# A Statistical Analysis of Cryptocurrencies

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## Abstract

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## 1. Introduction

## 2. Data

**Bitcoin**is undoubtedly the most popular and prominent cryptocurrency. It was the first realisation of the idea of a new type of money, mentioned over two decades ago, that “uses cryptography to control its creation and transactions, rather than a central authority” (Bitcoin Project 2017). This decentralization means that the Bitcoin network is controlled and owned by all of its users, and as all users must adhere to the same set of rules, there is a great incentive to maintain the decentralized nature of the network. Bitcoin uses blockchain technology, which keeps a record of every single transaction, and the processing and authentication of transactions are carried out by the network of users (Bitcoin Project 2017). Although the decentralized nature offers many advantages, such as being free from government control and regulation, critics often argue that apart from its users, there is nobody overlooking the whole system and that the value of Bitcoin is unfounded. In return for contributing their computing power to the network to carry out some of the tasks mentioned above, also known as “mining”, users are rewarded with Bitcoins. These properties set Bitcoin apart from traditional currencies, which are controlled and backed by a central bank or governing body.

**Dash**(formerly known as Darkcoin and XCoin) is a “privacy-centric digital currency with instant transactions” (The Dash Network 2017). Although it is based upon Bitcoin’s foundations and shares similar properties, Dash’s network is two-tiered, improving upon that of Bitcoin’s. In contrast with Bitcoin, Dash is overseen by a decentralized network of servers—known as “Masternodes” (The Dash Network 2017) which alleviates the need for a third party governing body, and allows for functions such as financial privacy and instant transactions. On the other hand, users or “miners” in the network provide the computing power for basic functions such as sending and receiving currency, and the prevention of double spending. The advantage of utilizing Masternodes is that transactions can be confirmed almost in real time (compared with the Bitcoin network) because Masternodes are separate from miners, and the two have non-overlapping functions (The Dash Network 2017). Dash utilizes the X11 chained proof-of-work hashing algorithm which helps to distribute the processing evenly across the network while maintaining a similar coin distribution to Bitcoin. Using eleven different hashes increases security and reduces the uncertainty in Dash. Dash operates “Decentralized Governance by Blockchain” (The Dash Network 2017) which allows owners of Masternodes to make decisions, and provides a method for the platform to fund its own development.

**LiteCoin (LTC)**was created in 2011 by Charles Lee with support from the Bitcoin community. Based on the same peer-to-peer protocol used by Bitcoin, it is often cited as Bitcoin’s leading rival as it features improvements over the current implementation of Bitcoin. It has two main features which distinguish it from Bitcoin, its use of scrypt as a proof-of-work algorithm and a significantly faster confirmation time for transactions. The former enables standard computational hardware to verify transactions and reduces the incentive to use specially designed hardware, while the latter reduces transaction confirmation times to minutes rather than hours and is particularly attractive in time-critical situations (LiteCoin Project 2017).

**MaidSafeCoin**is a digital currency which powers the peer-to-peer Secure Access For Everyone (SAFE) network, which combines the computing power of all its users, and can be thought of as a “crowd-sourced internet” (MaidSafe 2017a). Each MadeSafe coin has a unique identity and there exists a hard upper limit of 4.3 billion coins as opposed to Bitcoin’s 21 million. As the currency is used to pay for services on the SAFE network, the currency will be recycled meaning that in theory the amount of MaidSafe coins will never be exhausted. The process of generating new currency is similar to other cryptocurrencies and in the case of the SAFE network it is known as “farming” (MaidSafe 2017b). Users contribute their computing power and storage space to the network and are rewarded with coins when the network accesses data from their store (MaidSafe 2017b).

**Monero (XMR)**is a “secure, private, untraceable currency” (Monero 2017) centred around decentralization and scalability that was launched in April 2014. The currency itself is completely donation-based, community driven and based entirely on proof-of-work. Whilst transactions in the network are private by default, users can set their level of privacy allowing as much or as little access to their transactions as they wish. Although it employs a proof-of-work algorithm, Monero is more similar to LiteCoin in that mining of the currency can be done by any modern computer and is not restricted to specially designed hardware. It arguably holds some advantages over other Bitcoin-based cryptocurrencies such as having a dynamic block size (overcoming the problem of scalability), and being a disinflationary currency meaning that there will always exist an incentive to produce the Monero currency (Monero 2017).

