115 | 0.0035 |
t2 | 0.0146 | 0.0075 | 0.0096 | 0.0075 | 0.0144 | 0.0074 | ||
t3 | 0.0113 | 0.0105 | 0.0086 | 0.0099 | 0.0157 | 0.0100 | ||
t4 | 0.0118 | 0.0242 | 0.0150 | 0.0212 | 0.0203 | 0.0232 | ||
50 | t1 | 0.0071 | 0.0019 | 0.0068 | 0.0019 | 0.0062 | 0.0018 | |
t2 | 0.0093 | 0.0040 | 0.0074 | 0.0039 | 0.0071 | 0.0039 | ||
t3 | 0.0055 | 0.0053 | 0.0069 | 0.0055 | 0.0050 | 0.0054 | ||
t4 | 0.0070 | 0.0110 | 0.0050 | 0.0115 | 0.0059 | 0.0118 | ||
100 | t1 | 0.0057 | 0.0010 | 0.0049 | 0.0009 | 0.0043 | 0.0010 | |
t2 | 0.0033 | 0.0021 | 0.0035 | 0.0022 | 0.0035 | 0.0020 | ||
t3 | 0.0017 | 0.0028 | 0.0058 | 0.0027 | 0.0022 | 0.0028 | ||
t4 | -0.0010 | 0.0064 | 0.0027 | 0.0059 | 0.0009 | 0.0058 | ||
Increasing | 25 | t1 | 0.0395 | 0.0032 | 0.0388 | 0.0031 | 0.0386 | 0.0030 |
t2 | 0.0480 | 0.0064 | 0.0466 | 0.0065 | 0.0503 | 0.0065 | ||
t3 | 0.0440 | 0.0082 | 0.0435 | 0.0079 | 0.0473 | 0.0086 | ||
t4 | 0.0370 | 0.0204 | 0.0392 | 0.0182 | 0.0455 | 0.0196 | ||
50 | t1 | 0.0224 | 0.0016 | 0.0232 | 0.0016 | 0.0238 | 0.0017 | |
t2 | 0.0286 | 0.0035 | 0.0282 | 0.0034 | 0.0268 | 0.0033 | ||
t3 | 0.0244 | 0.0043 | 0.0252 | 0.0047 | 0.0238 | 0.0045 | ||
t4 | 0.0192 | 0.0105 | 0.0175 | 0.0108 | 0.0163 | 0.0110 | ||
100 | t1 | 0.0145 | 0.0009 | 0.0141 | 0.0008 | 0.0139 | 0.0009 | |
t2 | 0.0140 | 0.0019 | 0.0141 | 0.0019 | 0.0148 | 0.0019 | ||
t3 | 0.0103 | 0.0026 | 0.0137 | 0.0026 | 0.0115 | 0.0025 | ||
t4 | 0.0063 | 0.0060 | 0.0058 | 0.0057 | 0.0082 | 0.0055 |
c | ||||||||
0.5 | 0.7 | 1.1 | ||||||
Estimator | n | point | B | MSE | B | MSE | B | MSE |
Usual | 25 | t1 | -0.0040 | 0.1866 | 0.0149 | 0.0686 | 0.0018 | 0.0242 |
t2 | 0.0030 | 0.2704 | -0.0067 | 0.1025 | -0.0042 | 0.0346 | ||
t3 | -0.0412 | 0.4076 | -0.0285 | 0.1508 | -0.0082 | 0.0469 | ||
t4 | -0.0591 | 0.5858 | -0.0349 | 0.2085 | -0.0156 | 0.0620 | ||
50 | t1 | 0.0017 | 0.0903 | -0.0100 | 0.0407 | 0.0040 | 0.0113 | |
t2 | -0.0152 | 0.1273 | -0.0178 | 0.0554 | 0.0005 | 0.0159 | ||
t3 | -0.0312 | 0.1913 | -0.0296 | 0.0816 | -0.0025 | 0.0221 | ||
t4 | -0.0534 | 0.3003 | -0.0345 | 0.1103 | -0.0004 | 0.0273 | ||
100 | t1 | -0.0086 | 0.0461 | -0.0017 | 0.0172 | 0.0008 | 0.0057 | |
t2 | -0.0089 | 0.0577 | -0.0030 | 0.0253 | -0.0010 | 0.0080 | ||
t3 | -0.0137 | 0.0823 | -0.0057 | 0.0343 | -0.0037 | 0.0107 | ||
t4 | -0.0222 | 0.1243 | -0.0086 | 0.0495 | -0.0035 | 0.0141 | ||
Increasing | 25 | t1 | 0.0004 | 0.1832 | 0.0271 | 0.0603 | 0.1125 | 0.0197 |
t2 | 0.0032 | 0.2702 | -0.0018 | 0.1008 | 0.0049 | 0.0304 | ||
t3 | -0.0411 | 0.4076 | -0.0285 | 0.1508 | -0.0043 | 0.0438 | ||
t4 | -0.0591 | 0.5858 | -0.0349 | 0.2085 | -0.0155 | 0.0620 | ||
50 | t1 | 0.0057 | 0.0897 | 0.0009 | 0.0373 | 0.0830 | 0.0087 | |
t2 | -0.0135 | 0.1261 | -0.0142 | 0.0539 | 0.0048 | 0.0151 | ||
t3 | -0.0307 | 0.1906 | -0.0286 | 0.0802 | -0.0012 | 0.0218 | ||
t4 | -0.0534 | 0.3002 | -0.0343 | 0.1103 | -0.0004 | 0.0273 | ||
100 | t1 | -0.0046 | 0.0447 | 0.0014 | 0.0166 | 0.0631 | 0.0046 | |
t2 | -0.0072 | 0.0570 | -0.0017 | 0.0252 | 0.0008 | 0.0077 | ||
t3 | -0.0137 | 0.0823 | -0.0042 | 0.0339 | -0.0029 | 0.0105 | ||
t4 | -0.0222 | 0.1243 | -0.0085 | 0.0495 | -0.0033 | 0.0140 |
Patient (i) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
T1i | 6.9 | 1.63 | 13.83 | 35.53 | 14.8 | 6.2 | 22 | 1.7 | 43.03 |
T2i | 20.17 | 10.27 | 5.67 | 5.90 | 33.9 | 1.73 | 30.2 | 1.7 | 1.77 |
Patient (i) | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
T1i | 6.53 | 42.17 | 48.43 | 9.6 | 7.6 | 1.8 | 9.9 | 13.77 | 0.83 |
T2i | 18.7 | 42.17 | 14.3 | 13.33 | 14.27 | 34.57 | 21.57 | 13.77 | 10.33 |
Patient (i) | 19 | 20 | 21 | 22 | 23 | 24 | |||
T1i | 1.97 | 11.3 | 30.4 | 19 | 5.43 | 46.63 | |||
T2i | 11.07 | 2.1 | 13.97 | 13.80 | 13.57 | 42.43 |