**Dogecoin**(Dogecoin 2017) originated from a popular internet meme in December 2013. Created by an Australian brand and marketing specialist, and a programmer in Portland, Oregon, it initially started off as a joke currency but quickly gained traction. It is a variation on Litecoin, running on the cryptographic scrypt enabling similar advantages over Bitcoin such as faster transaction processing times. Part of the attraction of Dogecoin is its light-hearted culture and lower barriers to entry to investing in or acquiring cryptocurrencies. One of the most popular uses for Dogecoin is the tipping of others on the internet who create or share interesting content, and can be thought of as the next level up from a “like” on social media or an “upvote” on internet forums. This in part has arisen from the fact that it has now become too expensive to tip using Bitcoin.

**Ripple**was originally developed in 2012 and is the first global real-time gross settlement network (RTGS) which “enables banks to send real-time international payments across networks” (Ripple 2017). The Ripple network is a blockchain network which incorporates a payment system, and a currency system known as XRP which is not based on proof-of-work like Monero and Dash. A unique property of Ripple is that XRP is not compulsory for transactions on the network, although it is encouraged as a bridge currency for more competitive cross border payments (Ripple 2017). The Ripple protocol is currently used by companies such as UBS, Santander, and Standard Chartered, and increasingly being used by the financial services industry as technology in settlements. Compared with Bitcoin, it has advantages such as greater control over the system as it is not subject to the price volatility of the underlying currencies, and it has a more secure distributed authentication process.

## 3. Distributions Fitted

- the Student’s t distribution (Gosset 1908) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle K\left(\nu \right)}}{{\displaystyle \sigma}}{\left[1+\frac{{\displaystyle {(x-\mu )}^{2}}}{{\displaystyle {\sigma}^{2}\nu}}\right]}^{-(1+\nu )/2}}\end{array}$$$$\begin{array}{c}\hfill {\displaystyle B(a,b)={\int}_{0}^{1}{t}^{a-1}{(1-t)}^{b-1}dt;}\end{array}$$
- the Laplace distribution (Laplace 1774) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle 1}}{{\displaystyle 2\sigma}}exp\left(-\frac{{\displaystyle \mid x-\mu \mid}}{{\displaystyle \sigma}}\right)}\end{array}$$
- the skew t distribution (Azzalini and Capitanio 2003) with$$\begin{array}{ccc}\hfill f\left(x\right)& =& {\displaystyle \frac{{\displaystyle K\left(\nu \right)}}{{\displaystyle \sigma}}{\left[1+\frac{{\displaystyle {(x-\mu )}^{2}}}{{\displaystyle {\sigma}^{2}\nu}}\right]}^{-(1+\nu )/2}}\hfill \\ & & {\displaystyle +\frac{{\displaystyle 2{K}^{2}\left(\nu \right)\lambda (x-\mu )}}{{\displaystyle {\sigma}^{2}}}{}_{2}{F}_{1}\left(\frac{{\displaystyle 1}}{{\displaystyle 2}},\frac{{\displaystyle 1+\nu}}{{\displaystyle 2}};\frac{{\displaystyle 3}}{{\displaystyle 2}};-\frac{{\displaystyle {\lambda}^{2}{(x-\mu )}^{2}}}{{\displaystyle {\sigma}^{2}\nu}}\right)}\hfill \end{array}$$$$\begin{array}{c}\hfill {\displaystyle {}_{2}{F}_{1}\left(a,b;c;x\right)=\sum _{k=0}^{\infty}\frac{{\displaystyle {\left(a\right)}_{k}{\left(b\right)}_{k}}}{{\displaystyle {\left(c\right)}_{k}}}\frac{{\displaystyle {x}^{k}}}{{\displaystyle k!}},}\end{array}$$
- the generalized t distribution (McDonald and Newey 1988) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle \tau}}{{\displaystyle 2\sigma {\nu}^{1/\nu}B\left(\nu ,1/\tau \right)}}{\left[1+\frac{{\displaystyle 1}}{{\displaystyle \nu}}{\left|\frac{{\displaystyle x-\mu}}{{\displaystyle \sigma}}\right|}^{\tau}\right]}^{-\left(\nu +1/\tau \right)}}\end{array}$$
- the skewed Student’s t distribution (Zhu and Galbraith 2010) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle K\left(\nu \right)}}{{\displaystyle \sigma}}\left\{\begin{array}{cc}{\displaystyle {\left\{1+\frac{{\displaystyle 1}}{{\displaystyle \nu}}{\left[\frac{{\displaystyle x-\mu}}{{\displaystyle 2\sigma \alpha}}\right]}^{2}\right\}}^{-\frac{\nu +1}{2}},}\hfill & \mathrm{if}x\le \mu ,\hfill \\ {\displaystyle {\left\{1+\frac{{\displaystyle 1}}{{\displaystyle \nu}}{\left[\frac{{\displaystyle x-\mu}}{{\displaystyle 2\sigma \left(1-\alpha \right)}}\right]}^{2}\right\}}^{-\frac{\nu +1}{2}},}\hfill & \mathrm{if}x\mu \hfill \end{array}\right.}\end{array}$$
- the asymmetric Student’s t distribution (Zhu and Galbraith 2010) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle 1}}{{\displaystyle \sigma}}\left\{\begin{array}{cc}{\displaystyle \frac{{\displaystyle \alpha}}{{\displaystyle {\alpha}^{\ast}}}K\left({\nu}_{1}\right){\left\{1+\frac{{\displaystyle 1}}{{\displaystyle {\nu}_{1}}}{\left[\frac{{\displaystyle x-\mu}}{{\displaystyle 2\sigma {\alpha}^{\ast}}}\right]}^{2}\right\}}^{-\frac{{\nu}_{1}+1}{2}},}\hfill & \mathrm{if}x\le \mu ,\hfill \\ {\displaystyle \frac{{\displaystyle 1-\alpha}}{{\displaystyle 1-{\alpha}^{\ast}}}K\left({\nu}_{2}\right){\left\{1+\frac{{\displaystyle 1}}{{\displaystyle {\nu}_{2}}}{\left[\frac{{\displaystyle x-\mu}}{{\displaystyle 2\sigma \left(1-{\alpha}^{\ast}\right)}}\right]}^{2}\right\}}^{-\frac{{\nu}_{2}+1}{2}},}\hfill & \mathrm{if}x\mu \hfill \end{array}\right.}\end{array}$$$$\begin{array}{c}\hfill {\displaystyle {\alpha}^{\ast}=\frac{{\displaystyle \alpha K\left({\nu}_{1}\right)}}{{\displaystyle \alpha K\left({\nu}_{1}\right)+(1-\alpha )K\left({\nu}_{2}\right)}};}\end{array}$$
- the normal inverse Gaussian distribution (Barndorff-Nielsen 1977) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle {\left(\gamma /\delta \right)}^{\lambda}\alpha}}{{\displaystyle \sqrt{2\pi}{K}_{-1/2}\left(\delta \gamma \right)}}exp\left[\beta (x-\mu )\right]{\left[{\delta}^{2}+{(x-\mu )}^{2}\right]}^{-1}{K}_{-1}\left(\alpha \sqrt{{\delta}^{2}+{(x-\mu )}^{2}}\right)}\end{array}$$$$\begin{array}{c}\hfill {\displaystyle {K}_{\nu}\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\displaystyle \pi \mathrm{csc}\left(\pi \nu \right)}}{{\displaystyle 2}}\left[{I}_{-\nu}\left(x\right)-{I}_{\nu}\left(x\right)\right],}\hfill & \mathrm{if}\nu \notin \mathbb{Z},\hfill \\ {\displaystyle \underset{\mu \to \nu}{lim}{K}_{\mu}\left(x\right),}\hfill & \mathrm{if}\nu \in \mathbb{Z},\hfill \end{array}\right.}\end{array}$$$$\begin{array}{c}\hfill {\displaystyle {I}_{\nu}\left(x\right)=\sum _{k=0}^{\infty}\frac{{\displaystyle 1}}{{\displaystyle \mathsf{\Gamma}(k+\nu +1)k!}}{\left(\frac{{\displaystyle x}}{{\displaystyle 2}}\right)}^{2k+\nu},}\end{array}$$$$\begin{array}{c}\hfill {\displaystyle \mathsf{\Gamma}\left(a\right)={\int}_{0}^{\infty}{t}^{a-1}exp(-t)dt;}\end{array}$$
- the generalized hyperbolic distribution (Barndorff-Nielsen 1977) with$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{\displaystyle {\left(\gamma /\delta \right)}^{\lambda}{\alpha}^{1/2-\lambda}}}{{\displaystyle \sqrt{2\pi}{K}_{\lambda}\left(\delta \gamma \right)}}exp\left[\beta (x-\mu )\right]{\left[{\delta}^{2}+{(x-\mu )}^{2}\right]}^{\lambda -1/2}{K}_{\lambda -1/2}\left(\alpha \sqrt{{\delta}^{2}+{(x-\mu )}^{2}}\right)}\end{array}$$

- the Akaike information criterion (Akaike 1974) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{AIC}=2k-2lnL\left(\widehat{\mathsf{\Theta}}\right);}\end{array}$$
- the Bayesian information criterion (Schwarz 1978) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{BIC}=klnn-2lnL\left(\widehat{\mathsf{\Theta}}\right);}\end{array}$$
- the consistent Akaike information criterion (CAIC) (Bozdogan 1987) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{CAIC}=-2lnL\left(\widehat{\mathsf{\Theta}}\right)+k\left(lnn+1\right);}\end{array}$$
- the corrected Akaike information criterion (AICc) (Hurvich and Tsai 1989) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{AICc}=\mathrm{AIC}+\frac{{\displaystyle 2k(k+1)}}{{\displaystyle n-k-1}};}\end{array}$$
- the Hannan-Quinn criterion (Hannan and Quinn 1979) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{HQC}=-2lnL\left(\widehat{\mathsf{\Theta}}\right)+2klnlnn.}\end{array}$$

- the Kolmogorov-Smirnov statistic (Kolmogorov 1933; Smirnov 1948) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{KS}=\underset{x}{sup}\left|\frac{1}{n}\sum _{i=1}^{n}I\left\{{x}_{i}\le x\right\}-\widehat{F}\left(x\right)\right|,}\end{array}$$
- the Anderson-Darling statistic (Anderson and Darling 1954) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{AD}=-n-\sum _{i=1}^{n}\left\{ln\widehat{F}\left({x}_{\left(i\right)}\right)+ln\left[1-\widehat{F}\left({x}_{(n+1-i)}\right)\right]\right\},}\end{array}$$
- the Cramer-von Mises statistic (Cramer 1928; Von Mises 1928) defined by$$\begin{array}{c}\hfill {\displaystyle \mathrm{CM}=\frac{1}{12n}+\sum _{i=1}^{n}\left[\frac{2i-1}{2n}-\widehat{F}\left({x}_{\left(i\right)}\right)\right].}\end{array}$$

## 4. Results

#### 4.1. Fitted Distributions and Results

#### 4.2. Q-Q Plots

#### 4.3. P-P Plots

#### 4.4. Goodness of Fit Tests

#### 4.5. VaR and ES Plots

#### 4.6. Kupiec’s test

#### 4.7. Dynamic Volatility

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Histograms of daily log returns of the exchange rates of the seven cryptocurrencies versus the U.S. Dollar from 23 June 2014 until 28 February 2017.

**Figure 2.**The Q-Q plots of the best fitting distributions for daily log returns of the exchange rates of Bitcoin (first row, left), Dash (first row, right), Dogecoin (second row, left), Litecoin (second row, right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until 28 February 2017.

**Figure 3.**The P-P plots of the best fitting distributions for daily log returns of the exchange rates of Bitcoin (first row, left), Dash (first row, right), Dogecoin (second row, left), Litecoin (second row, right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until 28 February 2017.

**Figure 4.**Value at risk for the best fitting distributions for daily log returns of the exchange rates of Bitcoin (first row, left), Dash (first row, right), Dogecoin (second row, left), Litecoin (second row, right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until 28 February 2017. Also shown are the values at risk for daily log returns of the exchange rates of the Euro computed using the same best fitting distributions.

**Figure 5.**Expected shortfall for the best fitting distributions for daily log returns of the exchange rates of Bitcoin (first row, left), Dash (first row, right), Dogecoin (second row, left), Litecoin (second row, right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until 28 February 2017. Also shown are the expected shortfall values for daily log returns of the exchange rates of the Euro computed using the same best fitting distributions.

**Figure 6.**Kupiec’s p-values for the best fitting distributions for daily log returns of the exchange rates of Bitcoin (first row, left), Dash (first row, right), Dogecoin (second row, left), Litecoin (second row, right), MaidSafeCoin (third row, left), Monero (third row, right) and Ripple (last row) from 23 June 2014 until 28 February 2017.

**Figure 7.**Histograms of standard deviations of daily log returns of the exchange rates of the seven cryptocurrencies over windows of width 20 days. Also shown is the histogram of standard deviations of daily log returns of the exchange rates of the Euro over windows of width 20 days.

**Table 1.**Summary statistics of daily exchange rates of Bitcoin, Dash, Dogecoin, Litecoin, MaidSafeCoin, Monero, Ripple and Euro, versus the U.S. Dollar from 23 June 2014 until 28 February 2017.

Statistics | Bitcoin | Dash | Dogecoin | Litecoin | MaidSafeCoin | Monero | Ripple | Euro |
---|---|---|---|---|---|---|---|---|

Minimum | 192.700 | 1.178 | 0.000 | 1.269 | 0.012 | 0.235 | 0.003 | 0.626 |

Q1 | 273.600 | 2.577 | 0.000 | 3.091 | 0.020 | 0.491 | 0.006 | 0.736 |

Median | 415.200 | 3.623 | 0.000 | 3.662 | 0.029 | 0.811 | 0.007 | 0.779 |

Mean | 447.400 | 5.385 | 0.000 | 3.659 | 0.046 | 2.355 | 0.008 | 0.830 |

Q3 | 593.000 | 7.921 | 0.000 | 4.021 | 0.074 | 1.970 | 0.008 | 0.856 |

Maximum | 1140.000 | 17.560 | 0.000 | 9.793 | 0.152 | 17.590 | 0.028 | 1.207 |

Skewness | 0.841 | 1.006 | 0.417 | 1.363 | 0.849 | 2.108 | 2.543 | 1.127 |

Kurtosis | 3.096 | 2.992 | 3.175 | 6.621 | 2.503 | 6.526 | 10.693 | 3.067 |

SD | 193.241 | 3.583 | 0.000 | 1.433 | 0.032 | 3.397 | 0.004 | 0.142 |

Variance | 37,342.159 | 12.838 | 0.000 | 2.053 | 0.001 | 11.543 | 0.000 | 0.020 |

CV | 0.432 | 0.665 | 0.294 | 0.392 | 0.695 | 1.443 | 0.471 | 0.171 |

Range | 946.938 | 16.385 | 0.000 | 8.524 | 0.140 | 17.358 | 0.025 | 0.581 |

IQR | 319.400 | 5.344 | 0.000 | 0.930 | 0.054 | 1.479 | 0.002 | 0.119 |

**Table 2.**Summary statistics of daily log returns of the exchange rates of Bitcoin, Dash, Dogecoin, Litecoin, MaidSafeCoin, Monero, Ripple and the Euro, versus the U.S. Dollar from 23 June 2014 until 28 February 2017.

Statistics | Bitcoin | Dash | Dogecoin | Litecoin | MaidSafeCoin | Monero | Ripple | Euro |
---|---|---|---|---|---|---|---|---|

Minimum | −0.159 | −0.580 | −0.385 | −0.278 | −0.404 | −0.560 | −0.299 | −0.046 |

Q1 | −0.011 | −0.019 | −0.009 | −0.010 | −0.026 | −0.026 | −0.014 | −0.004 |

Median | −0.001 | 0.003 | 0.002 | 0.000 | −0.001 | 0.002 | 0.002 | 0 |

Mean | −0.001 | −0.001 | 0.000 | 0.001 | −0.002 | −0.001 | −0.000 | −0.00004 |

Q3 | 0.008 | 0.020 | 0.015 | 0.009 | 0.023 | 0.028 | 0.017 | 0.003 |

Maximum | 0.205 | 0.411 | 0.188 | 0.433 | 0.241 | 0.277 | 0.288 | 0.038 |

Skewness | 0.758 | −1.487 | −2.506 | 0.756 | −0.478 | −1.414 | −0.401 | −0.145 |

Kurtosis | 11.568 | 26.805 | 24.434 | 22.385 | 8.520 | 13.954 | 13.818 | 2.662 |

SD | 0.028 | 0.051 | 0.042 | 0.042 | 0.054 | 0.062 | 0.046 | 0.006 |

Variance | 0.001 | 0.003 | 0.002 | 0.002 | 0.003 | 0.004 | 0.002 | 0.00004 |

CV | −47.976 | −84.519 | 89.782 | 45.619 | −21.499 | −54.548 | −96.585 | −143.498 |

Range | 0.364 | 0.991 | 0.573 | 0.711 | 0.645 | 0.837 | 0.587 | 0.085 |

IQR | 0.019 | 0.039 | 0.023 | 0.019 | 0.049 | 0.054 | 0.030 | 0.007 |

**Table 3.**Best fitting distributions and parameter estimates, with standard errors given in brackets.

Crytptocurrency | Best Fitting Distribution | Parameter Estimates and Standard Errors |
---|---|---|

Bitcoin | Generalized hyperbolic | $\widehat{\mu}=-0.001$ $\left(0.000\right)$, |

$\widehat{\delta}=0.003$ $\left(0.001\right)$, | ||

$\widehat{\alpha}=29.644$ $\left(3.707\right)$, | ||

$\widehat{\beta}=0.530$ $\left(1.305\right)$, | ||

$\widehat{\lambda}=0.220$ $\left(0.010\right)$. | ||

Dash | Normal inverse Gaussian | $\widehat{\mu}=0.004$, |

$\widehat{\delta}=0.025$, | ||

$\widehat{\alpha}=10.714$, | ||

$\widehat{\beta}=-2.100$. | ||

Dogecoin | Generalized t | $\widehat{\mu}=0.002$ $\left(0.000\right)$, |

$\widehat{\sigma}=0.014$ $\left(0.002\right)$, | ||

$\widehat{p}=0.893$ $\left(0.094\right)$, | ||

$\widehat{\nu}=3.768$ $\left(1.269\right)$. | ||

Litecoin | Generalized hyperbolic | $\widehat{\mu}=0.000$ $\left(0.000\right)$, |

$\widehat{\delta}=0.006$ $\left(0.001\right)$, | ||

$\widehat{\alpha}=10.517$ $\left(2.021\right)$, | ||

$\widehat{\beta}=0.412$ $\left(0.801\right)$, | ||

$\widehat{\lambda}=-0.186$ $\left(0.078\right)$. | ||

MaidSafeCoin | Laplace | $\widehat{\mu}=-0.001$ $\left(0.001\right)$, |

$\widehat{\sigma}=0.0368$ $\left(0.001\right)$. | ||

Monero | Normal inverse Gaussian | $\widehat{\mu}=0.005$, |

$\widehat{\delta}=0.040$, | ||

$\widehat{\alpha}=11.164$, | ||

$\widehat{\beta}=-1.705$. | ||

Ripple | Normal inverse Gaussian | $\widehat{\mu}=0.003$, |

$\widehat{\delta}=0.018$, | ||

$\widehat{\alpha}=8.729$, | ||

$\widehat{\beta}=-1.670$. |

**Table 4.**Fitted distributions and results for daily log returns of the exchange rates of Bitcoin from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −2303.9 | −4601.8 | −4601.8 | −4587.3 | −4596.3 | −4584.3 |

Laplace | −2289.3 | −4574.7 | −4574.7 | −4565.0 | −4571.0 | −4563.0 |

Skew t | −2304.0 | −4600.0 | −4599.9 | −4580.5 | −4592.6 | −4576.5 |

GT | −2325.7 | −4643.3 | −4643.3 | −4623.8 | −4635.9 | −4619.8 |

SST | −2304.0 | −4600.0 | −4599.9 | −4580.5 | −4592.6 | −4576.5 |

AST | −2304.3 | −4598.6 | −4598.6 | −4574.3 | −4589.4 | −4569.3 |

NIG | −2316.0 | −4623.9 | −4623.9 | −4604.5 | −4616.5 | −4600.5 |

GH | −2325.7 | −4641.5 | −4641.4 | −4617.1 | −4632.2 | −4612.1 |

**Table 5.**Fitted distributions and results for daily log returns of the exchange rates of Dash from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −1704.8 | −3403.6 | −3403.6 | −3389.0 | −3398.1 | −3386.0 |

Laplace | −1689.4 | −3374.8 | −3374.7 | −3365.0 | −3371.1 | −3363.0 |

Skew t | −1707.4 | −3406.8 | −3406.8 | −3387.3 | −3399.4 | −3383.3 |

GT | −1708.9 | −3409.7 | −3409.7 | −3390.3 | −3402.3 | −3386.3 |

SST | −1707.5 | −3407.0 | −3406.9 | −3387.5 | −3399.6 | −3383.5 |

AST | −1707.5 | −3405.1 | −3405.0 | −3380.8 | −3395.8 | −3375.8 |

NIG | −1710.4 | −3412.8 | −3412.8 | −3393.4 | −3405.4 | −3389.4 |

GH | −1710.5 | −3410.9 | −3410.8 | −3386.6 | −3401.6 | −3381.6 |

**Table 6.**Fitted distributions and results for daily log returns of the exchange rates of Dogecoin from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −2037.7 | −4069.4 | −4069.4 | −4054.8 | −4063.8 | −4051.8 |

Laplace | −1985.9 | −3967.8 | −3967.8 | −3958.1 | −3964.1 | −3956.1 |

Skew t | −2037.7 | −4067.5 | −4067.4 | −4048.0 | −4060.1 | −4044.0 |

GT | −2051.9 | −4095.8 | −4095.8 | −4076.4 | −4088.4 | −4072.4 |

SST | −2037.7 | −4067.4 | −4067.3 | −4047.9 | −4060.0 | −4043.9 |

AST | −2039.6 | −4069.1 | −4069.1 | −4044.8 | −4059.9 | −4039.8 |

NIG | −2048.1 | −4088.1 | −4088.1 | −4068.7 | −4080.7 | −4064.7 |

GH | −2052.2 | −4094.3 | −4094.2 | −4070.0 | −4085.0 | −4065.0 |

**Table 7.**Fitted distributions and results for daily log returns of the exchange rates of Litecoin from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −2113.3 | −4220.7 | −4220.7 | −4206.1 | −4215.1 | −4203.1 |

Laplace | −2020.0 | −4036.0 | −4036.0 | −4026.3 | −4032.3 | −4024.3 |

Skew t | −2113.6 | −4219.3 | −4219.3 | −4199.8 | −4211.9 | −4195.8 |

GT | −2125.7 | −4243.4 | −4243.4 | −4223.9 | −4236.0 | −4219.9 |

SST | −2113.5 | −4219.1 | −4219.1 | −4199.6 | −4211.7 | −4195.6 |

AST | −2113.6 | −4217.2 | −4217.1 | −4192.8 | −4207.9 | −4187.8 |

NIG | −2123.9 | −4239.9 | −4239.9 | −4220.4 | −4232.5 | −4216.4 |

GH | −2130.6 | −4251.2 | −4251.1 | −4226.9 | −4241.9 | −4221.9 |

**Table 8.**Fitted distributions and results for daily log returns of the exchange rates of MaidSafeCoin from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −1533.2 | −3060.3 | −3060.3 | −3045.7 | −3054.8 | −3042.7 |

Laplace | −1540.4 | −3076.8 | −3076.8 | −3067.1 | −3073.1 | −3065.1 |

Skew t | −1533.3 | −3058.6 | −3058.5 | −3039.1 | −3051.2 | −3035.1 |

GT | −1541.9 | −3075.9 | −3075.8 | −3056.4 | −3068.5 | −3052.4 |

SST | −1533.3 | −3058.5 | −3058.5 | −3039.1 | −3051.1 | −3035.1 |

AST | −1533.4 | −3056.7 | −3056.7 | −3032.4 | −3047.5 | −3027.4 |

NIG | −1539.0 | −3070.0 | −3070.0 | −3050.6 | −3062.6 | −3046.6 |

GH | −1542.4 | −3074.9 | −3074.8 | −3050.6 | −3065.6 | −3045.6 |

**Table 9.**Fitted distributions and results for daily log returns of the exchange rates of Monero from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −1438.6 | −2871.2 | −2871.2 | −2856.6 | −2865.6 | −2853.6 |

Laplace | −1432.2 | −2860.5 | −2860.5 | −2850.7 | −2856.8 | −2848.7 |

Skew t | −1440.0 | −2871.9 | −2871.9 | −2852.5 | −2864.5 | −2848.5 |

GT | −1440.5 | −2872.9 | −2872.9 | −2853.5 | −2865.5 | −2849.5 |

SST | −1439.7 | −2871.4 | −2871.4 | −2852.0 | −2864.0 | −2848.0 |

AST | −1440.6 | −2871.2 | −2871.1 | −2846.9 | −2861.9 | −2841.9 |

NIG | −1441.9 | −2875.8 | −2875.7 | −2856.3 | −2868.4 | −2852.3 |

GH | −1442.0 | −2874.0 | −2874.0 | −2849.7 | −2864.8 | −2844.7 |

**Table 10.**Fitted distributions and results for daily log returns of the exchange rates of Ripple from 23 June 2014 until 28 February 2017.

Distribution | $-ln\mathit{L}$ | AIC | AICC | BIC | HQC | CAIC |
---|---|---|---|---|---|---|

Student t | −1867.6 | −3729.1 | −3729.1 | −3714.5 | −3723.6 | −3711.5 |

Laplace | −1831.9 | −3659.9 | −3659.9 | −3650.2 | −3656.2 | −3648.2 |

Skew t | −1869.5 | −3731.1 | −3731.0 | −3711.6 | −3723.7 | −3707.6 |

GT | −1870.6 | −3733.1 | −3733.1 | −3713.7 | −3725.7 | −3709.7 |

SST | −1869.3 | −3730.5 | −3730.5 | −3711.1 | −3723.1 | −3707.1 |

AST | −1869.7 | −3729.4 | −3729.4 | −3705.1 | −3720.2 | −3700.1 |

NIG | −1875.6 | −3743.3 | −3743.2 | −3723.8 | −3735.8 | −3719.8 |

GH | −1875.9 | −3741.8 | −3741.8 | −3717.5 | −3732.6 | −3712.5 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chan, S.; Chu, J.; Nadarajah, S.; Osterrieder, J.
A Statistical Analysis of Cryptocurrencies. *J. Risk Financial Manag.* **2017**, *10*, 12.
https://doi.org/10.3390/jrfm10020012

**AMA Style**

Chan S, Chu J, Nadarajah S, Osterrieder J.
A Statistical Analysis of Cryptocurrencies. *Journal of Risk and Financial Management*. 2017; 10(2):12.
https://doi.org/10.3390/jrfm10020012

**Chicago/Turabian Style**

Chan, Stephen, Jeffrey Chu, Saralees Nadarajah, and Joerg Osterrieder.
2017. "A Statistical Analysis of Cryptocurrencies" *Journal of Risk and Financial Management* 10, no. 2: 12.
https://doi.org/10.3390/jrfm10020012